ChemWiki - User contributions [en]

Rep:TS:mhardst

2018-02-13T19:23:24Z

Mh4815: /* Orbital Overlap Integral */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - which are derived by plotting the electronic potential energy value (V) of the overall molecular system, for a particular nuclear geometry (Q) (where Q is a collective variable describing the particular nuclear configuration as a single vector).<ref name="Levine" /> This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, to construct a potential energy surface, the electronic Schrödinger equation is solved (for a particular nuclear configuration), and the resultant potential energy value plotted.<ref name="Hinchliffe" /> This is repeated across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

The electronic Hamiltonian operator has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

====Stationary Points====

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" /><ref name="Levine" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

====Distinguishing Minima and Transition States====

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

*A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

*A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

*A transition state is the maximum point on the minimum energy path (MEP) which connects two minima.<ref name="Levine" /> In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

====Frequency Calculations====

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.<ref name="Levine" />

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

====Semi-Empirical Technique====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms.<ref name="Levine" /> This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

====Density-Functional Theory Method====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly.<ref name="Levine" /><ref name="Hinchliffe" /> The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

====Comparison====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
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This is a symmetric molecular orbital.

=====LUMO=====
<jmol>
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This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
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This is an antisymmetric molecular orbital.

=====LUMO=====
<jmol>
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<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
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This is a symmetric molecular orbital.

====TS====

=====MO 16=====
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This is an antisymmetric molecular orbital.

=====MO 17=====
<jmol>
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This is a symmetric molecular orbital.

=====MO 18=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
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This is a symmetric molecular orbital.

=====MO 19=====
<jmol>
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<script>frame 14; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
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This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.<ref name="Fleming" />

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.<ref name="Fleming" />

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Atkins and Friedman" /><ref name="Levine" /><ref name="Moss and Coady" />

<jmol>
<jmolApplet>
<color>white</color>
<script>frame 15; vibration 1;</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
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The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/transition state geometries were optimised to the semi-empirical PM6 level, followed by subsequent reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

(Given that the frontier molecular orbitals of cyclohexadiene and 1,3-dioxole are analogous in shape to those of butadiene and ethene, respectively, this MO diagram uses the more simple butadiene/ethene MOs as schematic diagrams - simply for clarity).

=====MO 40=====
<jmol>
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<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
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This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
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This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
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</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
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<script>frame 20; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
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This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

(Given that the frontier molecular orbitals of cyclohexadiene and 1,3-dioxole are analogous in shape to those of butadiene and ethene, respectively, this MO diagram uses the more simple butadiene/ethene MOs as schematic diagrams - simply for clarity).

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
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</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
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This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
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</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
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</jmol>

This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene). Thus, an inverse electron demand Diels-Alder cycloaddition takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====HOMO of the Exo Transition State====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====HOMO of the Endo Transition State====

There are significant secondary orbital interactions in the transition state of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as an alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/transition state geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Bonding in the Ring===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the π-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6π-electron) 6-membered ring component in the transition state.<ref name="Clayden et. al." /> (The overall transition state itself is also aromatic, since it has 10π-electrons, i.e. obeys Hückel's (4n + 2) π-electron rule for aromaticity). <ref name="Clayden et. al." /> The C-C single bonds in the 6-membered ring all decrease in length over the course of the reaction, as they adopt a greater double bond character. <ref name="Clayden et. al." /> Eventually, this results in the formation a formal benzene ring component in all three products.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

Comparison of these values allows conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo Diels-Alder adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion, the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a transition state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" /><ref name="Levine" /> <ref name="Moss and Coady" /> This corresponds to a geometry with a single vibration with a negative frequency. <ref name="Levine" /> <ref name="Moss and Coady" />

In particular, the frontier molecular orbitals for the Diels-Alder reactions between butadiene and ethene, and a substituted butadiene derivative (cyclohexadiene) and an ethene derivative (1,3-dioxole) were analysed, and the symmetry of the interacting orbitals, and the molecular orbitals they produce in the transition state, were predicted. These predictions were subsequently matched to the computationally-derived molecular orbitals for each of the optimised transition states. These studies allowed conclusions to be drawn about the orbital symmetry requirements of such reactions - only orbitals of matching symmetry can interact.<ref name="Fleming" /> Also, the electronic nature of the observed Diels-Alder reactions was deduced, from comparison of the relative energy levels of the HOMO/LUMO (the frontier orbitals) of the reacting components. From this analysis, the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole was found to be an inverse electron demand Diels-Alder cycloaddition. <ref name="Clayden et. al." /><ref name="Bruckner" />

Furthermore, these optimisation studies also allowed relative energy values of the reactants, products and transition states of each of the different reactions to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, which in turn allowed deduction of the kinetically and thermodynamically favoured products for each type of reaction.<ref name="Clayden et. al." />

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T19:23:01Z

Mh4815: /* Symmetry Requirements */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - which are derived by plotting the electronic potential energy value (V) of the overall molecular system, for a particular nuclear geometry (Q) (where Q is a collective variable describing the particular nuclear configuration as a single vector).<ref name="Levine" /> This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, to construct a potential energy surface, the electronic Schrödinger equation is solved (for a particular nuclear configuration), and the resultant potential energy value plotted.<ref name="Hinchliffe" /> This is repeated across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

The electronic Hamiltonian operator has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

====Stationary Points====

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" /><ref name="Levine" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

====Distinguishing Minima and Transition States====

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

*A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

*A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

*A transition state is the maximum point on the minimum energy path (MEP) which connects two minima.<ref name="Levine" /> In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

====Frequency Calculations====

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.<ref name="Levine" />

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

====Semi-Empirical Technique====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms.<ref name="Levine" /> This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

====Density-Functional Theory Method====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly.<ref name="Levine" /><ref name="Hinchliffe" /> The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

====Comparison====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

====TS====

=====MO 16=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 17=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 18=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 19=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.<ref name="Fleming" />

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.<ref name="Fleming" />

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral<ref name="Fleming" /><ref name="Clayden et. al." />

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral<ref name="Fleming" /><ref name="Clayden et. al." />

*Symmetric-antisymmetric interaction: zero orbital overlap integral<ref name="Fleming" /><ref name="Clayden et. al." />

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Atkins and Friedman" /><ref name="Levine" /><ref name="Moss and Coady" />

<jmol>
<jmolApplet>
<color>white</color>
<script>frame 15; vibration 1;</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/transition state geometries were optimised to the semi-empirical PM6 level, followed by subsequent reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

(Given that the frontier molecular orbitals of cyclohexadiene and 1,3-dioxole are analogous in shape to those of butadiene and ethene, respectively, this MO diagram uses the more simple butadiene/ethene MOs as schematic diagrams - simply for clarity).

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

(Given that the frontier molecular orbitals of cyclohexadiene and 1,3-dioxole are analogous in shape to those of butadiene and ethene, respectively, this MO diagram uses the more simple butadiene/ethene MOs as schematic diagrams - simply for clarity).

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene). Thus, an inverse electron demand Diels-Alder cycloaddition takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====HOMO of the Exo Transition State====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====HOMO of the Endo Transition State====

There are significant secondary orbital interactions in the transition state of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as an alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/transition state geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Bonding in the Ring===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the π-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6π-electron) 6-membered ring component in the transition state.<ref name="Clayden et. al." /> (The overall transition state itself is also aromatic, since it has 10π-electrons, i.e. obeys Hückel's (4n + 2) π-electron rule for aromaticity). <ref name="Clayden et. al." /> The C-C single bonds in the 6-membered ring all decrease in length over the course of the reaction, as they adopt a greater double bond character. <ref name="Clayden et. al." /> Eventually, this results in the formation a formal benzene ring component in all three products.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

Comparison of these values allows conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo Diels-Alder adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion, the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a transition state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" /><ref name="Levine" /> <ref name="Moss and Coady" /> This corresponds to a geometry with a single vibration with a negative frequency. <ref name="Levine" /> <ref name="Moss and Coady" />

In particular, the frontier molecular orbitals for the Diels-Alder reactions between butadiene and ethene, and a substituted butadiene derivative (cyclohexadiene) and an ethene derivative (1,3-dioxole) were analysed, and the symmetry of the interacting orbitals, and the molecular orbitals they produce in the transition state, were predicted. These predictions were subsequently matched to the computationally-derived molecular orbitals for each of the optimised transition states. These studies allowed conclusions to be drawn about the orbital symmetry requirements of such reactions - only orbitals of matching symmetry can interact.<ref name="Fleming" /> Also, the electronic nature of the observed Diels-Alder reactions was deduced, from comparison of the relative energy levels of the HOMO/LUMO (the frontier orbitals) of the reacting components. From this analysis, the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole was found to be an inverse electron demand Diels-Alder cycloaddition. <ref name="Clayden et. al." /><ref name="Bruckner" />

Furthermore, these optimisation studies also allowed relative energy values of the reactants, products and transition states of each of the different reactions to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, which in turn allowed deduction of the kinetically and thermodynamically favoured products for each type of reaction.<ref name="Clayden et. al." />

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T19:21:10Z

Mh4815: /* Symmetry Requirements */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - which are derived by plotting the electronic potential energy value (V) of the overall molecular system, for a particular nuclear geometry (Q) (where Q is a collective variable describing the particular nuclear configuration as a single vector).<ref name="Levine" /> This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, to construct a potential energy surface, the electronic Schrödinger equation is solved (for a particular nuclear configuration), and the resultant potential energy value plotted.<ref name="Hinchliffe" /> This is repeated across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

The electronic Hamiltonian operator has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

====Stationary Points====

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" /><ref name="Levine" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

====Distinguishing Minima and Transition States====

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

*A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

*A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

*A transition state is the maximum point on the minimum energy path (MEP) which connects two minima.<ref name="Levine" /> In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

====Frequency Calculations====

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.<ref name="Levine" />

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

====Semi-Empirical Technique====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms.<ref name="Levine" /> This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

====Density-Functional Theory Method====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly.<ref name="Levine" /><ref name="Hinchliffe" /> The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

====Comparison====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

====TS====

=====MO 16=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 17=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 18=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 19=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.<ref name="Fleming" />

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.<ref name="Fleming" />

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral<ref name="Fleming" />

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral<ref name="Fleming" />

*Symmetric-antisymmetric interaction: zero orbital overlap integral<ref name="Fleming" />

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Atkins and Friedman" /><ref name="Levine" /><ref name="Moss and Coady" />

<jmol>
<jmolApplet>
<color>white</color>
<script>frame 15; vibration 1;</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/transition state geometries were optimised to the semi-empirical PM6 level, followed by subsequent reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

(Given that the frontier molecular orbitals of cyclohexadiene and 1,3-dioxole are analogous in shape to those of butadiene and ethene, respectively, this MO diagram uses the more simple butadiene/ethene MOs as schematic diagrams - simply for clarity).

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

(Given that the frontier molecular orbitals of cyclohexadiene and 1,3-dioxole are analogous in shape to those of butadiene and ethene, respectively, this MO diagram uses the more simple butadiene/ethene MOs as schematic diagrams - simply for clarity).

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene). Thus, an inverse electron demand Diels-Alder cycloaddition takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====HOMO of the Exo Transition State====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====HOMO of the Endo Transition State====

There are significant secondary orbital interactions in the transition state of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as an alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/transition state geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Bonding in the Ring===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the π-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6π-electron) 6-membered ring component in the transition state.<ref name="Clayden et. al." /> (The overall transition state itself is also aromatic, since it has 10π-electrons, i.e. obeys Hückel's (4n + 2) π-electron rule for aromaticity). <ref name="Clayden et. al." /> The C-C single bonds in the 6-membered ring all decrease in length over the course of the reaction, as they adopt a greater double bond character. <ref name="Clayden et. al." /> Eventually, this results in the formation a formal benzene ring component in all three products.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

Comparison of these values allows conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo Diels-Alder adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion, the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a transition state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" /><ref name="Levine" /> <ref name="Moss and Coady" /> This corresponds to a geometry with a single vibration with a negative frequency. <ref name="Levine" /> <ref name="Moss and Coady" />

In particular, the frontier molecular orbitals for the Diels-Alder reactions between butadiene and ethene, and a substituted butadiene derivative (cyclohexadiene) and an ethene derivative (1,3-dioxole) were analysed, and the symmetry of the interacting orbitals, and the molecular orbitals they produce in the transition state, were predicted. These predictions were subsequently matched to the computationally-derived molecular orbitals for each of the optimised transition states. These studies allowed conclusions to be drawn about the orbital symmetry requirements of such reactions - only orbitals of matching symmetry can interact.<ref name="Fleming" /> Also, the electronic nature of the observed Diels-Alder reactions was deduced, from comparison of the relative energy levels of the HOMO/LUMO (the frontier orbitals) of the reacting components. From this analysis, the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole was found to be an inverse electron demand Diels-Alder cycloaddition. <ref name="Clayden et. al." /><ref name="Bruckner" />

Furthermore, these optimisation studies also allowed relative energy values of the reactants, products and transition states of each of the different reactions to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, which in turn allowed deduction of the kinetically and thermodynamically favoured products for each type of reaction.<ref name="Clayden et. al." />

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T19:19:57Z

Mh4815: /* Conclusion */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - which are derived by plotting the electronic potential energy value (V) of the overall molecular system, for a particular nuclear geometry (Q) (where Q is a collective variable describing the particular nuclear configuration as a single vector).<ref name="Levine" /> This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, to construct a potential energy surface, the electronic Schrödinger equation is solved (for a particular nuclear configuration), and the resultant potential energy value plotted.<ref name="Hinchliffe" /> This is repeated across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

The electronic Hamiltonian operator has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

====Stationary Points====

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" /><ref name="Levine" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

====Distinguishing Minima and Transition States====

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

*A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

*A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

*A transition state is the maximum point on the minimum energy path (MEP) which connects two minima.<ref name="Levine" /> In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

====Frequency Calculations====

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.<ref name="Levine" />

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

====Semi-Empirical Technique====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms.<ref name="Levine" /> This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

====Density-Functional Theory Method====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly.<ref name="Levine" /><ref name="Hinchliffe" /> The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

====Comparison====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
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This is a symmetric molecular orbital.

=====LUMO=====
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This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
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This is an antisymmetric molecular orbital.

=====LUMO=====
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This is a symmetric molecular orbital.

====TS====

=====MO 16=====
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This is an antisymmetric molecular orbital.

=====MO 17=====
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This is a symmetric molecular orbital.

=====MO 18=====
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This is a symmetric molecular orbital.

=====MO 19=====
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This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Atkins and Friedman" /><ref name="Levine" /><ref name="Moss and Coady" />

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The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/transition state geometries were optimised to the semi-empirical PM6 level, followed by subsequent reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

(Given that the frontier molecular orbitals of cyclohexadiene and 1,3-dioxole are analogous in shape to those of butadiene and ethene, respectively, this MO diagram uses the more simple butadiene/ethene MOs as schematic diagrams - simply for clarity).

=====MO 40=====
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This is an antisymmetric molecular orbital.

=====MO 41=====
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This is a symmetric molecular orbital.

=====MO 42=====
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This is a symmetric molecular orbital.

=====MO 43=====
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This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

(Given that the frontier molecular orbitals of cyclohexadiene and 1,3-dioxole are analogous in shape to those of butadiene and ethene, respectively, this MO diagram uses the more simple butadiene/ethene MOs as schematic diagrams - simply for clarity).

=====MO 40=====
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This is an antisymmetric molecular orbital.

=====MO 41=====
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This is a symmetric molecular orbital.

=====MO 42=====
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This is a symmetric molecular orbital.

=====MO 43=====
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This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene). Thus, an inverse electron demand Diels-Alder cycloaddition takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====HOMO of the Exo Transition State====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====HOMO of the Endo Transition State====

There are significant secondary orbital interactions in the transition state of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as an alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/transition state geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Bonding in the Ring===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the π-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6π-electron) 6-membered ring component in the transition state.<ref name="Clayden et. al." /> (The overall transition state itself is also aromatic, since it has 10π-electrons, i.e. obeys Hückel's (4n + 2) π-electron rule for aromaticity). <ref name="Clayden et. al." /> The C-C single bonds in the 6-membered ring all decrease in length over the course of the reaction, as they adopt a greater double bond character. <ref name="Clayden et. al." /> Eventually, this results in the formation a formal benzene ring component in all three products.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

Comparison of these values allows conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo Diels-Alder adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion, the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a transition state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" /><ref name="Levine" /> <ref name="Moss and Coady" /> This corresponds to a geometry with a single vibration with a negative frequency. <ref name="Levine" /> <ref name="Moss and Coady" />

In particular, the frontier molecular orbitals for the Diels-Alder reactions between butadiene and ethene, and a substituted butadiene derivative (cyclohexadiene) and an ethene derivative (1,3-dioxole) were analysed, and the symmetry of the interacting orbitals, and the molecular orbitals they produce in the transition state, were predicted. These predictions were subsequently matched to the computationally-derived molecular orbitals for each of the optimised transition states. These studies allowed conclusions to be drawn about the orbital symmetry requirements of such reactions - only orbitals of matching symmetry can interact.<ref name="Fleming" /> Also, the electronic nature of the observed Diels-Alder reactions was deduced, from comparison of the relative energy levels of the HOMO/LUMO (the frontier orbitals) of the reacting components. From this analysis, the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole was found to be an inverse electron demand Diels-Alder cycloaddition. <ref name="Clayden et. al." /><ref name="Bruckner" />

Furthermore, these optimisation studies also allowed relative energy values of the reactants, products and transition states of each of the different reactions to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, which in turn allowed deduction of the kinetically and thermodynamically favoured products for each type of reaction.<ref name="Clayden et. al." />

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T19:16:08Z

Mh4815: /* Molecular Structure via Quantum Mechanics */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - which are derived by plotting the electronic potential energy value (V) of the overall molecular system, for a particular nuclear geometry (Q) (where Q is a collective variable describing the particular nuclear configuration as a single vector).<ref name="Levine" /> This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, to construct a potential energy surface, the electronic Schrödinger equation is solved (for a particular nuclear configuration), and the resultant potential energy value plotted.<ref name="Hinchliffe" /> This is repeated across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

The electronic Hamiltonian operator has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

====Stationary Points====

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" /><ref name="Levine" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

====Distinguishing Minima and Transition States====

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

*A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

*A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

*A transition state is the maximum point on the minimum energy path (MEP) which connects two minima.<ref name="Levine" /> In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

====Frequency Calculations====

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.<ref name="Levine" />

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

====Semi-Empirical Technique====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms.<ref name="Levine" /> This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

====Density-Functional Theory Method====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly.<ref name="Levine" /><ref name="Hinchliffe" /> The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

====Comparison====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

====TS====

=====MO 16=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 17=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 18=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 19=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Atkins and Friedman" /><ref name="Levine" /><ref name="Moss and Coady" />

<jmol>
<jmolApplet>
<color>white</color>
<script>frame 15; vibration 1;</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/transition state geometries were optimised to the semi-empirical PM6 level, followed by subsequent reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

(Given that the frontier molecular orbitals of cyclohexadiene and 1,3-dioxole are analogous in shape to those of butadiene and ethene, respectively, this MO diagram uses the more simple butadiene/ethene MOs as schematic diagrams - simply for clarity).

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

(Given that the frontier molecular orbitals of cyclohexadiene and 1,3-dioxole are analogous in shape to those of butadiene and ethene, respectively, this MO diagram uses the more simple butadiene/ethene MOs as schematic diagrams - simply for clarity).

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene). Thus, an inverse electron demand Diels-Alder cycloaddition takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====HOMO of the Exo Transition State====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====HOMO of the Endo Transition State====

There are significant secondary orbital interactions in the transition state of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as an alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/transition state geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Bonding in the Ring===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the π-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6π-electron) 6-membered ring component in the transition state.<ref name="Clayden et. al." /> (The overall transition state itself is also aromatic, since it has 10π-electrons, i.e. obeys Hückel's (4n + 2) π-electron rule for aromaticity). <ref name="Clayden et. al." /> The C-C single bonds in the 6-membered ring all decrease in length over the course of the reaction, as they adopt a greater double bond character. <ref name="Clayden et. al." /> Eventually, this results in the formation a formal benzene ring component in all three products.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

Comparison of these values allows conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo Diels-Alder adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion, the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a transition state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" /><ref name="Levine" /> <ref name="Moss and Coady" /> This corresponds to a geometry with a single vibration with a negative frequency. <ref name="Levine" /> <ref name="Moss and Coady" />

In particular, the frontier molecular orbitals for the Diels-Alder reactions between butadiene and ethene, and a substituted butadiene derivative (cyclohexadiene) and an ethene derivative (1,3-dioxole) were analysed, and the symmetry of the interacting orbitals, and the molecular orbitals they produce in the transition state, were predicted. These predictions were subsequently matched to the computationally-derived molecular orbitals for each of the optimised transition states. These studies allowed conclusions to be drawn about the orbital symmetry requirements of such reactions - only orbitals of matching symmetry can interact. Also, the electronic nature of the observed Diels-Alder reactions was deduced, from comparison of the relative energy levels of the HOMO/LUMO (the frontier orbitals) of the reacting components. From this analysis, the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole was found to be an inverse electron demand Diels-Alder cycloaddition. <ref name="Clayden et. al." /><ref name="Bruckner" />

Furthermore, these optimisation studies also allowed relative energy values of the reactants, products and transition states of each of the different reactions to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, which in turn allowed deduction of the kinetically and thermodynamically favoured products for each type of reaction.<ref name="Clayden et. al." />

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T19:15:40Z

Mh4815: /* Molecular Structure via Quantum Mechanics */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - which are derived by plotting the electronic potential energy value (V) of the overall molecular system, for a particular nuclear geometry (Q) (where Q is a collective variable describing the particular nuclear configuration as a single vector).<ref name="Levine" /> This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, to construct a potential energy surface, the electronic Schrödinger equation is solved (for a particular nuclear configuration), and the resultant potential energy value plotted.<ref name="Hinchliffe" /> This is repeated across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

The Hamiltonian equation has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

====Stationary Points====

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" /><ref name="Levine" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

====Distinguishing Minima and Transition States====

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

*A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

*A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

*A transition state is the maximum point on the minimum energy path (MEP) which connects two minima.<ref name="Levine" /> In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

====Frequency Calculations====

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.<ref name="Levine" />

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

====Semi-Empirical Technique====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms.<ref name="Levine" /> This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

====Density-Functional Theory Method====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly.<ref name="Levine" /><ref name="Hinchliffe" /> The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

====Comparison====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

====TS====

=====MO 16=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 17=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 18=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 19=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Atkins and Friedman" /><ref name="Levine" /><ref name="Moss and Coady" />

<jmol>
<jmolApplet>
<color>white</color>
<script>frame 15; vibration 1;</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/transition state geometries were optimised to the semi-empirical PM6 level, followed by subsequent reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

(Given that the frontier molecular orbitals of cyclohexadiene and 1,3-dioxole are analogous in shape to those of butadiene and ethene, respectively, this MO diagram uses the more simple butadiene/ethene MOs as schematic diagrams - simply for clarity).

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

(Given that the frontier molecular orbitals of cyclohexadiene and 1,3-dioxole are analogous in shape to those of butadiene and ethene, respectively, this MO diagram uses the more simple butadiene/ethene MOs as schematic diagrams - simply for clarity).

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene). Thus, an inverse electron demand Diels-Alder cycloaddition takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====HOMO of the Exo Transition State====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====HOMO of the Endo Transition State====

There are significant secondary orbital interactions in the transition state of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as an alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/transition state geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Bonding in the Ring===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the π-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6π-electron) 6-membered ring component in the transition state.<ref name="Clayden et. al." /> (The overall transition state itself is also aromatic, since it has 10π-electrons, i.e. obeys Hückel's (4n + 2) π-electron rule for aromaticity). <ref name="Clayden et. al." /> The C-C single bonds in the 6-membered ring all decrease in length over the course of the reaction, as they adopt a greater double bond character. <ref name="Clayden et. al." /> Eventually, this results in the formation a formal benzene ring component in all three products.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

Comparison of these values allows conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo Diels-Alder adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion, the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a transition state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" /><ref name="Levine" /> <ref name="Moss and Coady" /> This corresponds to a geometry with a single vibration with a negative frequency. <ref name="Levine" /> <ref name="Moss and Coady" />

In particular, the frontier molecular orbitals for the Diels-Alder reactions between butadiene and ethene, and a substituted butadiene derivative (cyclohexadiene) and an ethene derivative (1,3-dioxole) were analysed, and the symmetry of the interacting orbitals, and the molecular orbitals they produce in the transition state, were predicted. These predictions were subsequently matched to the computationally-derived molecular orbitals for each of the optimised transition states. These studies allowed conclusions to be drawn about the orbital symmetry requirements of such reactions - only orbitals of matching symmetry can interact. Also, the electronic nature of the observed Diels-Alder reactions was deduced, from comparison of the relative energy levels of the HOMO/LUMO (the frontier orbitals) of the reacting components. From this analysis, the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole was found to be an inverse electron demand Diels-Alder cycloaddition. <ref name="Clayden et. al." /><ref name="Bruckner" />

Furthermore, these optimisation studies also allowed relative energy values of the reactants, products and transition states of each of the different reactions to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, which in turn allowed deduction of the kinetically and thermodynamically favoured products for each type of reaction.<ref name="Clayden et. al." />

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T19:14:51Z

Mh4815: /* MO Diagram for the Endo Transition State */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - which are derived by plotting the electronic potential energy value (V) of the overall molecular system, for a particular nuclear geometry (Q) (where Q is a collective variable describing the particular nuclear configuration as a single vector).<ref name="Levine" /> This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, to construct a potential energy surface, the electronic Schrödinger equation is solved (for a particular nuclear configuration), and the resultant potential energy value plotted.<ref name="Hinchliffe" /> This is repeated across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

The Hamiltonian equation has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H} = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

====Stationary Points====

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" /><ref name="Levine" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

====Distinguishing Minima and Transition States====

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

*A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

*A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

*A transition state is the maximum point on the minimum energy path (MEP) which connects two minima.<ref name="Levine" /> In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

====Frequency Calculations====

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.<ref name="Levine" />

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

====Semi-Empirical Technique====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms.<ref name="Levine" /> This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

====Density-Functional Theory Method====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly.<ref name="Levine" /><ref name="Hinchliffe" /> The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

====Comparison====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
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This is a symmetric molecular orbital.

=====LUMO=====
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This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
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This is an antisymmetric molecular orbital.

=====LUMO=====
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This is a symmetric molecular orbital.

====TS====

=====MO 16=====
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This is an antisymmetric molecular orbital.

=====MO 17=====
<jmol>
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This is a symmetric molecular orbital.

=====MO 18=====
<jmol>
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<script>frame 14; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
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This is a symmetric molecular orbital.

=====MO 19=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
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This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Atkins and Friedman" /><ref name="Levine" /><ref name="Moss and Coady" />

<jmol>
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<script>frame 15; vibration 1;</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
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The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/transition state geometries were optimised to the semi-empirical PM6 level, followed by subsequent reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

(Given that the frontier molecular orbitals of cyclohexadiene and 1,3-dioxole are analogous in shape to those of butadiene and ethene, respectively, this MO diagram uses the more simple butadiene/ethene MOs as schematic diagrams - simply for clarity).

=====MO 40=====
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This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
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This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
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<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
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This is a symmetric molecular orbital.

=====MO 43=====
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This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

(Given that the frontier molecular orbitals of cyclohexadiene and 1,3-dioxole are analogous in shape to those of butadiene and ethene, respectively, this MO diagram uses the more simple butadiene/ethene MOs as schematic diagrams - simply for clarity).

=====MO 40=====
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This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
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<script>frame 32; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
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This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
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<script>frame 32; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
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This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
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<script>frame 32; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
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This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene). Thus, an inverse electron demand Diels-Alder cycloaddition takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====HOMO of the Exo Transition State====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====HOMO of the Endo Transition State====

There are significant secondary orbital interactions in the transition state of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as an alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/transition state geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Bonding in the Ring===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the π-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6π-electron) 6-membered ring component in the transition state.<ref name="Clayden et. al." /> (The overall transition state itself is also aromatic, since it has 10π-electrons, i.e. obeys Hückel's (4n + 2) π-electron rule for aromaticity). <ref name="Clayden et. al." /> The C-C single bonds in the 6-membered ring all decrease in length over the course of the reaction, as they adopt a greater double bond character. <ref name="Clayden et. al." /> Eventually, this results in the formation a formal benzene ring component in all three products.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

Comparison of these values allows conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo Diels-Alder adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion, the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a transition state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" /><ref name="Levine" /> <ref name="Moss and Coady" /> This corresponds to a geometry with a single vibration with a negative frequency. <ref name="Levine" /> <ref name="Moss and Coady" />

In particular, the frontier molecular orbitals for the Diels-Alder reactions between butadiene and ethene, and a substituted butadiene derivative (cyclohexadiene) and an ethene derivative (1,3-dioxole) were analysed, and the symmetry of the interacting orbitals, and the molecular orbitals they produce in the transition state, were predicted. These predictions were subsequently matched to the computationally-derived molecular orbitals for each of the optimised transition states. These studies allowed conclusions to be drawn about the orbital symmetry requirements of such reactions - only orbitals of matching symmetry can interact. Also, the electronic nature of the observed Diels-Alder reactions was deduced, from comparison of the relative energy levels of the HOMO/LUMO (the frontier orbitals) of the reacting components. From this analysis, the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole was found to be an inverse electron demand Diels-Alder cycloaddition. <ref name="Clayden et. al." /><ref name="Bruckner" />

Furthermore, these optimisation studies also allowed relative energy values of the reactants, products and transition states of each of the different reactions to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, which in turn allowed deduction of the kinetically and thermodynamically favoured products for each type of reaction.<ref name="Clayden et. al." />

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T19:13:59Z

Mh4815: /* MO Diagram for the Exo Transition State */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - which are derived by plotting the electronic potential energy value (V) of the overall molecular system, for a particular nuclear geometry (Q) (where Q is a collective variable describing the particular nuclear configuration as a single vector).<ref name="Levine" /> This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, to construct a potential energy surface, the electronic Schrödinger equation is solved (for a particular nuclear configuration), and the resultant potential energy value plotted.<ref name="Hinchliffe" /> This is repeated across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

The Hamiltonian equation has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H} = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

====Stationary Points====

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" /><ref name="Levine" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

====Distinguishing Minima and Transition States====

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

*A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

*A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

*A transition state is the maximum point on the minimum energy path (MEP) which connects two minima.<ref name="Levine" /> In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

====Frequency Calculations====

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.<ref name="Levine" />

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

====Semi-Empirical Technique====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms.<ref name="Levine" /> This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

====Density-Functional Theory Method====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly.<ref name="Levine" /><ref name="Hinchliffe" /> The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

====Comparison====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

====TS====

=====MO 16=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 17=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 18=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 19=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Atkins and Friedman" /><ref name="Levine" /><ref name="Moss and Coady" />

<jmol>
<jmolApplet>
<color>white</color>
<script>frame 15; vibration 1;</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/transition state geometries were optimised to the semi-empirical PM6 level, followed by subsequent reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

(Given that the frontier molecular orbitals of cyclohexadiene and 1,3-dioxole are analogous in shape to those of butadiene and ethene, respectively, this MO diagram uses the more simple butadiene/ethene MOs as schematic diagrams - simply for clarity).

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene). Thus, an inverse electron demand Diels-Alder cycloaddition takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====HOMO of the Exo Transition State====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====HOMO of the Endo Transition State====

There are significant secondary orbital interactions in the transition state of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as an alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/transition state geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Bonding in the Ring===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the π-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6π-electron) 6-membered ring component in the transition state.<ref name="Clayden et. al." /> (The overall transition state itself is also aromatic, since it has 10π-electrons, i.e. obeys Hückel's (4n + 2) π-electron rule for aromaticity). <ref name="Clayden et. al." /> The C-C single bonds in the 6-membered ring all decrease in length over the course of the reaction, as they adopt a greater double bond character. <ref name="Clayden et. al." /> Eventually, this results in the formation a formal benzene ring component in all three products.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

Comparison of these values allows conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo Diels-Alder adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion, the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a transition state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" /><ref name="Levine" /> <ref name="Moss and Coady" /> This corresponds to a geometry with a single vibration with a negative frequency. <ref name="Levine" /> <ref name="Moss and Coady" />

In particular, the frontier molecular orbitals for the Diels-Alder reactions between butadiene and ethene, and a substituted butadiene derivative (cyclohexadiene) and an ethene derivative (1,3-dioxole) were analysed, and the symmetry of the interacting orbitals, and the molecular orbitals they produce in the transition state, were predicted. These predictions were subsequently matched to the computationally-derived molecular orbitals for each of the optimised transition states. These studies allowed conclusions to be drawn about the orbital symmetry requirements of such reactions - only orbitals of matching symmetry can interact. Also, the electronic nature of the observed Diels-Alder reactions was deduced, from comparison of the relative energy levels of the HOMO/LUMO (the frontier orbitals) of the reacting components. From this analysis, the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole was found to be an inverse electron demand Diels-Alder cycloaddition. <ref name="Clayden et. al." /><ref name="Bruckner" />

Furthermore, these optimisation studies also allowed relative energy values of the reactants, products and transition states of each of the different reactions to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, which in turn allowed deduction of the kinetically and thermodynamically favoured products for each type of reaction.<ref name="Clayden et. al." />

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T19:13:33Z

Mh4815: /* MO Diagram for the Exo Transition State */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - which are derived by plotting the electronic potential energy value (V) of the overall molecular system, for a particular nuclear geometry (Q) (where Q is a collective variable describing the particular nuclear configuration as a single vector).<ref name="Levine" /> This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, to construct a potential energy surface, the electronic Schrödinger equation is solved (for a particular nuclear configuration), and the resultant potential energy value plotted.<ref name="Hinchliffe" /> This is repeated across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

The Hamiltonian equation has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H} = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

====Stationary Points====

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" /><ref name="Levine" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

====Distinguishing Minima and Transition States====

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

*A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

*A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

*A transition state is the maximum point on the minimum energy path (MEP) which connects two minima.<ref name="Levine" /> In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

====Frequency Calculations====

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.<ref name="Levine" />

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

====Semi-Empirical Technique====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms.<ref name="Levine" /> This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

====Density-Functional Theory Method====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly.<ref name="Levine" /><ref name="Hinchliffe" /> The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

====Comparison====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

====TS====

=====MO 16=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 17=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 18=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 19=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Atkins and Friedman" /><ref name="Levine" /><ref name="Moss and Coady" />

<jmol>
<jmolApplet>
<color>white</color>
<script>frame 15; vibration 1;</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/transition state geometries were optimised to the semi-empirical PM6 level, followed by subsequent reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

(Given that the frontier molecular orbitals of cyclohexadiene and 1,3-dioxole are analogous in shape to those of butadiene and ethene, respectively, these MO diagrams use the more simple butadiene/ethene MOs as schematic diagrams - simply for clarity).

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene). Thus, an inverse electron demand Diels-Alder cycloaddition takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====HOMO of the Exo Transition State====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====HOMO of the Endo Transition State====

There are significant secondary orbital interactions in the transition state of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as an alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/transition state geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Bonding in the Ring===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the π-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6π-electron) 6-membered ring component in the transition state.<ref name="Clayden et. al." /> (The overall transition state itself is also aromatic, since it has 10π-electrons, i.e. obeys Hückel's (4n + 2) π-electron rule for aromaticity). <ref name="Clayden et. al." /> The C-C single bonds in the 6-membered ring all decrease in length over the course of the reaction, as they adopt a greater double bond character. <ref name="Clayden et. al." /> Eventually, this results in the formation a formal benzene ring component in all three products.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

Comparison of these values allows conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo Diels-Alder adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion, the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a transition state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" /><ref name="Levine" /> <ref name="Moss and Coady" /> This corresponds to a geometry with a single vibration with a negative frequency. <ref name="Levine" /> <ref name="Moss and Coady" />

In particular, the frontier molecular orbitals for the Diels-Alder reactions between butadiene and ethene, and a substituted butadiene derivative (cyclohexadiene) and an ethene derivative (1,3-dioxole) were analysed, and the symmetry of the interacting orbitals, and the molecular orbitals they produce in the transition state, were predicted. These predictions were subsequently matched to the computationally-derived molecular orbitals for each of the optimised transition states. These studies allowed conclusions to be drawn about the orbital symmetry requirements of such reactions - only orbitals of matching symmetry can interact. Also, the electronic nature of the observed Diels-Alder reactions was deduced, from comparison of the relative energy levels of the HOMO/LUMO (the frontier orbitals) of the reacting components. From this analysis, the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole was found to be an inverse electron demand Diels-Alder cycloaddition. <ref name="Clayden et. al." /><ref name="Bruckner" />

Furthermore, these optimisation studies also allowed relative energy values of the reactants, products and transition states of each of the different reactions to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, which in turn allowed deduction of the kinetically and thermodynamically favoured products for each type of reaction.<ref name="Clayden et. al." />

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T19:06:48Z

Mh4815: /* Molecular Structure via Quantum Mechanics */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - which are derived by plotting the electronic potential energy value (V) of the overall molecular system, for a particular nuclear geometry (Q) (where Q is a collective variable describing the particular nuclear configuration as a single vector).<ref name="Levine" /> This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, to construct a potential energy surface, the electronic Schrödinger equation is solved (for a particular nuclear configuration), and the resultant potential energy value plotted.<ref name="Hinchliffe" /> This is repeated across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

The Hamiltonian equation has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H} = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

====Stationary Points====

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" /><ref name="Levine" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

====Distinguishing Minima and Transition States====

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

*A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

*A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

*A transition state is the maximum point on the minimum energy path (MEP) which connects two minima.<ref name="Levine" /> In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

====Frequency Calculations====

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.<ref name="Levine" />

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

====Semi-Empirical Technique====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms.<ref name="Levine" /> This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

====Density-Functional Theory Method====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly.<ref name="Levine" /><ref name="Hinchliffe" /> The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

====Comparison====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

====TS====

=====MO 16=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 17=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 18=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 19=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Atkins and Friedman" /><ref name="Levine" /><ref name="Moss and Coady" />

<jmol>
<jmolApplet>
<color>white</color>
<script>frame 15; vibration 1;</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/transition state geometries were optimised to the semi-empirical PM6 level, followed by subsequent reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene). Thus, an inverse electron demand Diels-Alder cycloaddition takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====HOMO of the Exo Transition State====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====HOMO of the Endo Transition State====

There are significant secondary orbital interactions in the transition state of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as an alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/transition state geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Bonding in the Ring===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the π-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6π-electron) 6-membered ring component in the transition state.<ref name="Clayden et. al." /> (The overall transition state itself is also aromatic, since it has 10π-electrons, i.e. obeys Hückel's (4n + 2) π-electron rule for aromaticity). <ref name="Clayden et. al." /> The C-C single bonds in the 6-membered ring all decrease in length over the course of the reaction, as they adopt a greater double bond character. <ref name="Clayden et. al." /> Eventually, this results in the formation a formal benzene ring component in all three products.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

Comparison of these values allows conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo Diels-Alder adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion, the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a transition state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" /><ref name="Levine" /> <ref name="Moss and Coady" /> This corresponds to a geometry with a single vibration with a negative frequency. <ref name="Levine" /> <ref name="Moss and Coady" />

In particular, the frontier molecular orbitals for the Diels-Alder reactions between butadiene and ethene, and a substituted butadiene derivative (cyclohexadiene) and an ethene derivative (1,3-dioxole) were analysed, and the symmetry of the interacting orbitals, and the molecular orbitals they produce in the transition state, were predicted. These predictions were subsequently matched to the computationally-derived molecular orbitals for each of the optimised transition states. These studies allowed conclusions to be drawn about the orbital symmetry requirements of such reactions - only orbitals of matching symmetry can interact. Also, the electronic nature of the observed Diels-Alder reactions was deduced, from comparison of the relative energy levels of the HOMO/LUMO (the frontier orbitals) of the reacting components. From this analysis, the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole was found to be an inverse electron demand Diels-Alder cycloaddition. <ref name="Clayden et. al." /><ref name="Bruckner" />

Furthermore, these optimisation studies also allowed relative energy values of the reactants, products and transition states of each of the different reactions to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, which in turn allowed deduction of the kinetically and thermodynamically favoured products for each type of reaction.<ref name="Clayden et. al." />

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T19:02:43Z

Mh4815: /* Molecular Structure via Quantum Mechanics */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - which are derived by plotting the electronic potential energy value (V) of the overall molecular system, for a particular nuclear geometry (Q) (where Q is a collective variable describing the particular nuclear configuration as a single vector).<ref name="Levine" /> This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, to construct a potential energy surface, the electronic Schrödinger equation is solved (for a particular nuclear configuration), and the resultant potential energy value plotted.<ref name="Hinchliffe" /> This is repeated across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

The Hamiltonian equation has the form:<ref name="Atkins and Friedman" />

<math> \hat{H} = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

====Stationary Points====

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" /><ref name="Levine" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

====Distinguishing Minima and Transition States====

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

*A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

*A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

*A transition state is the maximum point on the minimum energy path (MEP) which connects two minima.<ref name="Levine" /> In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

====Frequency Calculations====

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.<ref name="Levine" />

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

====Semi-Empirical Technique====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms.<ref name="Levine" /> This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

====Density-Functional Theory Method====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly.<ref name="Levine" /><ref name="Hinchliffe" /> The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

====Comparison====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
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This is a symmetric molecular orbital.

=====LUMO=====
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This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
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This is an antisymmetric molecular orbital.

=====LUMO=====
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This is a symmetric molecular orbital.

====TS====

=====MO 16=====
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This is an antisymmetric molecular orbital.

=====MO 17=====
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This is a symmetric molecular orbital.

=====MO 18=====
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This is a symmetric molecular orbital.

=====MO 19=====
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This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Atkins and Friedman" /><ref name="Levine" /><ref name="Moss and Coady" />

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The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/transition state geometries were optimised to the semi-empirical PM6 level, followed by subsequent reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

=====MO 40=====
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This is an antisymmetric molecular orbital.

=====MO 41=====
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This is a symmetric molecular orbital.

=====MO 42=====
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This is a symmetric molecular orbital.

=====MO 43=====
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This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

=====MO 40=====
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This is an antisymmetric molecular orbital.

=====MO 41=====
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This is a symmetric molecular orbital.

=====MO 42=====
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This is a symmetric molecular orbital.

=====MO 43=====
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This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene). Thus, an inverse electron demand Diels-Alder cycloaddition takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====HOMO of the Exo Transition State====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====HOMO of the Endo Transition State====

There are significant secondary orbital interactions in the transition state of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as an alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/transition state geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Bonding in the Ring===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the π-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6π-electron) 6-membered ring component in the transition state.<ref name="Clayden et. al." /> (The overall transition state itself is also aromatic, since it has 10π-electrons, i.e. obeys Hückel's (4n + 2) π-electron rule for aromaticity). <ref name="Clayden et. al." /> The C-C single bonds in the 6-membered ring all decrease in length over the course of the reaction, as they adopt a greater double bond character. <ref name="Clayden et. al." /> Eventually, this results in the formation a formal benzene ring component in all three products.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

Comparison of these values allows conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo Diels-Alder adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion, the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a transition state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" /><ref name="Levine" /> <ref name="Moss and Coady" /> This corresponds to a geometry with a single vibration with a negative frequency. <ref name="Levine" /> <ref name="Moss and Coady" />

In particular, the frontier molecular orbitals for the Diels-Alder reactions between butadiene and ethene, and a substituted butadiene derivative (cyclohexadiene) and an ethene derivative (1,3-dioxole) were analysed, and the symmetry of the interacting orbitals, and the molecular orbitals they produce in the transition state, were predicted. These predictions were subsequently matched to the computationally-derived molecular orbitals for each of the optimised transition states. These studies allowed conclusions to be drawn about the orbital symmetry requirements of such reactions - only orbitals of matching symmetry can interact. Also, the electronic nature of the observed Diels-Alder reactions was deduced, from comparison of the relative energy levels of the HOMO/LUMO (the frontier orbitals) of the reacting components. From this analysis, the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole was found to be an inverse electron demand Diels-Alder cycloaddition. <ref name="Clayden et. al." /><ref name="Bruckner" />

Furthermore, these optimisation studies also allowed relative energy values of the reactants, products and transition states of each of the different reactions to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, which in turn allowed deduction of the kinetically and thermodynamically favoured products for each type of reaction.<ref name="Clayden et. al." />

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T18:56:17Z

Mh4815: /* Molecular Structure via Quantum Mechanics */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - which are derived by plotting the electronic potential energy value (V) of the overall molecular system, for a particular nuclear geometry (Q) (where Q is a collective variable describing the particular nuclear configuration as a single vector).<ref name="Levine" /> This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, to construct a potential energy surface, the electronic Schrödinger equation is solved (for a particular nuclear configuration), and the resultant potential energy value plotted. This is repeated across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" />

The Hamiltonian equation has the form:<ref name="Atkins and Friedman" />

<math> \hat{H} = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

====Stationary Points====

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" /><ref name="Levine" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

====Distinguishing Minima and Transition States====

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

*A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

*A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

*A transition state is the maximum point on the minimum energy path (MEP) which connects two minima.<ref name="Levine" /> In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

====Frequency Calculations====

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.<ref name="Levine" />

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

====Semi-Empirical Technique====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms.<ref name="Levine" /> This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

====Density-Functional Theory Method====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly.<ref name="Levine" /><ref name="Hinchliffe" /> The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

====Comparison====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

====TS====

=====MO 16=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 17=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 18=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 19=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Atkins and Friedman" /><ref name="Levine" /><ref name="Moss and Coady" />

<jmol>
<jmolApplet>
<color>white</color>
<script>frame 15; vibration 1;</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/transition state geometries were optimised to the semi-empirical PM6 level, followed by subsequent reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene). Thus, an inverse electron demand Diels-Alder cycloaddition takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====HOMO of the Exo Transition State====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====HOMO of the Endo Transition State====

There are significant secondary orbital interactions in the transition state of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as an alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/transition state geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Bonding in the Ring===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the π-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6π-electron) 6-membered ring component in the transition state.<ref name="Clayden et. al." /> (The overall transition state itself is also aromatic, since it has 10π-electrons, i.e. obeys Hückel's (4n + 2) π-electron rule for aromaticity). <ref name="Clayden et. al." /> The C-C single bonds in the 6-membered ring all decrease in length over the course of the reaction, as they adopt a greater double bond character. <ref name="Clayden et. al." /> Eventually, this results in the formation a formal benzene ring component in all three products.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

Comparison of these values allows conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo Diels-Alder adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion, the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a transition state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" /><ref name="Levine" /> <ref name="Moss and Coady" /> This corresponds to a geometry with a single vibration with a negative frequency. <ref name="Levine" /> <ref name="Moss and Coady" />

In particular, the frontier molecular orbitals for the Diels-Alder reactions between butadiene and ethene, and a substituted butadiene derivative (cyclohexadiene) and an ethene derivative (1,3-dioxole) were analysed, and the symmetry of the interacting orbitals, and the molecular orbitals they produce in the transition state, were predicted. These predictions were subsequently matched to the computationally-derived molecular orbitals for each of the optimised transition states. These studies allowed conclusions to be drawn about the orbital symmetry requirements of such reactions - only orbitals of matching symmetry can interact. Also, the electronic nature of the observed Diels-Alder reactions was deduced, from comparison of the relative energy levels of the HOMO/LUMO (the frontier orbitals) of the reacting components. From this analysis, the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole was found to be an inverse electron demand Diels-Alder cycloaddition. <ref name="Clayden et. al." /><ref name="Bruckner" />

Furthermore, these optimisation studies also allowed relative energy values of the reactants, products and transition states of each of the different reactions to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, which in turn allowed deduction of the kinetically and thermodynamically favoured products for each type of reaction.<ref name="Clayden et. al." />

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T18:55:49Z

Mh4815: /* Molecular Structure via Quantum Mechanics */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - which are derived by plotting the electronic potential energy value (V) of the overall molecular system for a particular nuclear geometry (Q) (where Q is a collective variable describing the particular nuclear configuration as a single vector).<ref name="Levine" /> This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, to construct a potential energy surface, the electronic Schrödinger equation is solved (for a particular nuclear configuration), and the resultant potential energy value plotted. This is repeated across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" />

The Hamiltonian equation has the form:<ref name="Atkins and Friedman" />

<math> \hat{H} = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

====Stationary Points====

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" /><ref name="Levine" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

====Distinguishing Minima and Transition States====

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

*A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

*A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

*A transition state is the maximum point on the minimum energy path (MEP) which connects two minima.<ref name="Levine" /> In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

====Frequency Calculations====

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.<ref name="Levine" />

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

====Semi-Empirical Technique====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms.<ref name="Levine" /> This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

====Density-Functional Theory Method====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly.<ref name="Levine" /><ref name="Hinchliffe" /> The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

====Comparison====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

====TS====

=====MO 16=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 17=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 18=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 19=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Atkins and Friedman" /><ref name="Levine" /><ref name="Moss and Coady" />

<jmol>
<jmolApplet>
<color>white</color>
<script>frame 15; vibration 1;</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/transition state geometries were optimised to the semi-empirical PM6 level, followed by subsequent reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene). Thus, an inverse electron demand Diels-Alder cycloaddition takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====HOMO of the Exo Transition State====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====HOMO of the Endo Transition State====

There are significant secondary orbital interactions in the transition state of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as an alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/transition state geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Bonding in the Ring===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the π-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6π-electron) 6-membered ring component in the transition state.<ref name="Clayden et. al." /> (The overall transition state itself is also aromatic, since it has 10π-electrons, i.e. obeys Hückel's (4n + 2) π-electron rule for aromaticity). <ref name="Clayden et. al." /> The C-C single bonds in the 6-membered ring all decrease in length over the course of the reaction, as they adopt a greater double bond character. <ref name="Clayden et. al." /> Eventually, this results in the formation a formal benzene ring component in all three products.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

Comparison of these values allows conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo Diels-Alder adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion, the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a transition state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" /><ref name="Levine" /> <ref name="Moss and Coady" /> This corresponds to a geometry with a single vibration with a negative frequency. <ref name="Levine" /> <ref name="Moss and Coady" />

In particular, the frontier molecular orbitals for the Diels-Alder reactions between butadiene and ethene, and a substituted butadiene derivative (cyclohexadiene) and an ethene derivative (1,3-dioxole) were analysed, and the symmetry of the interacting orbitals, and the molecular orbitals they produce in the transition state, were predicted. These predictions were subsequently matched to the computationally-derived molecular orbitals for each of the optimised transition states. These studies allowed conclusions to be drawn about the orbital symmetry requirements of such reactions - only orbitals of matching symmetry can interact. Also, the electronic nature of the observed Diels-Alder reactions was deduced, from comparison of the relative energy levels of the HOMO/LUMO (the frontier orbitals) of the reacting components. From this analysis, the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole was found to be an inverse electron demand Diels-Alder cycloaddition. <ref name="Clayden et. al." /><ref name="Bruckner" />

Furthermore, these optimisation studies also allowed relative energy values of the reactants, products and transition states of each of the different reactions to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, which in turn allowed deduction of the kinetically and thermodynamically favoured products for each type of reaction.<ref name="Clayden et. al." />

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T18:52:29Z

Mh4815: /* Density-Functional Theory Method */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - found by plotting the electronic potential energy value (V) of the overall molecular system for a particular nuclear geometry (Q) (where Q is a collective variable describing the particular nuclear configuration as a single vector).<ref name="Levine" /> This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, to construct a potential energy surface, the electronic Schrödinger equation is solved (for a particular nuclear configuration), and the resultant potential energy value plotted. This is repeated across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" />

The Hamiltonian equation has the form:<ref name="Atkins and Friedman" />

<math> \hat{H} = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

====Stationary Points====

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" /><ref name="Levine" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

====Distinguishing Minima and Transition States====

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

*A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

*A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

*A transition state is the maximum point on the minimum energy path (MEP) which connects two minima.<ref name="Levine" /> In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

====Frequency Calculations====

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.<ref name="Levine" />

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

====Semi-Empirical Technique====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms.<ref name="Levine" /> This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

====Density-Functional Theory Method====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly.<ref name="Levine" /><ref name="Hinchliffe" /> The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

====Comparison====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

====TS====

=====MO 16=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 17=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 18=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 19=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Atkins and Friedman" /><ref name="Levine" /><ref name="Moss and Coady" />

<jmol>
<jmolApplet>
<color>white</color>
<script>frame 15; vibration 1;</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/transition state geometries were optimised to the semi-empirical PM6 level, followed by subsequent reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene). Thus, an inverse electron demand Diels-Alder cycloaddition takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====HOMO of the Exo Transition State====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====HOMO of the Endo Transition State====

There are significant secondary orbital interactions in the transition state of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as an alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/transition state geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Bonding in the Ring===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the π-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6π-electron) 6-membered ring component in the transition state.<ref name="Clayden et. al." /> (The overall transition state itself is also aromatic, since it has 10π-electrons, i.e. obeys Hückel's (4n + 2) π-electron rule for aromaticity). <ref name="Clayden et. al." /> The C-C single bonds in the 6-membered ring all decrease in length over the course of the reaction, as they adopt a greater double bond character. <ref name="Clayden et. al." /> Eventually, this results in the formation a formal benzene ring component in all three products.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

Comparison of these values allows conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo Diels-Alder adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion, the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a transition state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" /><ref name="Levine" /> <ref name="Moss and Coady" /> This corresponds to a geometry with a single vibration with a negative frequency. <ref name="Levine" /> <ref name="Moss and Coady" />

In particular, the frontier molecular orbitals for the Diels-Alder reactions between butadiene and ethene, and a substituted butadiene derivative (cyclohexadiene) and an ethene derivative (1,3-dioxole) were analysed, and the symmetry of the interacting orbitals, and the molecular orbitals they produce in the transition state, were predicted. These predictions were subsequently matched to the computationally-derived molecular orbitals for each of the optimised transition states. These studies allowed conclusions to be drawn about the orbital symmetry requirements of such reactions - only orbitals of matching symmetry can interact. Also, the electronic nature of the observed Diels-Alder reactions was deduced, from comparison of the relative energy levels of the HOMO/LUMO (the frontier orbitals) of the reacting components. From this analysis, the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole was found to be an inverse electron demand Diels-Alder cycloaddition. <ref name="Clayden et. al." /><ref name="Bruckner" />

Furthermore, these optimisation studies also allowed relative energy values of the reactants, products and transition states of each of the different reactions to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, which in turn allowed deduction of the kinetically and thermodynamically favoured products for each type of reaction.<ref name="Clayden et. al." />

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T18:52:05Z

Mh4815: /* Semi-Empirical Technique */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - found by plotting the electronic potential energy value (V) of the overall molecular system for a particular nuclear geometry (Q) (where Q is a collective variable describing the particular nuclear configuration as a single vector).<ref name="Levine" /> This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, to construct a potential energy surface, the electronic Schrödinger equation is solved (for a particular nuclear configuration), and the resultant potential energy value plotted. This is repeated across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" />

The Hamiltonian equation has the form:<ref name="Atkins and Friedman" />

<math> \hat{H} = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

====Stationary Points====

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" /><ref name="Levine" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

====Distinguishing Minima and Transition States====

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

*A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

*A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

*A transition state is the maximum point on the minimum energy path (MEP) which connects two minima.<ref name="Levine" /> In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

====Frequency Calculations====

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.<ref name="Levine" />

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

====Semi-Empirical Technique====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms.<ref name="Levine" /> This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

====Density-Functional Theory Method====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly. The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

====Comparison====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

====TS====

=====MO 16=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 17=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 18=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 19=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Atkins and Friedman" /><ref name="Levine" /><ref name="Moss and Coady" />

<jmol>
<jmolApplet>
<color>white</color>
<script>frame 15; vibration 1;</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/transition state geometries were optimised to the semi-empirical PM6 level, followed by subsequent reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene). Thus, an inverse electron demand Diels-Alder cycloaddition takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====HOMO of the Exo Transition State====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====HOMO of the Endo Transition State====

There are significant secondary orbital interactions in the transition state of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as an alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/transition state geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Bonding in the Ring===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the π-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6π-electron) 6-membered ring component in the transition state.<ref name="Clayden et. al." /> (The overall transition state itself is also aromatic, since it has 10π-electrons, i.e. obeys Hückel's (4n + 2) π-electron rule for aromaticity). <ref name="Clayden et. al." /> The C-C single bonds in the 6-membered ring all decrease in length over the course of the reaction, as they adopt a greater double bond character. <ref name="Clayden et. al." /> Eventually, this results in the formation a formal benzene ring component in all three products.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

Comparison of these values allows conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo Diels-Alder adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion, the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a transition state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" /><ref name="Levine" /> <ref name="Moss and Coady" /> This corresponds to a geometry with a single vibration with a negative frequency. <ref name="Levine" /> <ref name="Moss and Coady" />

In particular, the frontier molecular orbitals for the Diels-Alder reactions between butadiene and ethene, and a substituted butadiene derivative (cyclohexadiene) and an ethene derivative (1,3-dioxole) were analysed, and the symmetry of the interacting orbitals, and the molecular orbitals they produce in the transition state, were predicted. These predictions were subsequently matched to the computationally-derived molecular orbitals for each of the optimised transition states. These studies allowed conclusions to be drawn about the orbital symmetry requirements of such reactions - only orbitals of matching symmetry can interact. Also, the electronic nature of the observed Diels-Alder reactions was deduced, from comparison of the relative energy levels of the HOMO/LUMO (the frontier orbitals) of the reacting components. From this analysis, the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole was found to be an inverse electron demand Diels-Alder cycloaddition. <ref name="Clayden et. al." /><ref name="Bruckner" />

Furthermore, these optimisation studies also allowed relative energy values of the reactants, products and transition states of each of the different reactions to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, which in turn allowed deduction of the kinetically and thermodynamically favoured products for each type of reaction.<ref name="Clayden et. al." />

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T18:51:33Z

Mh4815: /* Frequency Calculations */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - found by plotting the electronic potential energy value (V) of the overall molecular system for a particular nuclear geometry (Q) (where Q is a collective variable describing the particular nuclear configuration as a single vector).<ref name="Levine" /> This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, to construct a potential energy surface, the electronic Schrödinger equation is solved (for a particular nuclear configuration), and the resultant potential energy value plotted. This is repeated across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" />

The Hamiltonian equation has the form:<ref name="Atkins and Friedman" />

<math> \hat{H} = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

====Stationary Points====

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" /><ref name="Levine" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

====Distinguishing Minima and Transition States====

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

*A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

*A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

*A transition state is the maximum point on the minimum energy path (MEP) which connects two minima.<ref name="Levine" /> In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

====Frequency Calculations====

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.<ref name="Levine" />

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

====Semi-Empirical Technique====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms. This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

====Density-Functional Theory Method====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly. The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

====Comparison====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

====TS====

=====MO 16=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 17=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 18=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 19=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Atkins and Friedman" /><ref name="Levine" /><ref name="Moss and Coady" />

<jmol>
<jmolApplet>
<color>white</color>
<script>frame 15; vibration 1;</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/transition state geometries were optimised to the semi-empirical PM6 level, followed by subsequent reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene). Thus, an inverse electron demand Diels-Alder cycloaddition takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====HOMO of the Exo Transition State====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====HOMO of the Endo Transition State====

There are significant secondary orbital interactions in the transition state of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as an alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/transition state geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Bonding in the Ring===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the π-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6π-electron) 6-membered ring component in the transition state.<ref name="Clayden et. al." /> (The overall transition state itself is also aromatic, since it has 10π-electrons, i.e. obeys Hückel's (4n + 2) π-electron rule for aromaticity). <ref name="Clayden et. al." /> The C-C single bonds in the 6-membered ring all decrease in length over the course of the reaction, as they adopt a greater double bond character. <ref name="Clayden et. al." /> Eventually, this results in the formation a formal benzene ring component in all three products.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

Comparison of these values allows conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo Diels-Alder adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion, the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a transition state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" /><ref name="Levine" /> <ref name="Moss and Coady" /> This corresponds to a geometry with a single vibration with a negative frequency. <ref name="Levine" /> <ref name="Moss and Coady" />

In particular, the frontier molecular orbitals for the Diels-Alder reactions between butadiene and ethene, and a substituted butadiene derivative (cyclohexadiene) and an ethene derivative (1,3-dioxole) were analysed, and the symmetry of the interacting orbitals, and the molecular orbitals they produce in the transition state, were predicted. These predictions were subsequently matched to the computationally-derived molecular orbitals for each of the optimised transition states. These studies allowed conclusions to be drawn about the orbital symmetry requirements of such reactions - only orbitals of matching symmetry can interact. Also, the electronic nature of the observed Diels-Alder reactions was deduced, from comparison of the relative energy levels of the HOMO/LUMO (the frontier orbitals) of the reacting components. From this analysis, the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole was found to be an inverse electron demand Diels-Alder cycloaddition. <ref name="Clayden et. al." /><ref name="Bruckner" />

Furthermore, these optimisation studies also allowed relative energy values of the reactants, products and transition states of each of the different reactions to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, which in turn allowed deduction of the kinetically and thermodynamically favoured products for each type of reaction.<ref name="Clayden et. al." />

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T18:48:21Z

Mh4815: /* Molecular Structure via Quantum Mechanics */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - found by plotting the electronic potential energy value (V) of the overall molecular system for a particular nuclear geometry (Q) (where Q is a collective variable describing the particular nuclear configuration as a single vector).<ref name="Levine" /> This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, to construct a potential energy surface, the electronic Schrödinger equation is solved (for a particular nuclear configuration), and the resultant potential energy value plotted. This is repeated across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" />

The Hamiltonian equation has the form:<ref name="Atkins and Friedman" />

<math> \hat{H} = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

====Stationary Points====

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" /><ref name="Levine" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

====Distinguishing Minima and Transition States====

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

*A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

*A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

*A transition state is the maximum point on the minimum energy path (MEP) which connects two minima.<ref name="Levine" /> In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

====Frequency Calculations====

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

====Semi-Empirical Technique====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms. This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

====Density-Functional Theory Method====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly. The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

====Comparison====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

====TS====

=====MO 16=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 17=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 18=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 19=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Atkins and Friedman" /><ref name="Levine" /><ref name="Moss and Coady" />

<jmol>
<jmolApplet>
<color>white</color>
<script>frame 15; vibration 1;</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/transition state geometries were optimised to the semi-empirical PM6 level, followed by subsequent reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene). Thus, an inverse electron demand Diels-Alder cycloaddition takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====HOMO of the Exo Transition State====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====HOMO of the Endo Transition State====

There are significant secondary orbital interactions in the transition state of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as an alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/transition state geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Bonding in the Ring===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the π-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6π-electron) 6-membered ring component in the transition state.<ref name="Clayden et. al." /> (The overall transition state itself is also aromatic, since it has 10π-electrons, i.e. obeys Hückel's (4n + 2) π-electron rule for aromaticity). <ref name="Clayden et. al." /> The C-C single bonds in the 6-membered ring all decrease in length over the course of the reaction, as they adopt a greater double bond character. <ref name="Clayden et. al." /> Eventually, this results in the formation a formal benzene ring component in all three products.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

Comparison of these values allows conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo Diels-Alder adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion, the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a transition state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" /><ref name="Levine" /> <ref name="Moss and Coady" /> This corresponds to a geometry with a single vibration with a negative frequency. <ref name="Levine" /> <ref name="Moss and Coady" />

In particular, the frontier molecular orbitals for the Diels-Alder reactions between butadiene and ethene, and a substituted butadiene derivative (cyclohexadiene) and an ethene derivative (1,3-dioxole) were analysed, and the symmetry of the interacting orbitals, and the molecular orbitals they produce in the transition state, were predicted. These predictions were subsequently matched to the computationally-derived molecular orbitals for each of the optimised transition states. These studies allowed conclusions to be drawn about the orbital symmetry requirements of such reactions - only orbitals of matching symmetry can interact. Also, the electronic nature of the observed Diels-Alder reactions was deduced, from comparison of the relative energy levels of the HOMO/LUMO (the frontier orbitals) of the reacting components. From this analysis, the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole was found to be an inverse electron demand Diels-Alder cycloaddition. <ref name="Clayden et. al." /><ref name="Bruckner" />

Furthermore, these optimisation studies also allowed relative energy values of the reactants, products and transition states of each of the different reactions to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, which in turn allowed deduction of the kinetically and thermodynamically favoured products for each type of reaction.<ref name="Clayden et. al." />

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T18:42:46Z

Mh4815: /* Molecular Potential Energy Surfaces */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - found by plotting the electronic potential energy value (V) of the overall molecular system for a particular nuclear geometry (Q). This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, to construct a potential energy surface, the electronic Schrödinger equation is solved (for a particular nuclear configuration), and the resultant potential energy value plotted. This is repeated across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" />

The Hamiltonian equation has the form:<ref name="Atkins and Friedman" />

<math> \hat{H} = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

====Stationary Points====

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" /><ref name="Levine" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

====Distinguishing Minima and Transition States====

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

*A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

*A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

*A transition state is the maximum point on the minimum energy path (MEP) which connects two minima.<ref name="Levine" /> In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

====Frequency Calculations====

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

====Semi-Empirical Technique====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms. This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

====Density-Functional Theory Method====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly. The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

====Comparison====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
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This is a symmetric molecular orbital.

=====LUMO=====
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This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
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This is an antisymmetric molecular orbital.

=====LUMO=====
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This is a symmetric molecular orbital.

====TS====

=====MO 16=====
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This is an antisymmetric molecular orbital.

=====MO 17=====
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This is a symmetric molecular orbital.

=====MO 18=====
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This is a symmetric molecular orbital.

=====MO 19=====
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This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Atkins and Friedman" /><ref name="Levine" /><ref name="Moss and Coady" />

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The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/transition state geometries were optimised to the semi-empirical PM6 level, followed by subsequent reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

=====MO 40=====
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This is an antisymmetric molecular orbital.

=====MO 41=====
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This is a symmetric molecular orbital.

=====MO 42=====
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This is a symmetric molecular orbital.

=====MO 43=====
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This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

=====MO 40=====
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This is an antisymmetric molecular orbital.

=====MO 41=====
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This is a symmetric molecular orbital.

=====MO 42=====
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This is a symmetric molecular orbital.

=====MO 43=====
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This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene). Thus, an inverse electron demand Diels-Alder cycloaddition takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====HOMO of the Exo Transition State====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====HOMO of the Endo Transition State====

There are significant secondary orbital interactions in the transition state of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as an alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/transition state geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Bonding in the Ring===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the π-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6π-electron) 6-membered ring component in the transition state.<ref name="Clayden et. al." /> (The overall transition state itself is also aromatic, since it has 10π-electrons, i.e. obeys Hückel's (4n + 2) π-electron rule for aromaticity). <ref name="Clayden et. al." /> The C-C single bonds in the 6-membered ring all decrease in length over the course of the reaction, as they adopt a greater double bond character. <ref name="Clayden et. al." /> Eventually, this results in the formation a formal benzene ring component in all three products.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

Comparison of these values allows conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo Diels-Alder adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion, the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a transition state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" /><ref name="Levine" /> <ref name="Moss and Coady" /> This corresponds to a geometry with a single vibration with a negative frequency. <ref name="Levine" /> <ref name="Moss and Coady" />

In particular, the frontier molecular orbitals for the Diels-Alder reactions between butadiene and ethene, and a substituted butadiene derivative (cyclohexadiene) and an ethene derivative (1,3-dioxole) were analysed, and the symmetry of the interacting orbitals, and the molecular orbitals they produce in the transition state, were predicted. These predictions were subsequently matched to the computationally-derived molecular orbitals for each of the optimised transition states. These studies allowed conclusions to be drawn about the orbital symmetry requirements of such reactions - only orbitals of matching symmetry can interact. Also, the electronic nature of the observed Diels-Alder reactions was deduced, from comparison of the relative energy levels of the HOMO/LUMO (the frontier orbitals) of the reacting components. From this analysis, the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole was found to be an inverse electron demand Diels-Alder cycloaddition. <ref name="Clayden et. al." /><ref name="Bruckner" />

Furthermore, these optimisation studies also allowed relative energy values of the reactants, products and transition states of each of the different reactions to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, which in turn allowed deduction of the kinetically and thermodynamically favoured products for each type of reaction.<ref name="Clayden et. al." />

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T18:41:10Z

Mh4815: /* Molecular Potential Energy Surfaces */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - found by plotting the electronic potential energy value (V) of the overall molecular system for a particular nuclear geometry (Q). This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, to construct a potential energy surface, the electronic Schrödinger equation is solved (for a particular nuclear configuration), and the resultant potential energy value plotted. This is repeated across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" />

The Hamiltonian equation has the form:<ref name="Atkins and Friedman" />

<math> \hat{H} = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

====Stationary Points====

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

====Distinguishing Minima and Transition States====

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

*A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

*A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

*A transition state is the maximum point on the minimum energy path (MEP) which connects two minima.<ref name="Levine" /> In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

====Frequency Calculations====

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

====Semi-Empirical Technique====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms. This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

====Density-Functional Theory Method====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly. The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

====Comparison====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

====TS====

=====MO 16=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 17=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 18=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 19=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Atkins and Friedman" /><ref name="Levine" /><ref name="Moss and Coady" />

<jmol>
<jmolApplet>
<color>white</color>
<script>frame 15; vibration 1;</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/transition state geometries were optimised to the semi-empirical PM6 level, followed by subsequent reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene). Thus, an inverse electron demand Diels-Alder cycloaddition takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====HOMO of the Exo Transition State====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====HOMO of the Endo Transition State====

There are significant secondary orbital interactions in the transition state of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as an alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/transition state geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Bonding in the Ring===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the π-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6π-electron) 6-membered ring component in the transition state.<ref name="Clayden et. al." /> (The overall transition state itself is also aromatic, since it has 10π-electrons, i.e. obeys Hückel's (4n + 2) π-electron rule for aromaticity). <ref name="Clayden et. al." /> The C-C single bonds in the 6-membered ring all decrease in length over the course of the reaction, as they adopt a greater double bond character. <ref name="Clayden et. al." /> Eventually, this results in the formation a formal benzene ring component in all three products.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

Comparison of these values allows conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo Diels-Alder adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion, the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a transition state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" /><ref name="Levine" /> <ref name="Moss and Coady" /> This corresponds to a geometry with a single vibration with a negative frequency. <ref name="Levine" /> <ref name="Moss and Coady" />

In particular, the frontier molecular orbitals for the Diels-Alder reactions between butadiene and ethene, and a substituted butadiene derivative (cyclohexadiene) and an ethene derivative (1,3-dioxole) were analysed, and the symmetry of the interacting orbitals, and the molecular orbitals they produce in the transition state, were predicted. These predictions were subsequently matched to the computationally-derived molecular orbitals for each of the optimised transition states. These studies allowed conclusions to be drawn about the orbital symmetry requirements of such reactions - only orbitals of matching symmetry can interact. Also, the electronic nature of the observed Diels-Alder reactions was deduced, from comparison of the relative energy levels of the HOMO/LUMO (the frontier orbitals) of the reacting components. From this analysis, the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole was found to be an inverse electron demand Diels-Alder cycloaddition. <ref name="Clayden et. al." /><ref name="Bruckner" />

Furthermore, these optimisation studies also allowed relative energy values of the reactants, products and transition states of each of the different reactions to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, which in turn allowed deduction of the kinetically and thermodynamically favoured products for each type of reaction.<ref name="Clayden et. al." />

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T18:37:23Z

Mh4815: /* Conclusion */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - found by plotting the electronic potential energy value (V) of the overall molecular system for a particular nuclear geometry (Q). This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, to construct a potential energy surface, the electronic Schrödinger equation is solved (for a particular nuclear configuration), and the resultant potential energy value plotted. This is repeated across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" />

The Hamiltonian equation has the form:<ref name="Atkins and Friedman" />

<math> \hat{H} = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

====Stationary Points====

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

====Distinguishing Minima and Transition States====

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

*A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

*A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

*A transition state is the maximum point on the minimum energy path (MEP) which connects two minima. In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

====Frequency Calculations====

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

====Semi-Empirical Technique====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms. This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

====Density-Functional Theory Method====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly. The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

====Comparison====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

====TS====

=====MO 16=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 17=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 18=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 19=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Atkins and Friedman" /><ref name="Levine" /><ref name="Moss and Coady" />

<jmol>
<jmolApplet>
<color>white</color>
<script>frame 15; vibration 1;</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/transition state geometries were optimised to the semi-empirical PM6 level, followed by subsequent reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene). Thus, an inverse electron demand Diels-Alder cycloaddition takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====HOMO of the Exo Transition State====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====HOMO of the Endo Transition State====

There are significant secondary orbital interactions in the transition state of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as an alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/transition state geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Bonding in the Ring===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the π-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6π-electron) 6-membered ring component in the transition state.<ref name="Clayden et. al." /> (The overall transition state itself is also aromatic, since it has 10π-electrons, i.e. obeys Hückel's (4n + 2) π-electron rule for aromaticity). <ref name="Clayden et. al." /> The C-C single bonds in the 6-membered ring all decrease in length over the course of the reaction, as they adopt a greater double bond character. <ref name="Clayden et. al." /> Eventually, this results in the formation a formal benzene ring component in all three products.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

Comparison of these values allows conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo Diels-Alder adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion, the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a transition state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" /><ref name="Levine" /> <ref name="Moss and Coady" /> This corresponds to a geometry with a single vibration with a negative frequency. <ref name="Levine" /> <ref name="Moss and Coady" />

In particular, the frontier molecular orbitals for the Diels-Alder reactions between butadiene and ethene, and a substituted butadiene derivative (cyclohexadiene) and an ethene derivative (1,3-dioxole) were analysed, and the symmetry of the interacting orbitals, and the molecular orbitals they produce in the transition state, were predicted. These predictions were subsequently matched to the computationally-derived molecular orbitals for each of the optimised transition states. These studies allowed conclusions to be drawn about the orbital symmetry requirements of such reactions - only orbitals of matching symmetry can interact. Also, the electronic nature of the observed Diels-Alder reactions was deduced, from comparison of the relative energy levels of the HOMO/LUMO (the frontier orbitals) of the reacting components. From this analysis, the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole was found to be an inverse electron demand Diels-Alder cycloaddition. <ref name="Clayden et. al." /><ref name="Bruckner" />

Furthermore, these optimisation studies also allowed relative energy values of the reactants, products and transition states of each of the different reactions to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, which in turn allowed deduction of the kinetically and thermodynamically favoured products for each type of reaction.<ref name="Clayden et. al." />

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T18:37:01Z

Mh4815: /* Conclusion */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - found by plotting the electronic potential energy value (V) of the overall molecular system for a particular nuclear geometry (Q). This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, to construct a potential energy surface, the electronic Schrödinger equation is solved (for a particular nuclear configuration), and the resultant potential energy value plotted. This is repeated across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" />

The Hamiltonian equation has the form:<ref name="Atkins and Friedman" />

<math> \hat{H} = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

====Stationary Points====

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

====Distinguishing Minima and Transition States====

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

*A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

*A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

*A transition state is the maximum point on the minimum energy path (MEP) which connects two minima. In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

====Frequency Calculations====

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

====Semi-Empirical Technique====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms. This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

====Density-Functional Theory Method====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly. The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

====Comparison====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
<jmol>
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<script>frame 6; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
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</jmol>

This is a symmetric molecular orbital.

====TS====

=====MO 16=====
<jmol>
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<color>white</color>
<script>frame 14; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 17=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
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</jmol>

This is a symmetric molecular orbital.

=====MO 18=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
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</jmol>

This is a symmetric molecular orbital.

=====MO 19=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
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This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Atkins and Friedman" /><ref name="Levine" /><ref name="Moss and Coady" />

<jmol>
<jmolApplet>
<color>white</color>
<script>frame 15; vibration 1;</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
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</jmol>

The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/transition state geometries were optimised to the semi-empirical PM6 level, followed by subsequent reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
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<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
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</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
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</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene). Thus, an inverse electron demand Diels-Alder cycloaddition takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====HOMO of the Exo Transition State====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====HOMO of the Endo Transition State====

There are significant secondary orbital interactions in the transition state of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as an alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/transition state geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Bonding in the Ring===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the π-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6π-electron) 6-membered ring component in the transition state.<ref name="Clayden et. al." /> (The overall transition state itself is also aromatic, since it has 10π-electrons, i.e. obeys Hückel's (4n + 2) π-electron rule for aromaticity). <ref name="Clayden et. al." /> The C-C single bonds in the 6-membered ring all decrease in length over the course of the reaction, as they adopt a greater double bond character. <ref name="Clayden et. al." /> Eventually, this results in the formation a formal benzene ring component in all three products.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

Comparison of these values allows conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo Diels-Alder adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion, the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a transition state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" /><ref name="Levine" /> <ref name="Moss and Coady" /> This corresponds to a geometry with a single vibration with a negative frequency. <ref name="Levine" /> <ref name="Moss and Coady" />

In particular, the frontier molecular orbitals for the Diels-Alder reactions between butadiene and ethene, and a substituted butadiene derivative (cyclohexadiene) and an ethene derivative (1,3-dioxole) were analysed, and the symmetry of the interacting orbitals, and the molecular orbitals they produce in the transition state, were predicted. These predictions were subsequently matched to the computationally-derived molecular orbitals for each of the optimised transition states. These studies allowed conclusions to be drawn about the orbital symmetry requirements of such reactions - only orbitals of matching symmetry can interact. Also, the electronic nature of the observed Diels-Alder reactions was deduced, from comparison of the relative energy levels of the HOMO/LUMO (the frontier orbitals) of the reacting components. From this analysis, the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole was found to be an inverse electron demand Diels-Alder cycloaddition. <ref name="Clayden et. al." /><ref name="Bruckner" />

Furthermore, these optimisation studies also allowed relative energy values of the reactants, products and transition states of each of the different reactions to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, which in turn allowed deduction of the kinetically and thermodynamically favoured products for each type of reaction.

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T18:36:20Z

Mh4815: /* Conclusion */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - found by plotting the electronic potential energy value (V) of the overall molecular system for a particular nuclear geometry (Q). This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, to construct a potential energy surface, the electronic Schrödinger equation is solved (for a particular nuclear configuration), and the resultant potential energy value plotted. This is repeated across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" />

The Hamiltonian equation has the form:<ref name="Atkins and Friedman" />

<math> \hat{H} = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

====Stationary Points====

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

====Distinguishing Minima and Transition States====

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

*A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

*A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

*A transition state is the maximum point on the minimum energy path (MEP) which connects two minima. In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

====Frequency Calculations====

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

====Semi-Empirical Technique====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms. This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

====Density-Functional Theory Method====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly. The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

====Comparison====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
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This is a symmetric molecular orbital.

=====LUMO=====
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This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
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This is an antisymmetric molecular orbital.

=====LUMO=====
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This is a symmetric molecular orbital.

====TS====

=====MO 16=====
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This is an antisymmetric molecular orbital.

=====MO 17=====
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This is a symmetric molecular orbital.

=====MO 18=====
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This is a symmetric molecular orbital.

=====MO 19=====
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This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Atkins and Friedman" /><ref name="Levine" /><ref name="Moss and Coady" />

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The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/transition state geometries were optimised to the semi-empirical PM6 level, followed by subsequent reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

=====MO 40=====
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This is an antisymmetric molecular orbital.

=====MO 41=====
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This is a symmetric molecular orbital.

=====MO 42=====
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This is a symmetric molecular orbital.

=====MO 43=====
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This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

=====MO 40=====
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This is an antisymmetric molecular orbital.

=====MO 41=====
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This is a symmetric molecular orbital.

=====MO 42=====
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This is a symmetric molecular orbital.

=====MO 43=====
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This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene). Thus, an inverse electron demand Diels-Alder cycloaddition takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====HOMO of the Exo Transition State====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====HOMO of the Endo Transition State====

There are significant secondary orbital interactions in the transition state of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as an alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/transition state geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Bonding in the Ring===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the π-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6π-electron) 6-membered ring component in the transition state.<ref name="Clayden et. al." /> (The overall transition state itself is also aromatic, since it has 10π-electrons, i.e. obeys Hückel's (4n + 2) π-electron rule for aromaticity). <ref name="Clayden et. al." /> The C-C single bonds in the 6-membered ring all decrease in length over the course of the reaction, as they adopt a greater double bond character. <ref name="Clayden et. al." /> Eventually, this results in the formation a formal benzene ring component in all three products.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

Comparison of these values allows conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo Diels-Alder adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion, the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a transition state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" /><ref name="Levine" /> <ref name="Moss and Coady" /> This corresponds to a geometry with a single vibration with a negative frequency. <ref name="Levine" /> <ref name="Moss and Coady" />

In particular, the frontier molecular orbitals for the Diels-Alder reactions between butadiene and ethene, and a substituted butadiene derivative (cyclohexadiene) and an ethene derivative (1,3-dioxole) were analysed, and the symmetry of the interacting orbitals, and the molecular orbitals they produce in the transition state, were predicted. These predictions were subsequently matched to the computationally-derived molecular orbitals for each of the optimised transition states. These studies allowed conclusions to be drawn about the orbital symmetry requirements of such reactions - only orbitals of matching symmetry can interact. Also, the electronic nature of the observed Diels-Alder reactions was deduced, from comparison of the relative energy levels of the HOMO/LUMO (the frontier orbitals) of the reacting components. From this analysis, the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole was found to be an inverse electron demand Diels-Alder cycloaddition. <ref name="Clayden et. al." /><ref name="Bruckner" />

Furthermore, these optimisation studies also allowed relative energy values of the reactants, products and transition states of each of the different reactions to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, which in turn allowed deduction of the kinetically and thermodynamically favoured products of each type of reactions.

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T18:28:01Z

Mh4815: /* Kinetic and Thermodynamic Analysis */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - found by plotting the electronic potential energy value (V) of the overall molecular system for a particular nuclear geometry (Q). This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, to construct a potential energy surface, the electronic Schrödinger equation is solved (for a particular nuclear configuration), and the resultant potential energy value plotted. This is repeated across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" />

The Hamiltonian equation has the form:<ref name="Atkins and Friedman" />

<math> \hat{H} = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

====Stationary Points====

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

====Distinguishing Minima and Transition States====

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

*A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

*A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

*A transition state is the maximum point on the minimum energy path (MEP) which connects two minima. In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

====Frequency Calculations====

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

====Semi-Empirical Technique====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms. This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

====Density-Functional Theory Method====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly. The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

====Comparison====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

====TS====

=====MO 16=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 17=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 18=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 19=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Atkins and Friedman" /><ref name="Levine" /><ref name="Moss and Coady" />

<jmol>
<jmolApplet>
<color>white</color>
<script>frame 15; vibration 1;</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/transition state geometries were optimised to the semi-empirical PM6 level, followed by subsequent reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene). Thus, an inverse electron demand Diels-Alder cycloaddition takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====HOMO of the Exo Transition State====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====HOMO of the Endo Transition State====

There are significant secondary orbital interactions in the transition state of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as an alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/transition state geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Bonding in the Ring===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the π-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6π-electron) 6-membered ring component in the transition state.<ref name="Clayden et. al." /> (The overall transition state itself is also aromatic, since it has 10π-electrons, i.e. obeys Hückel's (4n + 2) π-electron rule for aromaticity). <ref name="Clayden et. al." /> The C-C single bonds in the 6-membered ring all decrease in length over the course of the reaction, as they adopt a greater double bond character. <ref name="Clayden et. al." /> Eventually, this results in the formation a formal benzene ring component in all three products.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

Comparison of these values allows conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo Diels-Alder adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion, the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a transition state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" /><ref name="Levine" /> <ref name="Moss and Coady" /> This corresponds to a geometry with a single vibration with a negative frequency. <ref name="Levine" /> <ref name="Moss and Coady" />

In particular, the frontier molecular orbitals for the Diels-Alder reactions between butadiene and ethene, and a substituted butadiene derivative (cyclohexadiene) and an ethene derivative (1,3-dioxole) were analysed, and the symmetry of the interacting orbitals, and the molecular orbitals they produce in the transition state, were predicted. These predictions were subsequently matched to the computationally-derived molecular orbitals for each of the optimised transition states. These studies allowed conclusions to be drawn about the orbital symmetry requirements of such reactions - only orbitals of matching symmetry can interact. Also, the electronic nature of the observed Diels-Alder reactions was deduced, from comparison of the relative energy levels of the HOMO/LUMO (the frontier orbitals) of the reacting components. From this analysis, the specific Diels-Alder reaction between cyclohexadiene and 1,3-dioxole was found to be an inverse electron demand Diels-Alder cycloaddition. <ref name="Clayden et. al." /><ref name="Bruckner" />

Furthermore, these optimisation studies also allowed relative energy values of the reactants, products and transition states of each of the different reactions to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, which in turn allowed deduction of the kinetically and thermodynamically favoured products of each type of reactions.

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T18:26:31Z

Mh4815: /* Kinetic and Thermodynamic Analysis */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - found by plotting the electronic potential energy value (V) of the overall molecular system for a particular nuclear geometry (Q). This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, to construct a potential energy surface, the electronic Schrödinger equation is solved (for a particular nuclear configuration), and the resultant potential energy value plotted. This is repeated across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" />

The Hamiltonian equation has the form:<ref name="Atkins and Friedman" />

<math> \hat{H} = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

====Stationary Points====

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

====Distinguishing Minima and Transition States====

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

*A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

*A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

*A transition state is the maximum point on the minimum energy path (MEP) which connects two minima. In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

====Frequency Calculations====

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

====Semi-Empirical Technique====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms. This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

====Density-Functional Theory Method====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly. The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

====Comparison====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

====TS====

=====MO 16=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 17=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 18=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 19=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Atkins and Friedman" /><ref name="Levine" /><ref name="Moss and Coady" />

<jmol>
<jmolApplet>
<color>white</color>
<script>frame 15; vibration 1;</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/transition state geometries were optimised to the semi-empirical PM6 level, followed by subsequent reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene). Thus, an inverse electron demand Diels-Alder cycloaddition takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====HOMO of the Exo Transition State====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====HOMO of the Endo Transition State====

There are significant secondary orbital interactions in the transition state of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as an alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/transition state geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Bonding in the Ring===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the π-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6π-electron) 6-membered ring component in the transition state.<ref name="Clayden et. al." /> (The overall transition state itself is also aromatic, since it has 10π-electrons, i.e. obeys Hückel's (4n + 2) π-electron rule for aromaticity). <ref name="Clayden et. al." /> The C-C single bonds in the 6-membered ring all decrease in length over the course of the reaction, as they adopt a greater double bond character. <ref name="Clayden et. al." /> Eventually, this results in the formation a formal benzene ring component in all three products.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

Comparison of these values allows conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion, the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a transition state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" /><ref name="Levine" /> <ref name="Moss and Coady" /> This corresponds to a geometry with a single vibration with a negative frequency. <ref name="Levine" /> <ref name="Moss and Coady" />

In particular, the frontier molecular orbitals for the Diels-Alder reactions between butadiene and ethene, and a substituted butadiene derivative (cyclohexadiene) and an ethene derivative (1,3-dioxole) were analysed, and the symmetry of the interacting orbitals, and the molecular orbitals they produce in the transition state, were predicted. These predictions were subsequently matched to the computationally-derived molecular orbitals for each of the optimised transition states. These studies allowed conclusions to be drawn about the orbital symmetry requirements of such reactions - only orbitals of matching symmetry can interact. Also, the electronic nature of the observed Diels-Alder reactions was deduced, from comparison of the relative energy levels of the HOMO/LUMO (the frontier orbitals) of the reacting components. From this analysis, the specific Diels-Alder reaction between cyclohexadiene and 1,3-dioxole was found to be an inverse electron demand Diels-Alder cycloaddition. <ref name="Clayden et. al." /><ref name="Bruckner" />

Furthermore, these optimisation studies also allowed relative energy values of the reactants, products and transition states of each of the different reactions to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, which in turn allowed deduction of the kinetically and thermodynamically favoured products of each type of reactions.

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T18:25:39Z

Mh4815: /* Exercise 3: Diels-Alder vs. Cheletropic */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - found by plotting the electronic potential energy value (V) of the overall molecular system for a particular nuclear geometry (Q). This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, to construct a potential energy surface, the electronic Schrödinger equation is solved (for a particular nuclear configuration), and the resultant potential energy value plotted. This is repeated across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" />

The Hamiltonian equation has the form:<ref name="Atkins and Friedman" />

<math> \hat{H} = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

====Stationary Points====

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

====Distinguishing Minima and Transition States====

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

*A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

*A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

*A transition state is the maximum point on the minimum energy path (MEP) which connects two minima. In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

====Frequency Calculations====

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

====Semi-Empirical Technique====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms. This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

====Density-Functional Theory Method====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly. The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

====Comparison====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

====TS====

=====MO 16=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 17=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 18=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 19=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Atkins and Friedman" /><ref name="Levine" /><ref name="Moss and Coady" />

<jmol>
<jmolApplet>
<color>white</color>
<script>frame 15; vibration 1;</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/transition state geometries were optimised to the semi-empirical PM6 level, followed by subsequent reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene). Thus, an inverse electron demand Diels-Alder cycloaddition takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====HOMO of the Exo Transition State====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====HOMO of the Endo Transition State====

There are significant secondary orbital interactions in the transition state of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as an alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/transition state geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Bonding in the Ring===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the π-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6π-electron) 6-membered ring component in the transition state.<ref name="Clayden et. al." /> (The overall transition state itself is also aromatic, since it has 10π-electrons, i.e. obeys Hückel's (4n + 2) π-electron rule for aromaticity). <ref name="Clayden et. al." /> The C-C single bonds in the 6-membered ring all decrease in length over the course of the reaction, as they adopt a greater double bond character. <ref name="Clayden et. al." /> Eventually, this results in the formation a formal benzene ring component in all three products.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

From this comparing these values, conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion, the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a transition state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" /><ref name="Levine" /> <ref name="Moss and Coady" /> This corresponds to a geometry with a single vibration with a negative frequency. <ref name="Levine" /> <ref name="Moss and Coady" />

In particular, the frontier molecular orbitals for the Diels-Alder reactions between butadiene and ethene, and a substituted butadiene derivative (cyclohexadiene) and an ethene derivative (1,3-dioxole) were analysed, and the symmetry of the interacting orbitals, and the molecular orbitals they produce in the transition state, were predicted. These predictions were subsequently matched to the computationally-derived molecular orbitals for each of the optimised transition states. These studies allowed conclusions to be drawn about the orbital symmetry requirements of such reactions - only orbitals of matching symmetry can interact. Also, the electronic nature of the observed Diels-Alder reactions was deduced, from comparison of the relative energy levels of the HOMO/LUMO (the frontier orbitals) of the reacting components. From this analysis, the specific Diels-Alder reaction between cyclohexadiene and 1,3-dioxole was found to be an inverse electron demand Diels-Alder cycloaddition. <ref name="Clayden et. al." /><ref name="Bruckner" />

Furthermore, these optimisation studies also allowed relative energy values of the reactants, products and transition states of each of the different reactions to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, which in turn allowed deduction of the kinetically and thermodynamically favoured products of each type of reactions.

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T18:25:01Z

Mh4815: /* Secondary Orbital Interactions */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - found by plotting the electronic potential energy value (V) of the overall molecular system for a particular nuclear geometry (Q). This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, to construct a potential energy surface, the electronic Schrödinger equation is solved (for a particular nuclear configuration), and the resultant potential energy value plotted. This is repeated across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" />

The Hamiltonian equation has the form:<ref name="Atkins and Friedman" />

<math> \hat{H} = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

====Stationary Points====

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

====Distinguishing Minima and Transition States====

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

*A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

*A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

*A transition state is the maximum point on the minimum energy path (MEP) which connects two minima. In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

====Frequency Calculations====

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

====Semi-Empirical Technique====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms. This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

====Density-Functional Theory Method====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly. The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

====Comparison====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

====TS====

=====MO 16=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 17=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 18=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 19=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Atkins and Friedman" /><ref name="Levine" /><ref name="Moss and Coady" />

<jmol>
<jmolApplet>
<color>white</color>
<script>frame 15; vibration 1;</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/transition state geometries were optimised to the semi-empirical PM6 level, followed by subsequent reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene). Thus, an inverse electron demand Diels-Alder cycloaddition takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====HOMO of the Exo Transition State====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====HOMO of the Endo Transition State====

There are significant secondary orbital interactions in the transition state of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as the alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/TS geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Bonding in the Ring===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the π-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6π-electron) 6-membered ring component in the transition state.<ref name="Clayden et. al." /> (The overall transition state itself is also aromatic, since it has 10π-electrons, i.e. obeys Hückel's (4n + 2) π-electron rule for aromaticity). <ref name="Clayden et. al." /> The C-C single bonds in the 6-membered ring all decrease in length over the course of the reaction, as they adopt a greater double bond character. <ref name="Clayden et. al." /> Eventually, this results in the formation a formal benzene ring component in all three products.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

From this comparing these values, conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion, the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a transition state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" /><ref name="Levine" /> <ref name="Moss and Coady" /> This corresponds to a geometry with a single vibration with a negative frequency. <ref name="Levine" /> <ref name="Moss and Coady" />

In particular, the frontier molecular orbitals for the Diels-Alder reactions between butadiene and ethene, and a substituted butadiene derivative (cyclohexadiene) and an ethene derivative (1,3-dioxole) were analysed, and the symmetry of the interacting orbitals, and the molecular orbitals they produce in the transition state, were predicted. These predictions were subsequently matched to the computationally-derived molecular orbitals for each of the optimised transition states. These studies allowed conclusions to be drawn about the orbital symmetry requirements of such reactions - only orbitals of matching symmetry can interact. Also, the electronic nature of the observed Diels-Alder reactions was deduced, from comparison of the relative energy levels of the HOMO/LUMO (the frontier orbitals) of the reacting components. From this analysis, the specific Diels-Alder reaction between cyclohexadiene and 1,3-dioxole was found to be an inverse electron demand Diels-Alder cycloaddition. <ref name="Clayden et. al." /><ref name="Bruckner" />

Furthermore, these optimisation studies also allowed relative energy values of the reactants, products and transition states of each of the different reactions to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, which in turn allowed deduction of the kinetically and thermodynamically favoured products of each type of reactions.

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T18:22:23Z

Mh4815: /* Type of Diels-Alder Reaction */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - found by plotting the electronic potential energy value (V) of the overall molecular system for a particular nuclear geometry (Q). This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, to construct a potential energy surface, the electronic Schrödinger equation is solved (for a particular nuclear configuration), and the resultant potential energy value plotted. This is repeated across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" />

The Hamiltonian equation has the form:<ref name="Atkins and Friedman" />

<math> \hat{H} = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

====Stationary Points====

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

====Distinguishing Minima and Transition States====

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

*A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

*A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

*A transition state is the maximum point on the minimum energy path (MEP) which connects two minima. In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

====Frequency Calculations====

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

====Semi-Empirical Technique====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms. This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

====Density-Functional Theory Method====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly. The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

====Comparison====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

====TS====

=====MO 16=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 17=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 18=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 19=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Atkins and Friedman" /><ref name="Levine" /><ref name="Moss and Coady" />

<jmol>
<jmolApplet>
<color>white</color>
<script>frame 15; vibration 1;</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/transition state geometries were optimised to the semi-empirical PM6 level, followed by subsequent reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene). Thus, an inverse electron demand Diels-Alder cycloaddition takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====Exo TS====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====Endo TS====

There are significant secondary orbital interactions in the TS of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as the alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/TS geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Bonding in the Ring===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the π-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6π-electron) 6-membered ring component in the transition state.<ref name="Clayden et. al." /> (The overall transition state itself is also aromatic, since it has 10π-electrons, i.e. obeys Hückel's (4n + 2) π-electron rule for aromaticity). <ref name="Clayden et. al." /> The C-C single bonds in the 6-membered ring all decrease in length over the course of the reaction, as they adopt a greater double bond character. <ref name="Clayden et. al." /> Eventually, this results in the formation a formal benzene ring component in all three products.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

From this comparing these values, conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion, the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a transition state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" /><ref name="Levine" /> <ref name="Moss and Coady" /> This corresponds to a geometry with a single vibration with a negative frequency. <ref name="Levine" /> <ref name="Moss and Coady" />

In particular, the frontier molecular orbitals for the Diels-Alder reactions between butadiene and ethene, and a substituted butadiene derivative (cyclohexadiene) and an ethene derivative (1,3-dioxole) were analysed, and the symmetry of the interacting orbitals, and the molecular orbitals they produce in the transition state, were predicted. These predictions were subsequently matched to the computationally-derived molecular orbitals for each of the optimised transition states. These studies allowed conclusions to be drawn about the orbital symmetry requirements of such reactions - only orbitals of matching symmetry can interact. Also, the electronic nature of the observed Diels-Alder reactions was deduced, from comparison of the relative energy levels of the HOMO/LUMO (the frontier orbitals) of the reacting components. From this analysis, the specific Diels-Alder reaction between cyclohexadiene and 1,3-dioxole was found to be an inverse electron demand Diels-Alder cycloaddition. <ref name="Clayden et. al." /><ref name="Bruckner" />

Furthermore, these optimisation studies also allowed relative energy values of the reactants, products and transition states of each of the different reactions to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, which in turn allowed deduction of the kinetically and thermodynamically favoured products of each type of reactions.

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T18:16:01Z

Mh4815: /* Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - found by plotting the electronic potential energy value (V) of the overall molecular system for a particular nuclear geometry (Q). This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, to construct a potential energy surface, the electronic Schrödinger equation is solved (for a particular nuclear configuration), and the resultant potential energy value plotted. This is repeated across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" />

The Hamiltonian equation has the form:<ref name="Atkins and Friedman" />

<math> \hat{H} = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

====Stationary Points====

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

====Distinguishing Minima and Transition States====

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

*A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

*A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

*A transition state is the maximum point on the minimum energy path (MEP) which connects two minima. In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

====Frequency Calculations====

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

====Semi-Empirical Technique====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms. This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

====Density-Functional Theory Method====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly. The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

====Comparison====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

====TS====

=====MO 16=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 17=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 18=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 19=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Atkins and Friedman" /><ref name="Levine" /><ref name="Moss and Coady" />

<jmol>
<jmolApplet>
<color>white</color>
<script>frame 15; vibration 1;</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/transition state geometries were optimised to the semi-empirical PM6 level, followed by subsequent reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene), an an inverse electron demand Diels-Alder cycloaddition therefore takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====Exo TS====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====Endo TS====

There are significant secondary orbital interactions in the TS of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as the alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/TS geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Bonding in the Ring===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the π-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6π-electron) 6-membered ring component in the transition state.<ref name="Clayden et. al." /> (The overall transition state itself is also aromatic, since it has 10π-electrons, i.e. obeys Hückel's (4n + 2) π-electron rule for aromaticity). <ref name="Clayden et. al." /> The C-C single bonds in the 6-membered ring all decrease in length over the course of the reaction, as they adopt a greater double bond character. <ref name="Clayden et. al." /> Eventually, this results in the formation a formal benzene ring component in all three products.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

From this comparing these values, conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion, the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a transition state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" /><ref name="Levine" /> <ref name="Moss and Coady" /> This corresponds to a geometry with a single vibration with a negative frequency. <ref name="Levine" /> <ref name="Moss and Coady" />

In particular, the frontier molecular orbitals for the Diels-Alder reactions between butadiene and ethene, and a substituted butadiene derivative (cyclohexadiene) and an ethene derivative (1,3-dioxole) were analysed, and the symmetry of the interacting orbitals, and the molecular orbitals they produce in the transition state, were predicted. These predictions were subsequently matched to the computationally-derived molecular orbitals for each of the optimised transition states. These studies allowed conclusions to be drawn about the orbital symmetry requirements of such reactions - only orbitals of matching symmetry can interact. Also, the electronic nature of the observed Diels-Alder reactions was deduced, from comparison of the relative energy levels of the HOMO/LUMO (the frontier orbitals) of the reacting components. From this analysis, the specific Diels-Alder reaction between cyclohexadiene and 1,3-dioxole was found to be an inverse electron demand Diels-Alder cycloaddition. <ref name="Clayden et. al." /><ref name="Bruckner" />

Furthermore, these optimisation studies also allowed relative energy values of the reactants, products and transition states of each of the different reactions to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, which in turn allowed deduction of the kinetically and thermodynamically favoured products of each type of reactions.

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T18:14:22Z

Mh4815: /* Transition State Reaction Path Vibration */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - found by plotting the electronic potential energy value (V) of the overall molecular system for a particular nuclear geometry (Q). This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, to construct a potential energy surface, the electronic Schrödinger equation is solved (for a particular nuclear configuration), and the resultant potential energy value plotted. This is repeated across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" />

The Hamiltonian equation has the form:<ref name="Atkins and Friedman" />

<math> \hat{H} = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

====Stationary Points====

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

====Distinguishing Minima and Transition States====

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

*A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

*A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

*A transition state is the maximum point on the minimum energy path (MEP) which connects two minima. In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

====Frequency Calculations====

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

====Semi-Empirical Technique====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms. This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

====Density-Functional Theory Method====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly. The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

====Comparison====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
<jmol>
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<script>frame 10; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
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</jmol>

This is a symmetric molecular orbital.

=====LUMO=====
<jmol>
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<color>white</color>
<script>frame 10; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
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</jmol>

This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

====TS====

=====MO 16=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 17=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 18=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 19=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Atkins and Friedman" /><ref name="Levine" /><ref name="Moss and Coady" />

<jmol>
<jmolApplet>
<color>white</color>
<script>frame 15; vibration 1;</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/TS geometries were optimised to the semi-empirical PM6 level, followed by reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene), an an inverse electron demand Diels-Alder cycloaddition therefore takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====Exo TS====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====Endo TS====

There are significant secondary orbital interactions in the TS of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as the alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/TS geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Bonding in the Ring===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the π-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6π-electron) 6-membered ring component in the transition state.<ref name="Clayden et. al." /> (The overall transition state itself is also aromatic, since it has 10π-electrons, i.e. obeys Hückel's (4n + 2) π-electron rule for aromaticity). <ref name="Clayden et. al." /> The C-C single bonds in the 6-membered ring all decrease in length over the course of the reaction, as they adopt a greater double bond character. <ref name="Clayden et. al." /> Eventually, this results in the formation a formal benzene ring component in all three products.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

From this comparing these values, conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion, the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a transition state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" /><ref name="Levine" /> <ref name="Moss and Coady" /> This corresponds to a geometry with a single vibration with a negative frequency. <ref name="Levine" /> <ref name="Moss and Coady" />

In particular, the frontier molecular orbitals for the Diels-Alder reactions between butadiene and ethene, and a substituted butadiene derivative (cyclohexadiene) and an ethene derivative (1,3-dioxole) were analysed, and the symmetry of the interacting orbitals, and the molecular orbitals they produce in the transition state, were predicted. These predictions were subsequently matched to the computationally-derived molecular orbitals for each of the optimised transition states. These studies allowed conclusions to be drawn about the orbital symmetry requirements of such reactions - only orbitals of matching symmetry can interact. Also, the electronic nature of the observed Diels-Alder reactions was deduced, from comparison of the relative energy levels of the HOMO/LUMO (the frontier orbitals) of the reacting components. From this analysis, the specific Diels-Alder reaction between cyclohexadiene and 1,3-dioxole was found to be an inverse electron demand Diels-Alder cycloaddition. <ref name="Clayden et. al." /><ref name="Bruckner" />

Furthermore, these optimisation studies also allowed relative energy values of the reactants, products and transition states of each of the different reactions to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, which in turn allowed deduction of the kinetically and thermodynamically favoured products of each type of reactions.

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T18:00:34Z

Mh4815: /* Bonding in the Ring */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - found by plotting the electronic potential energy value (V) of the overall molecular system for a particular nuclear geometry (Q). This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, to construct a potential energy surface, the electronic Schrödinger equation is solved (for a particular nuclear configuration), and the resultant potential energy value plotted. This is repeated across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" />

The Hamiltonian equation has the form:<ref name="Atkins and Friedman" />

<math> \hat{H} = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

====Stationary Points====

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

====Distinguishing Minima and Transition States====

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

*A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

*A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

*A transition state is the maximum point on the minimum energy path (MEP) which connects two minima. In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

====Frequency Calculations====

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

====Semi-Empirical Technique====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms. This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

====Density-Functional Theory Method====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly. The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

====Comparison====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

====TS====

=====MO 16=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 17=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 18=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 19=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Levine" />

<jmol>
<jmolApplet>
<color>white</color>
<script>frame 15; vibration 1;</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/TS geometries were optimised to the semi-empirical PM6 level, followed by reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene), an an inverse electron demand Diels-Alder cycloaddition therefore takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====Exo TS====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====Endo TS====

There are significant secondary orbital interactions in the TS of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as the alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/TS geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Bonding in the Ring===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the π-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6π-electron) 6-membered ring component in the transition state.<ref name="Clayden et. al." /> (The overall transition state itself is also aromatic, since it has 10π-electrons, i.e. obeys Hückel's (4n + 2) π-electron rule for aromaticity). <ref name="Clayden et. al." /> The C-C single bonds in the 6-membered ring all decrease in length over the course of the reaction, as they adopt a greater double bond character. <ref name="Clayden et. al." /> Eventually, this results in the formation a formal benzene ring component in all three products.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

From this comparing these values, conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion, the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a transition state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" /><ref name="Levine" /> <ref name="Moss and Coady" /> This corresponds to a geometry with a single vibration with a negative frequency. <ref name="Levine" /> <ref name="Moss and Coady" />

In particular, the frontier molecular orbitals for the Diels-Alder reactions between butadiene and ethene, and a substituted butadiene derivative (cyclohexadiene) and an ethene derivative (1,3-dioxole) were analysed, and the symmetry of the interacting orbitals, and the molecular orbitals they produce in the transition state, were predicted. These predictions were subsequently matched to the computationally-derived molecular orbitals for each of the optimised transition states. These studies allowed conclusions to be drawn about the orbital symmetry requirements of such reactions - only orbitals of matching symmetry can interact. Also, the electronic nature of the observed Diels-Alder reactions was deduced, from comparison of the relative energy levels of the HOMO/LUMO (the frontier orbitals) of the reacting components. From this analysis, the specific Diels-Alder reaction between cyclohexadiene and 1,3-dioxole was found to be an inverse electron demand Diels-Alder cycloaddition. <ref name="Clayden et. al." /><ref name="Bruckner" />

Furthermore, these optimisation studies also allowed relative energy values of the reactants, products and transition states of each of the different reactions to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, which in turn allowed deduction of the kinetically and thermodynamically favoured products of each type of reactions.

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T17:58:38Z

Mh4815: /* Bonding in the Ring */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - found by plotting the electronic potential energy value (V) of the overall molecular system for a particular nuclear geometry (Q). This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, to construct a potential energy surface, the electronic Schrödinger equation is solved (for a particular nuclear configuration), and the resultant potential energy value plotted. This is repeated across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" />

The Hamiltonian equation has the form:<ref name="Atkins and Friedman" />

<math> \hat{H} = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

====Stationary Points====

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

====Distinguishing Minima and Transition States====

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

*A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

*A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

*A transition state is the maximum point on the minimum energy path (MEP) which connects two minima. In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

====Frequency Calculations====

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

====Semi-Empirical Technique====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms. This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

====Density-Functional Theory Method====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly. The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

====Comparison====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

====TS====

=====MO 16=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 17=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 18=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 19=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Levine" />

<jmol>
<jmolApplet>
<color>white</color>
<script>frame 15; vibration 1;</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/TS geometries were optimised to the semi-empirical PM6 level, followed by reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene), an an inverse electron demand Diels-Alder cycloaddition therefore takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====Exo TS====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====Endo TS====

There are significant secondary orbital interactions in the TS of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as the alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/TS geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Bonding in the Ring===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the π-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6π-electron) 6-membered ring component in the transition state.<ref name="Clayden et. al." />(The overall transition state itself is also aromatic, since it has 10π-electrons, i.e. obeys Hückel's (4n + 2)π-electron rule for aromaticity). <ref name="Clayden et. al." /> The C-C single bonds in the 6-membered ring all decrease in length over the course of the reaction, as they adopt a greater double bond-type character. <ref name="Clayden et. al." />Eventually, this results in the formation a formal benzene ring component in all three products.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

From this comparing these values, conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion, the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a transition state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" /><ref name="Levine" /> <ref name="Moss and Coady" /> This corresponds to a geometry with a single vibration with a negative frequency. <ref name="Levine" /> <ref name="Moss and Coady" />

In particular, the frontier molecular orbitals for the Diels-Alder reactions between butadiene and ethene, and a substituted butadiene derivative (cyclohexadiene) and an ethene derivative (1,3-dioxole) were analysed, and the symmetry of the interacting orbitals, and the molecular orbitals they produce in the transition state, were predicted. These predictions were subsequently matched to the computationally-derived molecular orbitals for each of the optimised transition states. These studies allowed conclusions to be drawn about the orbital symmetry requirements of such reactions - only orbitals of matching symmetry can interact. Also, the electronic nature of the observed Diels-Alder reactions was deduced, from comparison of the relative energy levels of the HOMO/LUMO (the frontier orbitals) of the reacting components. From this analysis, the specific Diels-Alder reaction between cyclohexadiene and 1,3-dioxole was found to be an inverse electron demand Diels-Alder cycloaddition. <ref name="Clayden et. al." /><ref name="Bruckner" />

Furthermore, these optimisation studies also allowed relative energy values of the reactants, products and transition states of each of the different reactions to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, which in turn allowed deduction of the kinetically and thermodynamically favoured products of each type of reactions.

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T17:57:52Z

Mh4815: /* Bonding in the Ring */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - found by plotting the electronic potential energy value (V) of the overall molecular system for a particular nuclear geometry (Q). This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, to construct a potential energy surface, the electronic Schrödinger equation is solved (for a particular nuclear configuration), and the resultant potential energy value plotted. This is repeated across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" />

The Hamiltonian equation has the form:<ref name="Atkins and Friedman" />

<math> \hat{H} = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

====Stationary Points====

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

====Distinguishing Minima and Transition States====

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

*A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

*A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

*A transition state is the maximum point on the minimum energy path (MEP) which connects two minima. In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

====Frequency Calculations====

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

====Semi-Empirical Technique====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms. This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

====Density-Functional Theory Method====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly. The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

====Comparison====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

====TS====

=====MO 16=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 17=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 18=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 19=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Levine" />

<jmol>
<jmolApplet>
<color>white</color>
<script>frame 15; vibration 1;</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/TS geometries were optimised to the semi-empirical PM6 level, followed by reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene), an an inverse electron demand Diels-Alder cycloaddition therefore takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====Exo TS====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====Endo TS====

There are significant secondary orbital interactions in the TS of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as the alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/TS geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Bonding in the Ring===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the π-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6π-electron) ring component in the transition state.<ref name="Clayden et. al." />(The overall transition state itself is aromatic, since it has 10π-electrons, i.e. obeys Hüchel's (4n + 2)π-electron rule for aromaticity). <ref name="Clayden et. al." /> The C-C single bonds in the 6-membered ring all decrease in length over the course of the reaction, as they adopt a greater double bond-type character. <ref name="Clayden et. al." />Eventually, this results in the formation a formal benzene ring component in all three products.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

From this comparing these values, conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion, the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a transition state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" /><ref name="Levine" /> <ref name="Moss and Coady" /> This corresponds to a geometry with a single vibration with a negative frequency. <ref name="Levine" /> <ref name="Moss and Coady" />

In particular, the frontier molecular orbitals for the Diels-Alder reactions between butadiene and ethene, and a substituted butadiene derivative (cyclohexadiene) and an ethene derivative (1,3-dioxole) were analysed, and the symmetry of the interacting orbitals, and the molecular orbitals they produce in the transition state, were predicted. These predictions were subsequently matched to the computationally-derived molecular orbitals for each of the optimised transition states. These studies allowed conclusions to be drawn about the orbital symmetry requirements of such reactions - only orbitals of matching symmetry can interact. Also, the electronic nature of the observed Diels-Alder reactions was deduced, from comparison of the relative energy levels of the HOMO/LUMO (the frontier orbitals) of the reacting components. From this analysis, the specific Diels-Alder reaction between cyclohexadiene and 1,3-dioxole was found to be an inverse electron demand Diels-Alder cycloaddition. <ref name="Clayden et. al." /><ref name="Bruckner" />

Furthermore, these optimisation studies also allowed relative energy values of the reactants, products and transition states of each of the different reactions to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, which in turn allowed deduction of the kinetically and thermodynamically favoured products of each type of reactions.

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T17:56:41Z

Mh4815: /* Bonding in the Ring */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - found by plotting the electronic potential energy value (V) of the overall molecular system for a particular nuclear geometry (Q). This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, to construct a potential energy surface, the electronic Schrödinger equation is solved (for a particular nuclear configuration), and the resultant potential energy value plotted. This is repeated across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" />

The Hamiltonian equation has the form:<ref name="Atkins and Friedman" />

<math> \hat{H} = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

====Stationary Points====

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

====Distinguishing Minima and Transition States====

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

*A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

*A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

*A transition state is the maximum point on the minimum energy path (MEP) which connects two minima. In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

====Frequency Calculations====

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

====Semi-Empirical Technique====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms. This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

====Density-Functional Theory Method====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly. The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

====Comparison====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
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This is a symmetric molecular orbital.

=====LUMO=====
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This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
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This is an antisymmetric molecular orbital.

=====LUMO=====
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This is a symmetric molecular orbital.

====TS====

=====MO 16=====
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This is an antisymmetric molecular orbital.

=====MO 17=====
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This is a symmetric molecular orbital.

=====MO 18=====
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This is a symmetric molecular orbital.

=====MO 19=====
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This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Levine" />

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The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/TS geometries were optimised to the semi-empirical PM6 level, followed by reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

=====MO 40=====
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This is an antisymmetric molecular orbital.

=====MO 41=====
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This is a symmetric molecular orbital.

=====MO 42=====
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This is a symmetric molecular orbital.

=====MO 43=====
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This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

=====MO 40=====
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This is an antisymmetric molecular orbital.

=====MO 41=====
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This is a symmetric molecular orbital.

=====MO 42=====
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This is a symmetric molecular orbital.

=====MO 43=====
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This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene), an an inverse electron demand Diels-Alder cycloaddition therefore takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====Exo TS====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====Endo TS====

There are significant secondary orbital interactions in the TS of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as the alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/TS geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Bonding in the Ring===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the pi-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6π-electron) ring component in the transition state.<ref name="Clayden et. al." />(The overall transition state itself is aromatic, since it has 10π-electrons, i.e. obeys Hüchel's (4n + 2)π-electron rule for aromaticity). <ref name="Clayden et. al." /> The C-C single bonds in the 6-membered ring all decrease in length over the course of the reaction, as they adopt a greater double bond-type character. <ref name="Clayden et. al." />Eventually, this results in the formation a formal benzene ring component in all three products.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

From this comparing these values, conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion, the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a transition state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" /><ref name="Levine" /> <ref name="Moss and Coady" /> This corresponds to a geometry with a single vibration with a negative frequency. <ref name="Levine" /> <ref name="Moss and Coady" />

In particular, the frontier molecular orbitals for the Diels-Alder reactions between butadiene and ethene, and a substituted butadiene derivative (cyclohexadiene) and an ethene derivative (1,3-dioxole) were analysed, and the symmetry of the interacting orbitals, and the molecular orbitals they produce in the transition state, were predicted. These predictions were subsequently matched to the computationally-derived molecular orbitals for each of the optimised transition states. These studies allowed conclusions to be drawn about the orbital symmetry requirements of such reactions - only orbitals of matching symmetry can interact. Also, the electronic nature of the observed Diels-Alder reactions was deduced, from comparison of the relative energy levels of the HOMO/LUMO (the frontier orbitals) of the reacting components. From this analysis, the specific Diels-Alder reaction between cyclohexadiene and 1,3-dioxole was found to be an inverse electron demand Diels-Alder cycloaddition. <ref name="Clayden et. al." /><ref name="Bruckner" />

Furthermore, these optimisation studies also allowed relative energy values of the reactants, products and transition states of each of the different reactions to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, which in turn allowed deduction of the kinetically and thermodynamically favoured products of each type of reactions.

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T17:50:24Z

Mh4815: /* Conclusion */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - found by plotting the electronic potential energy value (V) of the overall molecular system for a particular nuclear geometry (Q). This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, to construct a potential energy surface, the electronic Schrödinger equation is solved (for a particular nuclear configuration), and the resultant potential energy value plotted. This is repeated across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" />

The Hamiltonian equation has the form:<ref name="Atkins and Friedman" />

<math> \hat{H} = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

====Stationary Points====

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

====Distinguishing Minima and Transition States====

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

*A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

*A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

*A transition state is the maximum point on the minimum energy path (MEP) which connects two minima. In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

====Frequency Calculations====

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

====Semi-Empirical Technique====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms. This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

====Density-Functional Theory Method====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly. The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

====Comparison====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

====TS====

=====MO 16=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 17=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 18=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 19=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Levine" />

<jmol>
<jmolApplet>
<color>white</color>
<script>frame 15; vibration 1;</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/TS geometries were optimised to the semi-empirical PM6 level, followed by reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene), an an inverse electron demand Diels-Alder cycloaddition therefore takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====Exo TS====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====Endo TS====

There are significant secondary orbital interactions in the TS of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as the alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/TS geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Bonding in the Ring===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the pi-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6π-electron) ring component in the transition state.<ref name="Clayden et. al." />(The overall transition state itself is aromatic, since it has 10π-electrons, i.e. obeys the (4n + 2)π-electron rule for aromaticity). <ref name="Clayden et. al." /> The C-C single bonds in the 6-membered ring all decrease in length over the course of the reaction, as they adopt a greater double bond-type character. <ref name="Clayden et. al." />Eventually, this results in the formation a formal benzene ring component in all three products.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

From this comparing these values, conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion, the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a transition state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" /><ref name="Levine" /> <ref name="Moss and Coady" /> This corresponds to a geometry with a single vibration with a negative frequency. <ref name="Levine" /> <ref name="Moss and Coady" />

In particular, the frontier molecular orbitals for the Diels-Alder reactions between butadiene and ethene, and a substituted butadiene derivative (cyclohexadiene) and an ethene derivative (1,3-dioxole) were analysed, and the symmetry of the interacting orbitals, and the molecular orbitals they produce in the transition state, were predicted. These predictions were subsequently matched to the computationally-derived molecular orbitals for each of the optimised transition states. These studies allowed conclusions to be drawn about the orbital symmetry requirements of such reactions - only orbitals of matching symmetry can interact. Also, the electronic nature of the observed Diels-Alder reactions was deduced, from comparison of the relative energy levels of the HOMO/LUMO (the frontier orbitals) of the reacting components. From this analysis, the specific Diels-Alder reaction between cyclohexadiene and 1,3-dioxole was found to be an inverse electron demand Diels-Alder cycloaddition. <ref name="Clayden et. al." /><ref name="Bruckner" />

Furthermore, these optimisation studies also allowed relative energy values of the reactants, products and transition states of each of the different reactions to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, which in turn allowed deduction of the kinetically and thermodynamically favoured products of each type of reactions.

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T15:46:20Z

Mh4815: /* Conclusion */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - found by plotting the electronic potential energy value (V) of the overall molecular system for a particular nuclear geometry (Q). This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, to construct a potential energy surface, the electronic Schrödinger equation is solved (for a particular nuclear configuration), and the resultant potential energy value plotted. This is repeated across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" />

The Hamiltonian equation has the form:<ref name="Atkins and Friedman" />

<math> \hat{H} = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

====Stationary Points====

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

====Distinguishing Minima and Transition States====

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

*A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

*A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

*A transition state is the maximum point on the minimum energy path (MEP) which connects two minima. In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

====Frequency Calculations====

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

====Semi-Empirical Technique====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms. This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

====Density-Functional Theory Method====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly. The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

====Comparison====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

====TS====

=====MO 16=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 17=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 18=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 19=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Levine" />

<jmol>
<jmolApplet>
<color>white</color>
<script>frame 15; vibration 1;</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/TS geometries were optimised to the semi-empirical PM6 level, followed by reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene), an an inverse electron demand Diels-Alder cycloaddition therefore takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====Exo TS====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====Endo TS====

There are significant secondary orbital interactions in the TS of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as the alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/TS geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Bonding in the Ring===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the pi-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6π-electron) ring component in the transition state.<ref name="Clayden et. al." />(The overall transition state itself is aromatic, since it has 10π-electrons, i.e. obeys the (4n + 2)π-electron rule for aromaticity). <ref name="Clayden et. al." /> The C-C single bonds in the 6-membered ring all decrease in length over the course of the reaction, as they adopt a greater double bond-type character. <ref name="Clayden et. al." />Eventually, this results in the formation a formal benzene ring component in all three products.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

From this comparing these values, conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion, the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a transition state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. <ref name="Levine" /> <ref name="Moss and Coady" /> This corresponds to a geometry with a single vibration with a negative frequency. <ref name="Levine" /> <ref name="Moss and Coady" />

In particular, the frontier molecular orbitals for the Diels-Alder reaction between a substituted butadiene derivative and ethene derivative were analysed, and the symmetry of the interacting orbitals, and the MOs they produce in the transition state, were predicted. These were subsequently matched to the computationally-derived MOs produced for each of the optimised transition states. These studies allowed conclusions to be drawn about the symmetry requirements of such reactions, and the electronic nature of the observed Diels-Alder reaction (from comparison of relative energy levels of the HOMO/LUMO of the reacting components).From this analysis, the specific Diels-Alder reaction between cyclohexadiene and 1,3-dioxole was found to be an inverse electron demand Diels-Alder cycloaddition. <ref name="Clayden et. al." /><ref name="Bruckner" />

Furthermore, these optimisation studies also allowed relative energy values of the reactants/products/transition states to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, allowing deduction of the kinetically and thermodynamically favoured products of each type of reactions.

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T15:44:21Z

Mh4815: /* Conclusion */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - found by plotting the electronic potential energy value (V) of the overall molecular system for a particular nuclear geometry (Q). This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, to construct a potential energy surface, the electronic Schrödinger equation is solved (for a particular nuclear configuration), and the resultant potential energy value plotted. This is repeated across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" />

The Hamiltonian equation has the form:<ref name="Atkins and Friedman" />

<math> \hat{H} = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

====Stationary Points====

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

====Distinguishing Minima and Transition States====

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

*A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

*A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

*A transition state is the maximum point on the minimum energy path (MEP) which connects two minima. In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

====Frequency Calculations====

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

====Semi-Empirical Technique====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms. This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

====Density-Functional Theory Method====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly. The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

====Comparison====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====LUMO=====
<jmol>
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<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
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</jmol>

This is a symmetric molecular orbital.

====TS====

=====MO 16=====
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This is an antisymmetric molecular orbital.

=====MO 17=====
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This is a symmetric molecular orbital.

=====MO 18=====
<jmol>
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This is a symmetric molecular orbital.

=====MO 19=====
<jmol>
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This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Levine" />

<jmol>
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<script>frame 15; vibration 1;</script>
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The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/TS geometries were optimised to the semi-empirical PM6 level, followed by reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

=====MO 40=====
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<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
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</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
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<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
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This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
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<script>frame 20; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
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This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
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<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
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This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

=====MO 40=====
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<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
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This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
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<color>white</color>
<script>frame 32; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
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This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
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<script>frame 32; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
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This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
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This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene), an an inverse electron demand Diels-Alder cycloaddition therefore takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====Exo TS====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====Endo TS====

There are significant secondary orbital interactions in the TS of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as the alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/TS geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Bonding in the Ring===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the pi-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6π-electron) ring component in the transition state.<ref name="Clayden et. al." />(The overall transition state itself is aromatic, since it has 10π-electrons, i.e. obeys the (4n + 2)π-electron rule for aromaticity). <ref name="Clayden et. al." /> The C-C single bonds in the 6-membered ring all decrease in length over the course of the reaction, as they adopt a greater double bond-type character. <ref name="Clayden et. al." />Eventually, this results in the formation a formal benzene ring component in all three products.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

From this comparing these values, conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion, the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a transition state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. <ref name="Levine" /> <ref name="Moss and Coady" /> This corresponds to a geometry with a single vibration with a negative frequency. <ref name="Levine" /> <ref name="Moss and Coady" />

In particular, the frontier molecular orbitals for the Diels-Alder reaction between a substituted butadiene derivative and ethene derivative were analysed, and the symmetry of the interacting orbitals, and the MOs they produce in the transition state, were predicted. These were subsequently matched to the computationally-derived MOs produced for each of the optimised transition states. These studies allowed conclusions to be drawn about the symmetry requirements of such reactions, and the electronic nature of the observed Diels-Alder reaction (from comparison of relative energy levels of the HOMO/LUMO of the reacting components).From this analysis, the specific Diels-Alder reaction between cyclohexadiene and 1,3-dioxole was found to be an inverse electron demand Diels-Alder cycloaddition.

Furthermore, these optimisation studies allowed relative energy values of the reactants/products/transition states to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, allowing deduction of the kinetic and thermodynamic products of each type of reactions.

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T15:40:37Z

Mh4815: /* Conclusion */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - found by plotting the electronic potential energy value (V) of the overall molecular system for a particular nuclear geometry (Q). This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, to construct a potential energy surface, the electronic Schrödinger equation is solved (for a particular nuclear configuration), and the resultant potential energy value plotted. This is repeated across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" />

The Hamiltonian equation has the form:<ref name="Atkins and Friedman" />

<math> \hat{H} = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

====Stationary Points====

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

====Distinguishing Minima and Transition States====

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

*A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

*A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

*A transition state is the maximum point on the minimum energy path (MEP) which connects two minima. In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

====Frequency Calculations====

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

====Semi-Empirical Technique====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms. This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

====Density-Functional Theory Method====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly. The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

====Comparison====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
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This is a symmetric molecular orbital.

=====LUMO=====
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This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
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This is an antisymmetric molecular orbital.

=====LUMO=====
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This is a symmetric molecular orbital.

====TS====

=====MO 16=====
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This is an antisymmetric molecular orbital.

=====MO 17=====
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This is a symmetric molecular orbital.

=====MO 18=====
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This is a symmetric molecular orbital.

=====MO 19=====
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This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Levine" />

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The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/TS geometries were optimised to the semi-empirical PM6 level, followed by reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

=====MO 40=====
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This is an antisymmetric molecular orbital.

=====MO 41=====
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This is a symmetric molecular orbital.

=====MO 42=====
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This is a symmetric molecular orbital.

=====MO 43=====
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This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

=====MO 40=====
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This is an antisymmetric molecular orbital.

=====MO 41=====
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This is a symmetric molecular orbital.

=====MO 42=====
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This is a symmetric molecular orbital.

=====MO 43=====
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This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene), an an inverse electron demand Diels-Alder cycloaddition therefore takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====Exo TS====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====Endo TS====

There are significant secondary orbital interactions in the TS of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as the alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/TS geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Bonding in the Ring===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the pi-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6π-electron) ring component in the transition state.<ref name="Clayden et. al." />(The overall transition state itself is aromatic, since it has 10π-electrons, i.e. obeys the (4n + 2)π-electron rule for aromaticity). <ref name="Clayden et. al." /> The C-C single bonds in the 6-membered ring all decrease in length over the course of the reaction, as they adopt a greater double bond-type character. <ref name="Clayden et. al." />Eventually, this results in the formation a formal benzene ring component in all three products.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

From this comparing these values, conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion, the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a transition state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. This corresponds to a geometry with a single vibration with a negative frequency.

In particular, the frontier molecular orbitals for the Diels-Alder reaction between a substituted butadiene derivative and ethene derivative were analysed, and the symmetry of the interacting orbitals, and the MOs they produce in the transition state, were predicted. These were subsequently matched to the computationally-derived MOs produced for each of the optimised transition states. These studies allowed conclusions to be drawn about the symmetry requirements of such reactions, and the electronic nature of the observed Diels-Alder reaction (from comparison of relative energy levels of the HOMO/LUMO of the reacting components).

Furthermore, these optimisation studies allowed relative energy values of the reactants/products/transition states to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, allowing deduction of the kinetic and thermodynamic products of each type of reactions.

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T15:38:52Z

Mh4815: /* Bonding in the Ring */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - found by plotting the electronic potential energy value (V) of the overall molecular system for a particular nuclear geometry (Q). This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, to construct a potential energy surface, the electronic Schrödinger equation is solved (for a particular nuclear configuration), and the resultant potential energy value plotted. This is repeated across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" />

The Hamiltonian equation has the form:<ref name="Atkins and Friedman" />

<math> \hat{H} = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

====Stationary Points====

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

====Distinguishing Minima and Transition States====

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

*A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

*A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

*A transition state is the maximum point on the minimum energy path (MEP) which connects two minima. In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

====Frequency Calculations====

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

====Semi-Empirical Technique====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms. This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

====Density-Functional Theory Method====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly. The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

====Comparison====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

====TS====

=====MO 16=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 17=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 18=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 19=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Levine" />

<jmol>
<jmolApplet>
<color>white</color>
<script>frame 15; vibration 1;</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/TS geometries were optimised to the semi-empirical PM6 level, followed by reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene), an an inverse electron demand Diels-Alder cycloaddition therefore takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====Exo TS====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====Endo TS====

There are significant secondary orbital interactions in the TS of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as the alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/TS geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Bonding in the Ring===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the pi-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6π-electron) ring component in the transition state.<ref name="Clayden et. al." />(The overall transition state itself is aromatic, since it has 10π-electrons, i.e. obeys the (4n + 2)π-electron rule for aromaticity). <ref name="Clayden et. al." /> The C-C single bonds in the 6-membered ring all decrease in length over the course of the reaction, as they adopt a greater double bond-type character. <ref name="Clayden et. al." />Eventually, this results in the formation a formal benzene ring component in all three products.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

From this comparing these values, conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion,the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a tranisiton state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. This corresponds to a geometry with a single vibration with a negative frequency.

In particular, the frontier molecular orbitals for the Diels-Alder reaction between a substituted butadiene derivative and ethene derivative were analysed, and the symmetry of the interacting orbitals, and the MOs they produce in the transition state, were predicted. These were subsequently matched to the computationally-derived MOs produced for each of the optimised transition states. These studies allowed conclusions to be drawn about the symmetry requirements of such reactions, and the electronic nature of the observed Diels-Alder reaction (from comparison of relative energy levels of the HOMO/LUMO of the reacting components).

Furthermore, these optimisation studies allowed relative energy values of the reactants/products/transition states to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, allowing deduction of the kinetic and thermodynamic products of each type of reactions.

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T15:37:57Z

Mh4815: /* Xylylene Stability */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - found by plotting the electronic potential energy value (V) of the overall molecular system for a particular nuclear geometry (Q). This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, to construct a potential energy surface, the electronic Schrödinger equation is solved (for a particular nuclear configuration), and the resultant potential energy value plotted. This is repeated across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" />

The Hamiltonian equation has the form:<ref name="Atkins and Friedman" />

<math> \hat{H} = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

====Stationary Points====

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

====Distinguishing Minima and Transition States====

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

*A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

*A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

*A transition state is the maximum point on the minimum energy path (MEP) which connects two minima. In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

====Frequency Calculations====

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

====Semi-Empirical Technique====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms. This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

====Density-Functional Theory Method====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly. The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

====Comparison====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

====TS====

=====MO 16=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 17=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 18=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 19=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Levine" />

<jmol>
<jmolApplet>
<color>white</color>
<script>frame 15; vibration 1;</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/TS geometries were optimised to the semi-empirical PM6 level, followed by reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene), an an inverse electron demand Diels-Alder cycloaddition therefore takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====Exo TS====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====Endo TS====

There are significant secondary orbital interactions in the TS of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as the alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/TS geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Bonding in the Ring===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the pi-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6π-electron) ring component in the transition state.(The overall transition state itself is aromatic, since it has 10π-electrons, i.e. obeys the (4n + 2)π-electron rule for aromaticity). The C-C single bonds in the 6-membered ring all decrease in length over the course of the reaction, as they adopt a greater double bond-type character. Eventually, this results in the formation a formal benzene ring component in all three products.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

From this comparing these values, conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion,the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a tranisiton state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. This corresponds to a geometry with a single vibration with a negative frequency.

In particular, the frontier molecular orbitals for the Diels-Alder reaction between a substituted butadiene derivative and ethene derivative were analysed, and the symmetry of the interacting orbitals, and the MOs they produce in the transition state, were predicted. These were subsequently matched to the computationally-derived MOs produced for each of the optimised transition states. These studies allowed conclusions to be drawn about the symmetry requirements of such reactions, and the electronic nature of the observed Diels-Alder reaction (from comparison of relative energy levels of the HOMO/LUMO of the reacting components).

Furthermore, these optimisation studies allowed relative energy values of the reactants/products/transition states to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, allowing deduction of the kinetic and thermodynamic products of each type of reactions.

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T15:19:13Z

Mh4815: /* Introduction */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - found by plotting the electronic potential energy value (V) of the overall molecular system for a particular nuclear geometry (Q). This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, to construct a potential energy surface, the electronic Schrödinger equation is solved (for a particular nuclear configuration), and the resultant potential energy value plotted. This is repeated across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" />

The Hamiltonian equation has the form:<ref name="Atkins and Friedman" />

<math> \hat{H} = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

====Stationary Points====

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

====Distinguishing Minima and Transition States====

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

*A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

*A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

*A transition state is the maximum point on the minimum energy path (MEP) which connects two minima. In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

====Frequency Calculations====

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

====Semi-Empirical Technique====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms. This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

====Density-Functional Theory Method====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly. The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

====Comparison====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

====TS====

=====MO 16=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 17=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 18=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 19=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Levine" />

<jmol>
<jmolApplet>
<color>white</color>
<script>frame 15; vibration 1;</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/TS geometries were optimised to the semi-empirical PM6 level, followed by reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene), an an inverse electron demand Diels-Alder cycloaddition therefore takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====Exo TS====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====Endo TS====

There are significant secondary orbital interactions in the TS of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as the alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/TS geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Xylylene Stability===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the pi-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6pi-electron) transition state, which formally results in a benzene-type component in all three aducts.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

From this comparing these values, conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion,the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a tranisiton state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. This corresponds to a geometry with a single vibration with a negative frequency.

In particular, the frontier molecular orbitals for the Diels-Alder reaction between a substituted butadiene derivative and ethene derivative were analysed, and the symmetry of the interacting orbitals, and the MOs they produce in the transition state, were predicted. These were subsequently matched to the computationally-derived MOs produced for each of the optimised transition states. These studies allowed conclusions to be drawn about the symmetry requirements of such reactions, and the electronic nature of the observed Diels-Alder reaction (from comparison of relative energy levels of the HOMO/LUMO of the reacting components).

Furthermore, these optimisation studies allowed relative energy values of the reactants/products/transition states to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, allowing deduction of the kinetic and thermodynamic products of each type of reactions.

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T15:17:35Z

Mh4815: /* Introduction */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - found by plotting the electronic potential energy value (V) of the overall molecular system for a particular nuclear geometry (Q). This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, to construct a potential energy surface, the electronic Schrödinger equation is solved (for a particular nuclear configuration), and the resultant potential energy value plotted. This is repeated across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" />

The Hamiltonian equation has the form:<ref name="Atkins and Friedman" />

<math> \hat{H} = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

====Stationary Points====

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

====Distinguishing Minima and Transition States====

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

*A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

*A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

*A transition state is the maximum point on the minimum energy path (MEP) which connects two minima. In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

====Frequency Calculations====

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

=====Semi-Empirical Technique=====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms. This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

=====Density-Functional Theory Method=====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly. The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

=====Comparison=====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
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This is a symmetric molecular orbital.

=====LUMO=====
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This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
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This is an antisymmetric molecular orbital.

=====LUMO=====
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This is a symmetric molecular orbital.

====TS====

=====MO 16=====
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This is an antisymmetric molecular orbital.

=====MO 17=====
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This is a symmetric molecular orbital.

=====MO 18=====
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This is a symmetric molecular orbital.

=====MO 19=====
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This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Levine" />

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The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/TS geometries were optimised to the semi-empirical PM6 level, followed by reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

=====MO 40=====
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This is an antisymmetric molecular orbital.

=====MO 41=====
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This is a symmetric molecular orbital.

=====MO 42=====
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This is a symmetric molecular orbital.

=====MO 43=====
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This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

=====MO 40=====
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This is an antisymmetric molecular orbital.

=====MO 41=====
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This is a symmetric molecular orbital.

=====MO 42=====
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This is a symmetric molecular orbital.

=====MO 43=====
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This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene), an an inverse electron demand Diels-Alder cycloaddition therefore takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====Exo TS====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====Endo TS====

There are significant secondary orbital interactions in the TS of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as the alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/TS geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Xylylene Stability===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the pi-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6pi-electron) transition state, which formally results in a benzene-type component in all three aducts.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

From this comparing these values, conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion,the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a tranisiton state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. This corresponds to a geometry with a single vibration with a negative frequency.

In particular, the frontier molecular orbitals for the Diels-Alder reaction between a substituted butadiene derivative and ethene derivative were analysed, and the symmetry of the interacting orbitals, and the MOs they produce in the transition state, were predicted. These were subsequently matched to the computationally-derived MOs produced for each of the optimised transition states. These studies allowed conclusions to be drawn about the symmetry requirements of such reactions, and the electronic nature of the observed Diels-Alder reaction (from comparison of relative energy levels of the HOMO/LUMO of the reacting components).

Furthermore, these optimisation studies allowed relative energy values of the reactants/products/transition states to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, allowing deduction of the kinetic and thermodynamic products of each type of reactions.

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T15:16:29Z

Mh4815: /* Molecular Potential Energy Surfaces */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - found by plotting the electronic potential energy value (V) of the overall molecular system for a particular nuclear geometry (Q). This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, to construct a potential energy surface, the electronic Schrödinger equation is solved (for a particular nuclear configuration), and the resultant potential energy value plotted. This is repeated across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" />

The Hamiltonian equation has the form:<ref name="Atkins and Friedman" />

<math> \hat{H} = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

===Stationary Points===

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

===Distinguishing Minima and Transition States===

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

*A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

*A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

*A transition state is the maximum point on the minimum energy path (MEP) which connects two minima. In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

===Frequency Calculations===

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

=====Semi-Empirical Technique=====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms. This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

=====Density-Functional Theory Method=====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly. The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

=====Comparison=====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

====TS====

=====MO 16=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 17=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 18=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 19=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Levine" />

<jmol>
<jmolApplet>
<color>white</color>
<script>frame 15; vibration 1;</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/TS geometries were optimised to the semi-empirical PM6 level, followed by reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene), an an inverse electron demand Diels-Alder cycloaddition therefore takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====Exo TS====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====Endo TS====

There are significant secondary orbital interactions in the TS of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as the alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/TS geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Xylylene Stability===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the pi-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6pi-electron) transition state, which formally results in a benzene-type component in all three aducts.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

From this comparing these values, conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion,the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a tranisiton state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. This corresponds to a geometry with a single vibration with a negative frequency.

In particular, the frontier molecular orbitals for the Diels-Alder reaction between a substituted butadiene derivative and ethene derivative were analysed, and the symmetry of the interacting orbitals, and the MOs they produce in the transition state, were predicted. These were subsequently matched to the computationally-derived MOs produced for each of the optimised transition states. These studies allowed conclusions to be drawn about the symmetry requirements of such reactions, and the electronic nature of the observed Diels-Alder reaction (from comparison of relative energy levels of the HOMO/LUMO of the reacting components).

Furthermore, these optimisation studies allowed relative energy values of the reactants/products/transition states to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, allowing deduction of the kinetic and thermodynamic products of each type of reactions.

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T15:15:50Z

Mh4815: /* Molecular Structure via Quantum Mechanics */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - found by plotting the electronic potential energy value (V) of the overall molecular system for a particular nuclear geometry (Q). This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, to construct a potential energy surface, the electronic Schrödinger equation is solved (for a particular nuclear configuration), and the resultant potential energy value plotted. This is repeated across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" />

The Hamiltonian equation has the form:<ref name="Atkins and Friedman" />

<math> \hat{H} = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

Under the Born-Oppenheimer approximation, a molecular potential surface is found by finding the electronic potential energy value (V) of the overall molecular system for a fixed nuclear geometry (Q). This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted, to yield the potential energy surface of the molecular system.<ref name="Atkins and Friedman" />

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

===Stationary Points===

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

===Distinguishing Minima and Transition States===

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

*A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

*A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

*A transition state is the maximum point on the minimum energy path (MEP) which connects two minima. In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

===Frequency Calculations===

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

=====Semi-Empirical Technique=====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms. This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

=====Density-Functional Theory Method=====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly. The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

=====Comparison=====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

====TS====

=====MO 16=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 17=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 18=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 19=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Levine" />

<jmol>
<jmolApplet>
<color>white</color>
<script>frame 15; vibration 1;</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/TS geometries were optimised to the semi-empirical PM6 level, followed by reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene), an an inverse electron demand Diels-Alder cycloaddition therefore takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====Exo TS====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====Endo TS====

There are significant secondary orbital interactions in the TS of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as the alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/TS geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Xylylene Stability===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the pi-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6pi-electron) transition state, which formally results in a benzene-type component in all three aducts.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

From this comparing these values, conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion,the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a tranisiton state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. This corresponds to a geometry with a single vibration with a negative frequency.

In particular, the frontier molecular orbitals for the Diels-Alder reaction between a substituted butadiene derivative and ethene derivative were analysed, and the symmetry of the interacting orbitals, and the MOs they produce in the transition state, were predicted. These were subsequently matched to the computationally-derived MOs produced for each of the optimised transition states. These studies allowed conclusions to be drawn about the symmetry requirements of such reactions, and the electronic nature of the observed Diels-Alder reaction (from comparison of relative energy levels of the HOMO/LUMO of the reacting components).

Furthermore, these optimisation studies allowed relative energy values of the reactants/products/transition states to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, allowing deduction of the kinetic and thermodynamic products of each type of reactions.

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T15:11:52Z

Mh4815: /* Introduction */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - found by plotting the electronic potential energy value (V) of the overall molecular system for a particular nuclear geometry (Q). This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted, to yield the potential energy surface of the molecular system.<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, we solve the electronic Schrödinger equation for a particular nuclear configuration, plot the resultant potnential energy value, and repeat across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" />

The Hamiltonian equation has the form:<ref name="Atkins and Friedman" />

<math> \hat{H} = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

Under the Born-Oppenheimer approximation, a molecular potential surface is found by finding the electronic potential energy value (V) of the overall molecular system for a fixed nuclear geometry (Q). This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted, to yield the potential energy surface of the molecular system.<ref name="Atkins and Friedman" />

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

===Stationary Points===

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

===Distinguishing Minima and Transition States===

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

*A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

*A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

*A transition state is the maximum point on the minimum energy path (MEP) which connects two minima. In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

===Frequency Calculations===

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

=====Semi-Empirical Technique=====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms. This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

=====Density-Functional Theory Method=====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly. The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

=====Comparison=====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

====TS====

=====MO 16=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 17=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 18=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
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This is a symmetric molecular orbital.

=====MO 19=====
<jmol>
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<color>white</color>
<script>frame 14; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
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This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Levine" />

<jmol>
<jmolApplet>
<color>white</color>
<script>frame 15; vibration 1;</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
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The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/TS geometries were optimised to the semi-empirical PM6 level, followed by reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
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<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
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</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
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</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
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This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
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</jmol>

This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
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<script>frame 32; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
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This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
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This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
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</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
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</jmol>

This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene), an an inverse electron demand Diels-Alder cycloaddition therefore takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====Exo TS====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====Endo TS====

There are significant secondary orbital interactions in the TS of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as the alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/TS geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Xylylene Stability===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the pi-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6pi-electron) transition state, which formally results in a benzene-type component in all three aducts.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

From this comparing these values, conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion,the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a tranisiton state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. This corresponds to a geometry with a single vibration with a negative frequency.

In particular, the frontier molecular orbitals for the Diels-Alder reaction between a substituted butadiene derivative and ethene derivative were analysed, and the symmetry of the interacting orbitals, and the MOs they produce in the transition state, were predicted. These were subsequently matched to the computationally-derived MOs produced for each of the optimised transition states. These studies allowed conclusions to be drawn about the symmetry requirements of such reactions, and the electronic nature of the observed Diels-Alder reaction (from comparison of relative energy levels of the HOMO/LUMO of the reacting components).

Furthermore, these optimisation studies allowed relative energy values of the reactants/products/transition states to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, allowing deduction of the kinetic and thermodynamic products of each type of reactions.

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T13:36:59Z

Mh4815: /* Distinguishing Minima and Transition States */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, but only for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - we plot electronic potential energy values at a specific nuclear configuration, across a range of particular nuclear geometries. <ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, we solve the electronic Schrödinger equation for a particular nuclear configuration, plot the resultant potnential energy value, and repeat across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" />

The Hamiltonian equation has the form:<ref name="Atkins and Friedman" />

<math> \hat{H} = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

Under the Born-Oppenheimer approximation, a molecular potential surface is found by finding the electronic potential energy value (V) of the overall molecular system for a fixed nuclear geometry (Q). This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted, to yield the potential energy surface of the molecular system.<ref name="Atkins and Friedman" />

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

===Stationary Points===

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

===Distinguishing Minima and Transition States===

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

*A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

*A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

*A transition state is the maximum point on the minimum energy path (MEP) which connects two minima. In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

===Frequency Calculations===

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

=====Semi-Empirical Technique=====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms. This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

=====Density-Functional Theory Method=====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly. The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

=====Comparison=====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
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This is a symmetric molecular orbital.

=====LUMO=====
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This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
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This is an antisymmetric molecular orbital.

=====LUMO=====
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This is a symmetric molecular orbital.

====TS====

=====MO 16=====
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This is an antisymmetric molecular orbital.

=====MO 17=====
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This is a symmetric molecular orbital.

=====MO 18=====
<jmol>
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This is a symmetric molecular orbital.

=====MO 19=====
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This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Levine" />

<jmol>
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The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/TS geometries were optimised to the semi-empirical PM6 level, followed by reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

=====MO 40=====
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This is an antisymmetric molecular orbital.

=====MO 41=====
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This is a symmetric molecular orbital.

=====MO 42=====
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This is a symmetric molecular orbital.

=====MO 43=====
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This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

=====MO 40=====
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This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
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This is a symmetric molecular orbital.

=====MO 42=====
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This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
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<script>frame 32; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
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This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene), an an inverse electron demand Diels-Alder cycloaddition therefore takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====Exo TS====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====Endo TS====

There are significant secondary orbital interactions in the TS of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as the alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/TS geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Xylylene Stability===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the pi-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6pi-electron) transition state, which formally results in a benzene-type component in all three aducts.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

From this comparing these values, conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion,the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a tranisiton state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. This corresponds to a geometry with a single vibration with a negative frequency.

In particular, the frontier molecular orbitals for the Diels-Alder reaction between a substituted butadiene derivative and ethene derivative were analysed, and the symmetry of the interacting orbitals, and the MOs they produce in the transition state, were predicted. These were subsequently matched to the computationally-derived MOs produced for each of the optimised transition states. These studies allowed conclusions to be drawn about the symmetry requirements of such reactions, and the electronic nature of the observed Diels-Alder reaction (from comparison of relative energy levels of the HOMO/LUMO of the reacting components).

Furthermore, these optimisation studies allowed relative energy values of the reactants/products/transition states to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, allowing deduction of the kinetic and thermodynamic products of each type of reactions.

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T13:36:32Z

Mh4815: /* Frequency Calculations */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, but only for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - we plot electronic potential energy values at a specific nuclear configuration, across a range of particular nuclear geometries. <ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, we solve the electronic Schrödinger equation for a particular nuclear configuration, plot the resultant potnential energy value, and repeat across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" />

The Hamiltonian equation has the form:<ref name="Atkins and Friedman" />

<math> \hat{H} = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

Under the Born-Oppenheimer approximation, a molecular potential surface is found by finding the electronic potential energy value (V) of the overall molecular system for a fixed nuclear geometry (Q). This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted, to yield the potential energy surface of the molecular system.<ref name="Atkins and Friedman" />

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

===Stationary Points===

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

===Distinguishing Minima and Transition States===

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

A transition state is the maximum point on the minimum energy path (MEP) which connects two minima. In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

===Frequency Calculations===

Physically, a single negative Hessian eigenvalue means that a transition structure can be identified as a molecular geometry with a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

=====Semi-Empirical Technique=====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms. This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

=====Density-Functional Theory Method=====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly. The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

=====Comparison=====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

====TS====

=====MO 16=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 17=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 18=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 19=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Levine" />

<jmol>
<jmolApplet>
<color>white</color>
<script>frame 15; vibration 1;</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/TS geometries were optimised to the semi-empirical PM6 level, followed by reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene), an an inverse electron demand Diels-Alder cycloaddition therefore takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====Exo TS====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====Endo TS====

There are significant secondary orbital interactions in the TS of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as the alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/TS geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Xylylene Stability===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the pi-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6pi-electron) transition state, which formally results in a benzene-type component in all three aducts.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

From this comparing these values, conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion,the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a tranisiton state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. This corresponds to a geometry with a single vibration with a negative frequency.

In particular, the frontier molecular orbitals for the Diels-Alder reaction between a substituted butadiene derivative and ethene derivative were analysed, and the symmetry of the interacting orbitals, and the MOs they produce in the transition state, were predicted. These were subsequently matched to the computationally-derived MOs produced for each of the optimised transition states. These studies allowed conclusions to be drawn about the symmetry requirements of such reactions, and the electronic nature of the observed Diels-Alder reaction (from comparison of relative energy levels of the HOMO/LUMO of the reacting components).

Furthermore, these optimisation studies allowed relative energy values of the reactants/products/transition states to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, allowing deduction of the kinetic and thermodynamic products of each type of reactions.

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T13:35:00Z

Mh4815: /* Stationary Points */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, but only for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - we plot electronic potential energy values at a specific nuclear configuration, across a range of particular nuclear geometries. <ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, we solve the electronic Schrödinger equation for a particular nuclear configuration, plot the resultant potnential energy value, and repeat across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" />

The Hamiltonian equation has the form:<ref name="Atkins and Friedman" />

<math> \hat{H} = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

Under the Born-Oppenheimer approximation, a molecular potential surface is found by finding the electronic potential energy value (V) of the overall molecular system for a fixed nuclear geometry (Q). This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted, to yield the potential energy surface of the molecular system.<ref name="Atkins and Friedman" />

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

===Stationary Points===

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy V with respect to variable Q equal to zero:<ref name="Atkins and Friedman" />

<math> (\frac{dV}{dQ}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

===Distinguishing Minima and Transition States===

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

A transition state is the maximum point on the minimum energy path (MEP) which connects two minima. In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

===Frequency Calculations===

Physically, a single negative Hessian eigenvalue translates as a true transition structure being identifiable as a molecular geometry which has a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

=====Semi-Empirical Technique=====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms. This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

=====Density-Functional Theory Method=====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly. The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

=====Comparison=====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

====TS====

=====MO 16=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 17=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 18=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 19=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Levine" />

<jmol>
<jmolApplet>
<color>white</color>
<script>frame 15; vibration 1;</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/TS geometries were optimised to the semi-empirical PM6 level, followed by reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene), an an inverse electron demand Diels-Alder cycloaddition therefore takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====Exo TS====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====Endo TS====

There are significant secondary orbital interactions in the TS of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as the alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/TS geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Xylylene Stability===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the pi-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6pi-electron) transition state, which formally results in a benzene-type component in all three aducts.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

From this comparing these values, conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion,the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a tranisiton state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. This corresponds to a geometry with a single vibration with a negative frequency.

In particular, the frontier molecular orbitals for the Diels-Alder reaction between a substituted butadiene derivative and ethene derivative were analysed, and the symmetry of the interacting orbitals, and the MOs they produce in the transition state, were predicted. These were subsequently matched to the computationally-derived MOs produced for each of the optimised transition states. These studies allowed conclusions to be drawn about the symmetry requirements of such reactions, and the electronic nature of the observed Diels-Alder reaction (from comparison of relative energy levels of the HOMO/LUMO of the reacting components).

Furthermore, these optimisation studies allowed relative energy values of the reactants/products/transition states to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, allowing deduction of the kinetic and thermodynamic products of each type of reactions.

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>

Rep:TS:mhardst

2018-02-13T13:33:43Z

Mh4815: /* Molecular Potential Energy Surfaces */

==Introduction==

===Molecular Structure via Quantum Mechanics===

Molecular structure of various systems can be investigated via quantum mechanical analysis. The Schrödinger equation must be solved, giving the molecular wavefunction of the system (Ψ).<ref name="Atkins and Friedman" /> The wavefunction of the system can be operated on by the Hamiltonian operator (Ĥ), which is specific for each individual molecular system, and the eigenvalue of the Hamiltonian equation corresponds to the total energy of the molecular system (E).<ref name="Atkins and Friedman" /> <ref name="Levine" />

The Born-Oppenheimer approximation assumes that since nuclei are so much more massive than electrons, electronic motion can take place within a stationary nuclear framework (i.e. nuclei do not have time to respond to electron motion).<ref name="Atkins and Friedman" /><ref name="Hinchliffe" /> Under the Born-Oppenheimer approximation, we can uncouple nuclear and electronic motion, and treat them separately via quantum mechanics - considering a distinct 'nuclear wavefunction' and 'electronic wavefunction'. <ref name="Hinchliffe" />

For molecular potential energy surfaces, we are interested only in the electronic wavefunction - which is found to depend on nuclear coordinates, i.e. we can write the electronic wavefunction of the molecule, and hence determine total electronic energy, but only for a specific nuclear configuration.<ref name="Hinchliffe" /> This is the basis of molecular potential energy surfaces - we plot electronic potential energy values at a specific nuclear configuration, across a range of particular nuclear geometries. <ref name="Hinchliffe" /> This results in the formation of a 'surface' which displays how potential energy varies as a function of nuclear configuration (which itself depends on internuclear separations/bond angles/etc.). <ref name="Hinchliffe" />

Therefore, we solve the electronic Schrödinger equation for a particular nuclear configuration, plot the resultant potnential energy value, and repeat across numerous nuclear configurations. <ref name="Hinchliffe" /> The electronic Schrödinger equation for an n-electron system has the form:<ref name="Atkins and Friedman" /><ref name="Hinchliffe" />

<math> \hat{H}_e\psi_e(r_1, r_2, ...r_n) = E_e\psi_e(r_1, r_2, ...r_n)</math>

Where r<sub>i</sub> is the vector describing the coordinates of electron i for a given 'fixed' nuclear framework.<ref name="Atkins and Friedman" />

The Hamiltonian equation has the form:<ref name="Atkins and Friedman" />

<math> \hat{H} = \sum_{i=1}^n(-\frac{\hbar^2}{2m_e}\nabla^2) + V </math>

===Molecular Potential Energy Surfaces===

A molecular potential energy surface contains an abundance of information regarding the molecular reaction dynamics of the given reaction that it describes.

Under the Born-Oppenheimer approximation, a molecular potential surface is found by finding the electronic potential energy value (V) of the overall molecular system for a fixed nuclear geometry (Q). This is repeated over many possible collective vectors Q, and each potential value at a given Q is plotted, to yield the potential energy surface of the molecular system.<ref name="Atkins and Friedman" />

A structure composed of N atoms will have (3N-6) degrees of freedom in its molecular potential energy surface. <ref name="Levine" /> Potential energy of the structure (V) is a function of these (3N-6) degrees of freedom, and so is embedded in a (3N-5) - dimensional space.<ref name="Levine" />

===Stationary Points===

A stationary point on any potential energy surface is defined as a point which has a first derivative of potential energy (V) with respect to variable Q<sub>i</sub> (a collective variable describing the vector of the nuclear coordinates of the reacting molecules at a given geometry) which is equal to zero:<ref name="Atkins and Friedman" />

<math> (\frac{dV}{dQ_i}) = 0 </math>

However, in order to differentiate between maxima/minima/saddle points, the sign of the second derivative at the point Q<sub>i</sub> must also be taken into account.<ref name="Atkins and Friedman" /> The set of second derivatives can be formatted into a matrix called the Hessian matrix. <ref name="Levine" />

===Distinguishing Minima and Transition States===

Using the Hessian matrix, minima and transition states (saddle points) on a given potential energy surface can be differentiated based on the number of negative Hessian eigenvalues of the matrix.<ref name="Atkins and Friedman" />

A minimum on a potential energy surface is characterized by having no negative Hessian eigenvalues (i.e. all of the Hessian eigenvalues are positive).<ref name="Atkins and Friedman" />

A maximum on a potential energy surface is characterized by having all negative Hessian eigenvalues.<ref name="Atkins and Friedman" />

A transition state is the maximum point on the minimum energy path (MEP) which connects two minima. In the context of the overall potential energy surface, a transition state is a saddle point on the potential energy surface.<ref name="Moss and Coady" /> Saddle points are characterized by having a single negative Hessian eigenvalue. <ref name="Atkins and Friedman" />

===Frequency Calculations===

Physically, a single negative Hessian eigenvalue translates as a true transition structure being identifiable as a molecular geometry which has a single imaginary frequency. <ref name="Levine" /> Therefore, throughout the following investigation, each time a transition structure was located and optimized, a frequency calculation was also performed in order to confirm that the resultant geometry was indeed a true transition structure. A single vibration with a negative frequency (whose vibrational motion corresponds to the (intuitively) expected molecular motion in the transition state) indicated a transition structure had been found.

===Analytical Optimization Techniques===

For molecular systems of an order greater than diatomic molecules, the quantum mechanics describing the molecular dynamics rapidly becomes extremely complex (as number of degrees of freedom in the wavefunction increases), and fully solving such a wavefunction is beyond the scope of modern-day computational power. <ref name="Levine" /> Therefore, current molecular geometry optimization techniques must make a series of assumptions in order to be possible within such computational constraints. There are several different models used, two of which were used in the investigation: a semi-empirical approach on the PM6 level, and a DFT (Density Functional Theory) approach at the B3LYP level, on a 6-31G(d) basis set.

=====Semi-Empirical Technique=====

For polyatomic systems, the Hamiltonian operator can be incredibly complex, and contain many different terms. This makes determination of the molecular wavefunction extremely difficult, and hence it is hard to find the correct functional form of the potential energy function - which is ultimately desired for molecular reaction dynamics analysis. Therefore, the semi-empirical approach is to use a highly simplified version of the Hamiltonian, as well as including experimentally-derived values as the coefficients within this Hamiltonian.<ref name="Levine" />

=====Density-Functional Theory Method=====

Density Functional Theory (DFT) presents an alternative route which avoids the molecular wavefunction altogether, but rather determines molecular potential energy indirectly. The technique attempts to find the electronic probability density function of the system, and calculates the potential energy value (at each individual geometry) from this.<ref name="Levine" /> However, this approach is only accurate for calculating the energy values of ground electronic states. <ref name="Hinchliffe" /> It is based on the quantum mechanical outcome that the ground state electronic energy of a given molecular system depends solely on its electronic probability density function. <ref name="Hinchliffe" />

=====Comparison=====

Typically, it is assumed that the DFT approach on the B3LYP level is a more accurate optimization technique than the semi-empirical approach on the PM6 level. However, given the more complex mathematical analysis involved, this computational calculation takes far longer to complete. Therefore, for computational investigations requiring geometry optimization techniques, the decision between which technique is chosen becomes a balance between accuracy and time.

==Exercise 1: Reaction of Butadiene with Ethene==

===Molecular Orbital Diagram===

[[File:Mhardst modiagram11.jpg |thumb|center|''Figure 1 - Simplified MO Diagram for TS of the Butadiene-Ethylene Diels-Alder Reaction''.]]

===Molecular Orbitals===

====Ethene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 10; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_ETHENE_PM6OPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

====Butadiene====

=====HOMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====LUMO=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 6; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_BUTADIENE_PM6OPT_ATTEMPT2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

====TS====

=====MO 16=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 16; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 17=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 17; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 18=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 18; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 19=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 14; mo 19; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Symmetry Requirements===

As can be seen from the above MO diagram:

*The antisymmetric HOMO of butadiene interacts with the antisymmetric LUMO of ethene, producing a bonding/antibonding pair of antisymmetric MOs (MOs 16 and 19).

*The symmetric LUMO of butadiene interacts with the symmetric HOMO of ethene, producing a bonding/antibonding pair of symmetric MOs (MOs 17 and 18).

From this example, it can be concluded that in order for two MOs to interact with one another, they must have the same symmetry, and they will produce a bonding/antibonding pair of MOs which also has this symmetry.

Therefore, this proves that for an 'allowed' reaction, interacting orbitals must have the same symmetry, and there will not be a change of symmetry in the MOs they produce over the course of the reaction.

====Orbital Overlap Integral====

*Symmetric-symmetric interaction: non-zero orbital overlap integral

*Antisymmetric-antisymmetric interaction: non-zero orbital overlap integral

*Symmetric-antisymmetric interaction: zero orbital overlap integral

===C-C Bond Lengths===

====Butadiene====

C(1)=C(2) double bond length: 1.333 A

C(2)-C(3) single bond length: 1.471 A

C(3)=C(4) double bond length: 1.333 A

====Ethylene====

C(5)=C(6) double bond length: 1.328 A

====Transition Structure====

C(1)-C(2) bond length: 1.380 A

C(2)-C(3) bond length: 1.411 A

C(3)-C(4) bond length: 1.380 A

C(4)-C(5) bond length: 2.115 A

C(5)-C(6) bond length: 1.382 A

C(6)-C(1) bond length: 2.114 A

====Cyclohexene====

C(1)-C(2) single bond length: 1.501 A

C(2)=C(3) double bond length: 1.337 A

C(3)-C(4) single bond length: 1.501 A

C(4)-C(5) single bond length: 1.537 A

C(5)-C(6) single bond length: 1.535 A

C(6)-C(1) single bond length: 1.537 A

====Change in Bond Length====

As the reaction progresses, the two C=C double bond lengths in butadiene increase from 1.333 A to 1.501 A, as they become C-C single bonds in the final product, cyclohexene. Conversely, the intermediate C-C single bond in butadiene shrinks from 1.471 A to 1.337 A as the double bond character emerges from the Diels-Alder reaction.

Meanwhile, the C=C double bond of ethylene increases from 1.328 A to 1.535 A over the course of the reaction, as it adopts a greater single bond character.

As the two components come together to react, the C-C distances between the terminal carbon atoms of both butadiene and ethylene decrease (from infinite separation) to 1.537 A.

====Typical Bond Lengths====

Typical sp<sup>3</sup> C-C Bond Length <ref name="Lide" /> = (1.525 -1.526) A

Typical sp<sup>2</sup> C=C Bond Length <ref name="Lide" /> = (1.330 - 1.335) A

Van der Waals' Radius of C <ref name="Mantina et. al." /> = 1.70 A

====Comparing TS vs. Typical Bond Lengths====

In the TS, the double bonds which are weakening/lengthening to become single bonds (the C(1)-C(2) and C(3)-C(4) bonds) have a bond length of 1.380 A - intermediate between typical C-C and C=C bond lengths. The single bond which becomes a double bond during the course of the Diels-Alder reaction (C(2)-C(3)) has a bond length of 1.411 A - also in between typical C-C and C=C bond lengths.

The single bonds which are partially formed between the terminal carbon atoms of the butadiene and ethylene reactant molecules (C(4)-C(5) and C(6)-C(1)) have bond lengths 2.115 A and 2.114 A respectively (which can be assumed as essentially identical, as they are accurate to a high precision of 1 x 10<sup>-1 </sup>A). This value is significantly larger than the typical C-C bond length of 1.525-1.526 A, indicating that bonding in the TS is far from complete. However, this length is also much less than twice the Van der Waals' radius of Carbon (3.40 A), thus proving that there is actually a significant extent of bonding between the terminal carbon atoms of each reactant molecule in the transition structure.

===Transition State Reaction Path Vibration===

The vibration which corresponds to the reaction path at the transition state is that which has an imaginary frequency (corresponding to the single negative Hessian eigenvalue).<ref name="Levine" />

<jmol>
<jmolApplet>
<color>white</color>
<script>frame 15; vibration 1;</script>
<uploadedFileContents>MHARDSTEX1_TS_METHOD3TSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

The C-C single bonds which form between the terminal carbon atoms of the butadiene and ethylene components form simultaneously, i.e. the C-C bond formation is synchronous.

==Exercise 2: Reaction of Cyclohexadiene and 1,3-Dioxole==

In this exercise, the Diels-Alder cycloaddition between cyclohexadiene and 1,3-dioxole was analysed. Both the exo and endo Diels-Alder cycloaddition pathways were examined.

All reactant/product/TS geometries were optimised to the semi-empirical PM6 level, followed by reoptimisation of the resultant structure to the DFT B3LYP level, on a 6-31G (d) basis set.

===MO Diagram for the Exo Transition State===

[[File:Mhardst exotsmodiagram2.jpg |thumb|center|''Figure 2 - MO Diagram for TS of the Exo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 20; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_EXOTS_B3LYPTSOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===MO Diagram for the Endo Transition State===

[[File:Mhardst endotsmodiagram2.jpg |thumb|center|''Figure 3 - MO Diagram for TS of the Endo Diels-Alder Reaction''.]]

=====MO 40=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 40; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

=====MO 41=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 42=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is a symmetric molecular orbital.

=====MO 43=====
<jmol>
<jmolApplet>
<color>white</color>
<script>frame 32; mo 43; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>MHARDST_EX2_ENDOTS_TSB3YLPOPT_ATTEMPT1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>

This is an antisymmetric molecular orbital.

===Type of Diels-Alder Reaction===

From analysis of the relative energy levels of the froniter molecular orbitals of both cyclohexadiene and 1,3-dioxole, it can be concluded that this particular Diels-Alder reaction is an inverse electron demand cycloaddition.<ref name="Clayden et. al." />

An inverse electron demand Diels-Alder reaction is defined as having the strongest interaction between the HOMO of the dienophile (1,3-dioxole), and the LUMO of the diene (cyclohexadiene) (i.e. this pair of frontier molecular orbitals are closest in energy, so have the strongest interaction).<ref name="Clayden et. al." /><ref name="Bruckner" />

The inverse electron demand nature of this Diels-Alder reaction can be easily justified by considering the electron-donating nature of the two ring-oxygen atoms in the 1,3-dioxole dienophile. Each oxygen atom donates electron density via an sp<sup>3</sup>-hybridised lone pair of electrons into the C=C π<sup>*</sup> antibonding orbital. <ref name="Clayden et. al." /> This interaction raises the energy of the 1,3-dioxole HOMO, by such an amount that it has a greater energy than the HOMO of the diene (cyclohexadiene), an an inverse electron demand Diels-Alder cycloaddition therefore takes place.<ref name="Clayden et. al." /><ref name="Bruckner" />

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+200.1</nowiki>
| style="text-align: center;" |<nowiki>-31.27</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+192.3</nowiki>
| style="text-align: center;" |<nowiki>-34.91</nowiki>
|}

''(All values given to 4 significant figures).''

These results allow some conclusions to be drawn about the kinetic and thermodynamic nature of the [4+2]-cycloaddition.

*The endo pathway has a lower activation energy than the exo pathway, and therefore, by the Arrhenius equation (k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), the endo pathway has a larger rate constant (k), so occurs more rapidly. Hence, the endo product is formed more rapidly and so is dubbed the 'kinetic product'.<ref name="Clayden et. al." />

*The endo pathway has a more negative reaction energy than the exo pathway, and so given that both reaction pathways begin at the same energy level (same reactants), the endo adduct must inherently be more thermodynamically stable, i.e. it is the 'thermodynamic product'.<ref name="Clayden et. al." />

Therefore, the endo adduct is both kinetically and thermodynamically favoured, so completely dominates the product distribution of the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole - regardless of whether the reaction takes place under kinetic (low temperature/short reaction times) or thermodynamic (high temperature/long reaction times) conditions.<ref name="Clayden et. al." />

===Secondary Orbital Interactions===

The 'Endo Rule' applies to all Diels-Alder reactions. It describes how the endo product dominates the product distribution (when the reaction is under kinetic control), despite often being the less thermodynamically stable product.<ref name="Clayden et. al." /> The endo adduct is always kinetically favoured because it has a lower energy transition state than the corresponding exo adduct. <ref name="Clayden et. al." />

The reason for the greater stability of the endo transition state lies in considering secondary orbital interactions in the HOMO.<ref name="Fleming" /> These are additional interactions which are not formally involved in bond formation between the diene and dienophile components.<ref name="Fleming" /> These secondary orbital interactions are the in-phase overlap of other p-orbitals in the dienophile (other than those involved in the C=C π-bond) with the newly forming intermediate C=C π-bond at the back of the diene component, in the transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

====Exo TS====

In the transition state for the exo pathway, there are no secondary orbital interactions, so there is no additional stabilising factor affecting the energy of the exo transition state. <ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO exo ts.jpg |thumb|center|''Figure 4 -Significant Orbital Interactions in the HOMO of the Exo Transition State''.]]

====Endo TS====

There are significant secondary orbital interactions in the TS of the endo pathway. <ref name="Fleming" /> These interactions are in-phase, so will provide additional stability to the transition state.<ref name="Clayden et. al." /><ref name="Fleming" /> This makes the transition state for the endo pathway more stable than that of the exo pathway, and hence, this explains why the endo pathway has a lower activation energy than the exo pathway.<ref name="Clayden et. al." /> <ref name="Fleming" /> This is the origin of the 'Endo Rule' in Diels-Alder reactions.<ref name="Clayden et. al." /><ref name="Fleming" />

[[File:MhardstHOMO endo ts.jpg |thumb|center|''Figure 5 -Significant Orbital Interactions in the HOMO of the Endo Transition State''.]]

==Exercise 3: Diels-Alder vs. Cheletropic==

In this exercise, three possible reaction pathways between a molecule of o-xylylene and sulfur dioxide were analysed. Both the exo and endo Diels-Alder cycloaddition pathways between the external cis-butadiene fragment of o-xylylene and sulfur dioxide were examined, as well as the alternative pericyclic reaction called a cheletropic reaction.

All reactant/product/TS geometries were optimised only to the semi-empirical PM6 level.

===Diels-Alder Reaction===

====Exo====

[[File:Mhardst exots.jpg|thumb|center|''Figure 6 - TS of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst exoproduct.jpg|thumb|center|''Figure 7 - Product of the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 exoproduct irc gif.gif|thumb|center|''Figure 8 - IRC for the Exo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

====Endo====

[[File:Mhardst endots.jpg|thumb|center|''Figure 9 - TS of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst endoproduct.jpg|thumb|center|''Figure 10 - Product of the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 endoproduct irc gif.gif|thumb|center|''Figure 11 - IRC for the Endo Diels-Alder Reaction Between o-Xylylene and Sulfur Dioxide''.]]

===Cheletropic===

[[File:Mhardst cheletropicts2.jpg|thumb|center|''Figure 12 - TS of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst cheletropicproduct.jpg|thumb|center|''Figure 13 - Product of the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

[[File:Mhardst ex3 cheletropic irc gif.gif|thumb|center|''Figure 14 - IRC for the Cheletropic Reaction Pathway Between o-Xylylene and Sulfur Dioxide''.]]

===Xylylene Stability===

During the course of each of the three reaction pathways studied, the internal cis-diene component (within the ring) conjugates with the newly-forming C=C double bond. This results in delocalisation of the pi-character of the orbitals over the entire 6-membered ring, forming an aromatic (planar; 6pi-electron) transition state, which formally results in a benzene-type component in all three aducts.

===Kinetic and Thermodynamic Analysis===

{| class="wikitable"
! style="background: #0D4F8B; color: white;" | Reaction Pathway
! style="background: #0D4F8B; color: white;" | Activation Energy/ kJmol<sup>-1</sup>
! style="background: #0D4F8B; color: white;" | Reaction Energy/ kJmol<sup>-1</sup>

|-
| style="text-align: center;" |Exo Diels-Alder
| style="text-align: center;" |<nowiki>+85.73</nowiki>
| style="text-align: center;" |<nowiki>-99.70</nowiki>
|-
| style="text-align: center;" |Endo Diels-Alder
| style="text-align: center;" |<nowiki>+81.74</nowiki>
| style="text-align: center;" |<nowiki>-99.05</nowiki>
|-
| style="text-align: center;" |Cheletropic
| style="text-align: center;" |<nowiki>+104.1</nowiki>
| style="text-align: center;" |<nowiki>-156.0</nowiki>
|}

''(All values given to 4 significant figures).''

From this comparing these values, conclusions can be drawn about the nature of each different reaction:

*The endo Diels-Alder cycloaddition has a smaller activation energy than the exo Diels-Alder pathway. Therefore, using the Arrhenius equation ((k=Ae<sup>(-E<sub>a</sub>/RT)</sup> where E<sub>a</sub> represents activation energy), it is readily observed that the endo pathway will have a larger value of k (rate constant).<ref name="Clayden et. al." /> Therefore, ''ceterus paribus'', the endo pathway will take place more rapidly, and the endo product will be formed fastest. The endo product is therefore the 'kinetic product' of the Diels-Alder reaction.<ref name="Clayden et. al." />

*The exo Diels-Alder cycloaddition has a more negative reaction energy, so is a more favourable pathway. Therefore, given that the exo and endo products both have identical starting materials, a more negative reaction energy means that the exo adduct is the more thermodynamically stable product. Therefore, the exo product is the 'thermodynamic product' of the Diels Alder pathway.<ref name="Clayden et. al." />

*The cheletropic pathway has a much larger activation energy than both Diels-Alder pathways, so will take place much more slowly. However, the product is also much more thermodynamically stable than either of the exo/endo adducts. Therefore, for a reaction under kinetic control (low temperature, short reaction times), the cheletropic pathway is unlikely to be observed, but under thermodynamic conditions (high temperature, long reaction times, i.e. equilibration conditions), the cheletropic product would dominate the product distribution.<ref name="Clayden et. al." />

All of this information can be clearly summarised in a schematic reaction profile, showing relative energy levels of the reactants, transition states structures, and products of each of the exo/endo Diels-Alder and cheletropic cycloadditions:

[[File:Mhardst reactionprofile.jpg|thumb|center|''Figure 15 - Reaction Profile for Three of the Possible Pathways of the Reaction Between the o-Xylylene and Sulfur Dioxide''.]]

==Conclusion==

In conclusion,the optimised transition states for the cycloaddition reactions of butadiene/ethene, cyclohexadiene/1,3-dioxole and o-xylylene/sulfur dioxide were all studied. Quantum mechanics provided a sound theoretical proof for the definitive location of a tranisiton state - a transition state corresponds to a saddle point on the potential energy surface for a given reaction, and hence has a single negative Hessian eigenvalue. This corresponds to a geometry with a single vibration with a negative frequency.

In particular, the frontier molecular orbitals for the Diels-Alder reaction between a substituted butadiene derivative and ethene derivative were analysed, and the symmetry of the interacting orbitals, and the MOs they produce in the transition state, were predicted. These were subsequently matched to the computationally-derived MOs produced for each of the optimised transition states. These studies allowed conclusions to be drawn about the symmetry requirements of such reactions, and the electronic nature of the observed Diels-Alder reaction (from comparison of relative energy levels of the HOMO/LUMO of the reacting components).

Furthermore, these optimisation studies allowed relative energy values of the reactants/products/transition states to be calculated. Hence, for each of the relevant reactions studied, the activation energies and reaction energies could be calculated, allowing deduction of the kinetic and thermodynamic products of each type of reactions.

==Appendix==

===Exercise 1===

IRC:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_METHOD3IRC_ATTEMPT1.LOG

Products:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDSTEX1_TS_CYCLOHEXENE_PM6OPTT_ATTEMPT1.LOG

===Exercise 2===

B3LYP Optimised Cyclohexadiene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_CYCLOHEXADIENE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised 1,3-Dioxole:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_13DIOXOLE_B3LYPOPT_ATTEMPT1.LOG

B3LYP Optimised Exo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOPRODUCT_B3LYP_ATTEMPT1.LOG

B3LYP Optimised Endo Product:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOPRODUCT_B3LYPOPT_ATTEMPT1.LOG

IRC Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_EXOTS_OPTIMISEDTS_IRC_ATTEMPT2.LOG

IRC Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX2_ENDOTS_TSOPTIMISED_IRC_ATTEMPT2.LOG

Single Point Energy Calculation for the Exo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EXOSINGLEPOINTENERGY_ATTEMPT1.LOG

Single Point Energy Calculation for the Endo Pathway:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_ENDOSINGLEPOINTENERGY_ATTEMPT1.LOG

===Exercise 3===

PM6 Optimised o-Xylylene:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARST_EX3_XYLYLENEREACTANT_PM6OPT_ATTEMPT4.LOG

PM6 Optimised Sulfur Dioxide:

https://wiki.ch.ic.ac.uk/wiki/index.php?title=File:MHARDST_EX3_SULFURDIOXIDEREACTANT_PM6OPT_ATTEMPT1.LOG

==References==
</references><ref name="Atkins and Friedman"> P. Atkins and R. Friedman, Molecular Quantum Mechanics, Oxford University Press Inc., New York, 2011 </ref>
</references>

<ref name="Levine"> I. N. Levine, Quantum Chemistry, Prentice-Hall Inc., New Jersey, 2000 </ref>

<ref name="Moss and Coady"> S. J. Moss and C. J. Coady, "Potential-Energy Surfaces and Transition-State Theory", J. Chem. Educ, 1983, 60 (6), pg. 455 - 461 </ref>
</references>

<ref name="Hinchliffe"> A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester, 1996, </ref>
</references>

<ref name="Lide "> D.R.Lide Jr., "A Survey of Carbon-Carbon Bond Lengths", Tetrahedron, 1962, 17 (3 - 4), pg. 125 - 134 </ref>
</references>

<ref name="Mantina et. al."> M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", Journal of Physical Chemistry A, 2009, 113 (19), pg. 5806 - 5812 </ref>
</references>

<ref name="Clayden et. al."> J. Clayden, N. Greeves, S. Warren and P. Wothers, Organic Chemistry, Oxford University Press Inc., New York, 2012</ref>
</references>

<ref name="Bruckner"> R. Bruckner, Advanced Organic Chemistry, Academic Press, San Diego, 2002</ref>
</references>

<ref name="Fleming"> I. Fleming, Frontier Orbitals and Organic Chemical Reactions, John Wiley & Sons, London, 1976</ref>
</references>