==Introduction==
Gaussian is a computational chemistry tool which enables efficient calculations of reaction coordinates across a potential energy surface. By differing the levels of approximations, Gaussian uses different quantum methods to calculate transition states to varying degrees of accuracy.

===Potential energy surface===
[[Image:SG PES image.JPG|right|thumb|Labelled potential energy surface <ref>D. S. Kaliakin, R. R. Zaari, S. A. Varganov, ''J. Chem. Educ.'', 2015, '''92''', 2106−2112.</ref>]]

A potential energy surface (PES) is a multi-dimensional plot of potential energy. The number of dimensions is the number of degrees of freedom, which are the normal modes. There are multiple possible reaction pathways to progress from reactants to products via a transition state, and deciding upon one relies on prior understanding of the approach trajectory and orientation of reactants.

=== Transition state ===
A transition state is the point of highest energy in the progression of a reaction. It is a second order saddle point with a gradient of zero, where the second derivative (corresponding to the force constant) is negative. This is relevant to computational theory because where a transition state is located, this vibrational frequency will appear negative. This is an imaginary number which resembles the vibration of the bonding atoms moving towards each other.

=== Minima ===
Potential energy surfaces contain both global and local minima. A local minimum is a low-energy geometry when compared to all surrounding points. This could be a reagent for example, which is generally a stable state of a compound, while not necessarily being the lowest energy substrate in the reaction. A global minimum point is the lowest energy point on the entire energy surface. This resembles the most stable state of any structure present in the reaction. At a minimum point, the surrounding gradient is zero and the second derivative is positive.

=== Method ===
In this experiment, three different methods were used. The first involved guessing the transition structure, which was optimised to give the transition state saddle point. Method 2 involved guessing the transition state of the reaction and freezing the atoms which form bonds in the process. This structure was then optimised to a minimum energy then used to find a transition structure by optimising. The third method began by drawing the either the reactants or products (the one with fewer atoms was chosen, here it was the product) and optimising. The bonds formed in the desired reaction were then broken and the atoms bonding were frozen. The process then continued as per method 2. For more complex structures, such as those in exercises 2 and 3, method 3 was the most successful in giving a good initial estimate for the transition state, saving time by not having to guess the structure.

Using Gaussian enables Slater functions to be replaced by a linear combination of Gaussian functions. This is similar to the method used to create molecular orbitals by linearly combining atomic orbitals. The simplification using Gaussian functions allows a relatively accurate calculation to be carried out more quickly.<ref>E. G. Lewars, ''Computational Chemistry: Introduction to the Theory And Applications of Molecular and Quantum Mechanics'', Springer, New York, 2003.</ref>

The concept of optimisation is that the computer runs a calculation to solve the Hamiltonian matrix. The geometry is then altered and the calculation is run again, until it gets closer to a solution in a step-wise fashion. PM6 is a semi-empirical method which works by using pre-set values for certain integrals. It therefore requires less calculations because it has made more approximations, making the tasks quicker to complete but less accurate. The other method used was a DFT-hybrid calculation, B3LYP, using the 6-31G(d) basis set. This is a more time-consuming but more accurate method as there are fewer approximations of integrals, which therefore produces results that are comparable to literature. DFT evaluates the Hamiltonian as a function of electron density. B3LYP uses both this and a Hartree-Fock calculation to calculate the exchange correlation contributions to the energy. It does not include correlations due to many-body interactions, but this level of complexity would make it more computationally expensive.<ref><nowiki>http://cmt.dur.ac.uk/sjc/thesis_prt/node23.html (viewed: 01/11/17)</nowiki></ref>

Exercises 1 and 3 used the PM6 method as this was sufficient for the relevant analysis. In exercise 2, where the required analysis was more quantitative (such as determining the energy of the reaction barrier), the B3LYP method was used after PM6 to further optimise the transition states to show the secondary orbital overlap more clearly. By doing these optimisations step-wise, it sped up the process. Crucially, the energies determined from either PM6 or B3LYP cannot be compared between the different levels of theory. IRC (intrinsic reaction coordinate) calculations were then used to confirm that the reaction has proceeded through the expected pathway to end up at the minimum point corresponding to the products.

== Exercise 1: Reaction of butadiene and ethene ==

Exercise 1 focussed on the reaction of butadiene with ethene in a <sub>π</sub>4<sub>s</sub> + <sub>π</sub>2<sub>s</sub> Diels-Alder cycloaddition, forming cyclohexene, as seen in the scheme below.

[[Image:SG ex1 Reaction Scheme.jpg|centre|thumb|Cycloaddition of butadiene with ethene]]

===MO analysis===
The MO diagram below shows the HOMO and LUMO MOs from each butadiene and ethene forming the four key MOs of the transition state. This was constructed by looking at the PM6 calculations of the reactants and the transition state. These are the HOMO-1, the HOMO, the LUMO and the LUMO+1. The HOMO-1 is slightly higher in energy than the HOMO of butadiene (despite being a new MO, which is usually lower in energy) because it is the transition state, which is at a maximum in terms of the reaction coordinate.

[[Image:SG4 Ex1 MO diagram.jpg|centre|thumb|MO diagram to show the formation of the transition state of the reaction between butadiene and ethene.]]

The Jmols below show that the HOMO of butadiene and the LUMO of ethene are both antisymmetric orbitals, and the LUMO of butadiene and the HOMO of ethene are both symmetric. Only orbitals of the same symmetry are “allowed” reactions because these have a non-zero orbital overlap integral. An overlap integral of zero (due to orbitals of different symmetry) means the reaction is “forbidden” and the orbitals therefore do not mix.
{| class="wikitable"
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|<nowiki><jmol>
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|-
|[[File:TS HOMO-1.png|thumb|TS HOMO-1 (orbital 5)]]
|[[File:SG TS HOMO.png|thumb|TS HOMO (orbital 6)]]
|-
|[[File:TS LUMO.png|thumb|TS LUMO (orbital 7)]]
|[[File:TS LUMO+1.png|thumb|TS LUMO+1 (orbital 8)]]
|}

Normal electron demand Diels-Alder reactions occur between an electron rich diene and an electron deficient dienophile and occur between the diene HOMO and the dienophile LUMO. Inverse electron demand is the opposite.

In MO diagrams, the molecular orbital has the greater contribution from the atomic orbital that is closer in energy to the newly formed MO. This same principle applies in this situation, which is demonstrated by the MOs in the Jmols. For example, the LUMO of butadiene is further in energy from the transition state HOMO than the ethene HOMO, so it’s orbital contribution is smaller.

===Bond length analysis===
{| class="wikitable"
! rowspan="2" |C-C bond
! colspan="3" |Bond length / Å
|-
!Reactants
!TS
!Products
|-
|C<sub>1</sub>-C<sub>2</sub>
|1.333
|1.380
|1.501
|-
|C<sub>2</sub>-C<sub>3</sub>
|1.471
|1.411
|1.337
|-
|C<sub>3</sub>-C<sub>4</sub>
|1.333
|1.380
|1.501
|-
|C<sub>4</sub>-C<sub>5</sub>
|N/A
|2.116
|1.537
|-
|C<sub>5</sub>-C<sub>6</sub>
|1.327
|1.382
|1.535
|-
|C<sub>6</sub>-C<sub>1</sub>
|N/A
|2.113
|1.537
|-
| colspan="4" |Budadiene C=C bond: 1.454 Å, butadiene C-C bond: 1.338 Å, ethene C=C: 1.331 Å.<ref>N. C. Craig, P. Groner, and D. C. McKean, ''J. Phys. Chem.'', 2006, '''110''', 7461.</ref>
|}

C1-C2 and C3-C4 were previously the double bonds in butadiene, which originally started at 1.333 Å. In the product they have become sp<sup>3</sup> carbons forming single bonds, which lengthen from the reagent, through the transition state, to the product. The C5-C6 bond was the double bond of ethene and underwent the same transition. The C2-C3 bond was the single bond in butadiene, which became the double bond in the product. This was reflected by the shortening of the bond length from 1.471 A to 1.337 Å.

The Van der Waals radius of carbon is 1.7 Å.<ref>S. S. Batsanov, ''Inorganic Materials'', 2001, '''37(9),''' 871–885.</ref> For a new bond to be forming, double the Van der Waals radii should be less than the “bond” length in the transition state. For carbon, this would be 3.4 Å and the newly forming bonds have an average distance of 2.114 Å in the transition state, showing the orbitals are interacting and a bond is forming. The bond length becomes 1.537 Å, which is the typical C-C carbon single bond length.

[[Image:SG Ex1 TS negative freq vibrations.gif|right|thumb|Animation of the "negative" frequency, showing the bonding atoms moving towards each other]]

By undertaking a frequency calculation, it is possible to clarify that the transition state has been optimised is the single negative vibration, which corresponds to an imaginary number. This is due to the negative force constant when moving over the maxima of the transition state (with respect to the reaction coordinate) between the two local minima of the reactants and the products. This vibrational mode is shown below and demonstrates the bonding atoms moving towards each other at the same time, which is concordant with the concerted nature of a cycloaddition.

===Key file links===

[[Media:SG ETHYLENE MINIMUM 2.LOG|Ethene]]

[[Media:SG BUTADIENE CIS MINIMUM 2.LOG|Butadiene]]

[[Media:SG CYCLOHEXENE TS.LOG|Transition state]]

[[Media:SG_CYCLOHEXENE_PRODUCT_MINIMUM_2.LOG|Cyclohexene]]

== Exercise 2: Reaction of cyclohexadiene and 1,3-dioxole ==

Exercise two investigated the reaction of cyclohexadiene and 1,3-dioxole, as shown in the scheme below. This is another [4+2] cycloaddition, which can result in an exo or endo product, depending on the trajectory of approach of the reactants.

The endo-transition state is lower in energy due to stabilisation via both primary and secondary orbital interactions. Primary orbital interactions are those depicted by the dotted lines in the transition states in the schemes below. The secondary orbital interactions depend on the location of the oxygen atoms. Each O has a lone pair in a p-orbital, which, in the endo-TS is able to stabilise the structure by overlapping with the forming pi-bond and reducing its energy. This causes the endo-product to be the kinetic product as the reaction runs faster through this TS. The exo-product is often lower in energy due to the reduced steric clash, however in this reaction the endo-product is lower as the Hs are further away from each other, which reduces the energy.<ref>I. Flemming, ''Frontier Orbitals and Organic Chemical Reactions, ''Wiley, London, 1978, 29-109.</ref>

[[File:SG Ex2 reaction scheme.PNG|center|thumb|Reaction scheme showing the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole and the exo and endo transition states and products]]

===MO analysis===

[[File:SG4 Ex2 MO diagram.PNG|centre|thumb|MO diagram showing the formation of the endo- and exo- transition state orbitals (numbered 5-8)]]
{| class="wikitable"
|
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|<nowiki><jmol></jmolApplet>
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Exo transition state HOMO (orbital 6)

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Exo transition state LUMO (orbital 7)

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Endo transition state HOMO (orbital 5)

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Endo transition state LUMO (orbital 8)
|}

Above is the MO diagram of the Diels-Alder exo- and endo- transition states and molecular orbitals for energy levels 5-8. The MO diagram has been constructed by running a single point energy calculation on the first frame of the IRC. This gives the energy levels of the orbitals of the fragments in relation to each other, enabling identification of the order of the fragment energies and so allowing confirmation that this is inverse electron demand.

The 1,3-dioxole orbitals have been raised in energy due to the oxygens donating their lone pairs to the double bond, making it a more electron rich dienophile. This causes the HOMO of the dienophile being a good energy match to interact with the LUMO of the diene, making the reaction inverse electron demand (which corresponds to the single point energy calculation). This overlap can be seen in the MO diagrams, where the LUMOs show a slightly bigger contribution from the diene LUMO, showing that this orbital is a little higher in energy than the HOMO of the dienophile.

In the Jmols of orbitals 5 and 6 (the endo- and exo-HOMOs), the secondary orbital interaction can only be seen in the endo-TS. It is between the oxygens and the pi-bond that is forming, and acts to stabilise this transition state greater than the exo-TS. The thermodynamics behind this has been explored in the section below.

===Thermodynamics analysis===
{| class="wikitable"
!Reaction
!Reaction barrier / kJ mol<sup>-1</sup>
!Reaction energy / kJ mol<sup>-1</sup>
|-
|Exo
|166.02
|<nowiki>-63.19</nowiki>
|-
|Endo
|158.26
|<nowiki>-66.75</nowiki>
|-
| colspan="3" |A table setting out the reaction barrier and the reaction energy for the formation of both the endo- and exo- products.
|}
The reaction barrier correlates to the activation energy of the process (the difference between the reactant and transition state energy) and the reaction energy is the change in free energy of the process. All numbers used are from B3LYP calculations because this enables comparison.

As explained the in introduction to exercise 2, the endo-TS is lower in energy than the exo-TS due to secondary orbital interactions. The endo-product is also lower in energy, which is expected due to its structure.

===Key file links===

[[Media:SG 13DIOXOLE MINIMUM BETTER.LOG|1,3-dioxole B3LYP]]

[[Media:SG CYCLOHEXADIENE MINIMUM BETTER.LOG|Cyclohexadiene B3LYP]]

[[Media:SG Ex2 EXO TS BETTER.LOG|Exo transition state B3LYP]]

[[Media:SG Ex2 ENDO TS BETTER.LOG|Endo transition state B3LYP]]

[[Media:SG_EXO_PRODUCT_MINIMUM_BETTER.LOG|Exo product B3LYP]]

[[Media:SG ENDO PRODUCT MINIMUM BETTER.LOG|Endo product B3LYP]]

== Exercise 3: Diels-Alder vs Cheletropic ==
[[File:SG Ex3 Reaction scheme.PNG|thumb|Possible reactions between Xylylene and SO<sub>2</sub>]]
This experiment aimed to investigate how the trajectory of approach affected the type of reaction that proceeds. The reaction scheme for the three potential pathways can be seen on the right. The possible products are the exo-, endo- and cheletropic.

According to IUPAC, a cheletropic reaction is a “cycloaddition across the terminal atoms of a fully conjugated system with formation of two new σ-bonds to a single atom of the reagent. There is a formal loss of one π-bond in the substrate and an increase in the coordination number of the relevant atom of the reagent.”<ref><nowiki>https://goldbook.iupac.org/html/C/C01014.html (viewed: 03/11/17)</nowiki></ref>

===IRC calculations===

Below are the IRC clips for the three reactions. The exo- and endo- reactions show that the C-O bond forms before the C-S bond. In the cheletropic reaction, the both bonds to the sulfur form at the same time, this is known as a synchronous reaction.
{| class="wikitable"
|[[Image:SG Chelotropic IRC.gif|thumb|centre|Animation of the cheletropic IRC (which shows the reaction progression)]]
|[[Image:SG Ex3 Exo IRC.gif|thumb|centre|Animation of the exo approach IRC (which shows the reaction progression)]]
|[[Image:SG Ex3 Endo IRC.gif|thumb|centre|Animation of the endo approach IRC (which shows the reaction progression)]]
|}

By looking at the IRCs, it is possible to see that the 6-membered ring that is present in the starting material becomes a structure where all Cs in the ring have a 1.5 bond order. This resembles an aromatic structure (as in the final product), which is extremely stable. This causes xylylene to be very unstable as it wants to react to form its aromatic counterpart, via a Diels-Alder reaction.

=== Thermodynamics ===
{| class="wikitable"
!Reaction
!Reaction barrier / kJ mol<sup>-1</sup>
!Reaction energy / kJ mol<sup>-1</sup>
|-
|Exo
|83.01
|<nowiki>-100.60</nowiki>
|-
|Endo
|79.07
|<nowiki>-99.96</nowiki>
|-
|Cheletropic
|101.17
|<nowiki>-156.38</nowiki>
|}

From the table, the kinetic and thermodynamic products can be determined. Here, the kinetic product is the endo-product because it has the lowest reaction barrier (i.e. the transition state energy is the lowest energy), again due to the secondary orbital overlap. The thermodynamic product is the product with the lowest energy, which is the cheletropic product. The chelotropic product is most stable because it still contains two S=O double bonds, which are very strong. The transition state for this reaction is higher in energy though due to the strain of forming a 5-membered ring (less favourable than a 6-membered ring).

This can be visualised by looking at this reaction profiles below.
{| class="wikitable"
|[[Image:SG Ex3 reaction profile.jpg|thumb|Energy profile for the exo- and endo- products of the Diels-Alder reaction]]
|
|}
[[Image:SG Ex3 cheletropic reaction profile.jpg|thumb|Energy profile for the cheletropic reaction]]

===Alternative reaction pathways===

Cis-butadiene has a second diene site that can potentially react. By examining it’s reaction barrier, it is possible to explain why it does not react here. The exo- reaction gives an activation energy of 116.75 kJ mol-1 and the endo- reaction gives a value of 108.99 kJ mol-1. The reason for this is the lack of formation of the aromatic ring in the transition state structures, meaning they are less favourable and so have higher in energy. This would always produce a high energy product (relative to reaction at the other site, due to the lack of pi-delocalisation).

===Key file links===

[[Media:SG REACTANT 1 MINIMUM.LOG|Xylylene]]

[[Media:SG SO2 MINIMUM.LOG|SO2]]

[[Media:SG Ex3 EXO TS.LOG|Exo transition state]]

[[Media:SG Ex3 ENDO TS.LOG|Endo transition state]]

[[Media:SG Ex3 CHELE TS.LOG|Cheletropic transition state]]

[[Media:SG Ex3 EXO PRODUCT MINIMUM.LOG|Exo product]]

[[Media:SG Ex3 ENDO PRODUCT MINIMUM.LOG|Endo product]]

[[Media:SG CHELE PRODUCT MINIMUM.LOG|Cheletropic product]]

[[Media:SG Ex3 EXO TS IRC.LOG|Second diene site exo IRC]]

[[Media:SG Ex3 ENDO TS IRC.LOG|Second diene site endo IRC]]

==Conclusion==

==References==

Rep:Mod:sg3415TS

2017-11-04T11:20:31Z

Sg3415: /* Thermodynamics */

==Introduction==
Gaussian is a computational chemistry tool which enables efficient calculations of reaction coordinates across a potential energy surface. By differing the levels of approximations, Gaussian uses different quantum methods to calculate transition states to varying degrees of accuracy.

===Potential energy surface===
[[Image:SG PES image.JPG|right|thumb|Labelled potential energy surface <ref>D. S. Kaliakin, R. R. Zaari, S. A. Varganov, ''J. Chem. Educ.'', 2015, '''92''', 2106−2112.</ref>]]

A potential energy surface (PES) is a multi-dimensional plot of potential energy. The number of dimensions is the number of degrees of freedom, which are the normal modes. There are multiple possible reaction pathways to progress from reactants to products via a transition state, and deciding upon one relies on prior understanding of the approach trajectory and orientation of reactants.

=== Transition state ===
A transition state is the point of highest energy in the progression of a reaction. It is a second order saddle point with a gradient of zero, where the second derivative (corresponding to the force constant) is negative. This is relevant to computational theory because where a transition state is located, this vibrational frequency will appear negative. This is an imaginary number which resembles the vibration of the bonding atoms moving towards each other.

=== Minima ===
Potential energy surfaces contain both global and local minima. A local minimum is a low-energy geometry when compared to all surrounding points. This could be a reagent for example, which is generally a stable state of a compound, while not necessarily being the lowest energy substrate in the reaction. A global minimum point is the lowest energy point on the entire energy surface. This resembles the most stable state of any structure present in the reaction. At a minimum point, the surrounding gradient is zero and the second derivative is positive.

=== Method ===
In this experiment, three different methods were used. The first involved guessing the transition structure, which was optimised to give the transition state saddle point. Method 2 involved guessing the transition state of the reaction and freezing the atoms which form bonds in the process. This structure was then optimised to a minimum energy then used to find a transition structure by optimising. The third method began by drawing the either the reactants or products (the one with fewer atoms was chosen, here it was the product) and optimising. The bonds formed in the desired reaction were then broken and the atoms bonding were frozen. The process then continued as per method 2. For more complex structures, such as those in exercises 2 and 3, method 3 was the most successful in giving a good initial estimate for the transition state, saving time by not having to guess the structure.

Using Gaussian enables Slater functions to be replaced by a linear combination of Gaussian functions. This is similar to the method used to create molecular orbitals by linearly combining atomic orbitals. The simplification using Gaussian functions allows a relatively accurate calculation to be carried out more quickly.<ref>E. G. Lewars, ''Computational Chemistry: Introduction to the Theory And Applications of Molecular and Quantum Mechanics'', Springer, New York, 2003.</ref>

The concept of optimisation is that the computer runs a calculation to solve the Hamiltonian matrix. The geometry is then altered and the calculation is run again, until it gets closer to a solution in a step-wise fashion. PM6 is a semi-empirical method which works by using pre-set values for certain integrals. It therefore requires less calculations because it has made more approximations, making the tasks quicker to complete but less accurate. The other method used was a DFT-hybrid calculation, B3LYP, using the 6-31G(d) basis set. This is a more time-consuming but more accurate method as there are fewer approximations of integrals, which therefore produces results that are comparable to literature. DFT evaluates the Hamiltonian as a function of electron density. B3LYP uses both this and a Hartree-Fock calculation to calculate the exchange correlation contributions to the energy. It does not include correlations due to many-body interactions, but this level of complexity would make it more computationally expensive.<ref><nowiki>http://cmt.dur.ac.uk/sjc/thesis_prt/node23.html (viewed: 01/11/17)</nowiki></ref>

Exercises 1 and 3 used the PM6 method as this was sufficient for the relevant analysis. In exercise 2, where the required analysis was more quantitative (such as determining the energy of the reaction barrier), the B3LYP method was used after PM6 to further optimise the transition states to show the secondary orbital overlap more clearly. By doing these optimisations step-wise, it sped up the process. Crucially, the energies determined from either PM6 or B3LYP cannot be compared between the different levels of theory. IRC (intrinsic reaction coordinate) calculations were then used to confirm that the reaction has proceeded through the expected pathway to end up at the minimum point corresponding to the products.

== Exercise 1: Reaction of butadiene and ethene ==

Exercise 1 focussed on the reaction of butadiene with ethene in a <sub>π</sub>4<sub>s</sub> + <sub>π</sub>2<sub>s</sub> Diels-Alder cycloaddition, forming cyclohexene, as seen in the scheme below.

[[Image:SG ex1 Reaction Scheme.jpg|centre|thumb|Cycloaddition of butadiene with ethene]]

===MO analysis===
The MO diagram below shows the HOMO and LUMO MOs from each butadiene and ethene forming the four key MOs of the transition state. This was constructed by looking at the PM6 calculations of the reactants and the transition state. These are the HOMO-1, the HOMO, the LUMO and the LUMO+1. The HOMO-1 is slightly higher in energy than the HOMO of butadiene (despite being a new MO, which is usually lower in energy) because it is the transition state, which is at a maximum in terms of the reaction coordinate.

[[Image:SG4 Ex1 MO diagram.jpg|centre|thumb|MO diagram to show the formation of the transition state of the reaction between butadiene and ethene.]]

The Jmols below show that the HOMO of butadiene and the LUMO of ethene are both antisymmetric orbitals, and the LUMO of butadiene and the HOMO of ethene are both symmetric. Only orbitals of the same symmetry are “allowed” reactions because these have a non-zero orbital overlap integral. An overlap integral of zero (due to orbitals of different symmetry) means the reaction is “forbidden” and the orbitals therefore do not mix.
{| class="wikitable"
|+A table setting out the HOMO and LUMO of butadiene and ethene
|<nowiki><jmol>
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|-
|[[File:TS HOMO-1.png|thumb|TS HOMO-1 (orbital 5)]]
|[[File:SG TS HOMO.png|thumb|TS HOMO (orbital 6)]]
|-
|[[File:TS LUMO.png|thumb|TS LUMO (orbital 7)]]
|[[File:TS LUMO+1.png|thumb|TS LUMO+1 (orbital 8)]]
|}

Normal electron demand Diels-Alder reactions occur between an electron rich diene and an electron deficient dienophile and occur between the diene HOMO and the dienophile LUMO. Inverse electron demand is the opposite.

In MO diagrams, the molecular orbital has the greater contribution from the atomic orbital that is closer in energy to the newly formed MO. This same principle applies in this situation, which is demonstrated by the MOs in the Jmols. For example, the LUMO of butadiene is further in energy from the transition state HOMO than the ethene HOMO, so it’s orbital contribution is smaller.

===Bond length analysis===
{| class="wikitable"
! rowspan="2" |C-C bond
! colspan="3" |Bond length / Å
|-
!Reactants
!TS
!Products
|-
|C<sub>1</sub>-C<sub>2</sub>
|1.333
|1.380
|1.501
|-
|C<sub>2</sub>-C<sub>3</sub>
|1.471
|1.411
|1.337
|-
|C<sub>3</sub>-C<sub>4</sub>
|1.333
|1.380
|1.501
|-
|C<sub>4</sub>-C<sub>5</sub>
|N/A
|2.116
|1.537
|-
|C<sub>5</sub>-C<sub>6</sub>
|1.327
|1.382
|1.535
|-
|C<sub>6</sub>-C<sub>1</sub>
|N/A
|2.113
|1.537
|-
| colspan="4" |Budadiene C=C bond: 1.454 Å, butadiene C-C bond: 1.338 Å, ethene C=C: 1.331 Å.<ref>N. C. Craig, P. Groner, and D. C. McKean, ''J. Phys. Chem.'', 2006, '''110''', 7461.</ref>
|}

C1-C2 and C3-C4 were previously the double bonds in butadiene, which originally started at 1.333 Å. In the product they have become sp<sup>3</sup> carbons forming single bonds, which lengthen from the reagent, through the transition state, to the product. The C5-C6 bond was the double bond of ethene and underwent the same transition. The C2-C3 bond was the single bond in butadiene, which became the double bond in the product. This was reflected by the shortening of the bond length from 1.471 A to 1.337 Å.

The Van der Waals radius of carbon is 1.7 Å.<ref>S. S. Batsanov, ''Inorganic Materials'', 2001, '''37(9),''' 871–885.</ref> For a new bond to be forming, double the Van der Waals radii should be less than the “bond” length in the transition state. For carbon, this would be 3.4 Å and the newly forming bonds have an average distance of 2.114 Å in the transition state, showing the orbitals are interacting and a bond is forming. The bond length becomes 1.537 Å, which is the typical C-C carbon single bond length.

[[Image:SG Ex1 TS negative freq vibrations.gif|right|thumb|Animation of the "negative" frequency, showing the bonding atoms moving towards each other]]

By undertaking a frequency calculation, it is possible to clarify that the transition state has been optimised is the single negative vibration, which corresponds to an imaginary number. This is due to the negative force constant when moving over the maxima of the transition state (with respect to the reaction coordinate) between the two local minima of the reactants and the products. This vibrational mode is shown below and demonstrates the bonding atoms moving towards each other at the same time, which is concordant with the concerted nature of a cycloaddition.

===Key file links===

[[Media:SG ETHYLENE MINIMUM 2.LOG|Ethene]]

[[Media:SG BUTADIENE CIS MINIMUM 2.LOG|Butadiene]]

[[Media:SG CYCLOHEXENE TS.LOG|Transition state]]

[[Media:SG_CYCLOHEXENE_PRODUCT_MINIMUM_2.LOG|Cyclohexene]]

== Exercise 2: Reaction of cyclohexadiene and 1,3-dioxole ==

Exercise two investigated the reaction of cyclohexadiene and 1,3-dioxole, as shown in the scheme below. This is another [4+2] cycloaddition, which can result in an exo or endo product, depending on the trajectory of approach of the reactants.

The endo-transition state is lower in energy due to stabilisation via both primary and secondary orbital interactions. Primary orbital interactions are those depicted by the dotted lines in the transition states in the schemes below. The secondary orbital interactions depend on the location of the oxygen atoms. Each O has a lone pair in a p-orbital, which, in the endo-TS is able to stabilise the structure by overlapping with the forming pi-bond and reducing its energy. This causes the endo-product to be the kinetic product as the reaction runs faster through this TS. The exo-product is often lower in energy due to the reduced steric clash, however in this reaction the endo-product is lower as the Hs are further away from each other, which reduces the energy.<ref>I. Flemming, ''Frontier Orbitals and Organic Chemical Reactions, ''Wiley, London, 1978, 29-109.</ref>

[[File:SG Ex2 reaction scheme.PNG|center|thumb|Reaction scheme showing the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole and the exo and endo transition states and products]]

===MO analysis===

[[File:SG4 Ex2 MO diagram.PNG|centre|thumb|MO diagram showing the formation of the endo- and exo- transition state orbitals (numbered 5-8)]]
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Exo transition state HOMO (orbital 6)

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Exo transition state LUMO (orbital 7)

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Endo transition state HOMO (orbital 5)

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Endo transition state LUMO (orbital 8)
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Above is the MO diagram of the Diels-Alder exo- and endo- transition states and molecular orbitals for energy levels 5-8. The MO diagram has been constructed by running a single point energy calculation on the first frame of the IRC. This gives the energy levels of the orbitals of the fragments in relation to each other, enabling identification of the order of the fragment energies and so allowing confirmation that this is inverse electron demand.

The 1,3-dioxole orbitals have been raised in energy due to the oxygens donating their lone pairs to the double bond, making it a more electron rich dienophile. This causes the HOMO of the dienophile being a good energy match to interact with the LUMO of the diene, making the reaction inverse electron demand (which corresponds to the single point energy calculation). This overlap can be seen in the MO diagrams, where the LUMOs show a slightly bigger contribution from the diene LUMO, showing that this orbital is a little higher in energy than the HOMO of the dienophile.

In the Jmols of orbitals 5 and 6 (the endo- and exo-HOMOs), the secondary orbital interaction can only be seen in the endo-TS. It is between the oxygens and the pi-bond that is forming, and acts to stabilise this transition state greater than the exo-TS. The thermodynamics behind this has been explored in the section below.

===Thermodynamics analysis===
{| class="wikitable"
!Reaction
!Reaction barrier / kJ mol<sup>-1</sup>
!Reaction energy / kJ mol<sup>-1</sup>
|-
|Exo
|166.02
|<nowiki>-63.19</nowiki>
|-
|Endo
|158.26
|<nowiki>-66.75</nowiki>
|-
| colspan="3" |A table setting out the reaction barrier and the reaction energy for the formation of both the endo- and exo- products.
|}
The reaction barrier correlates to the activation energy of the process (the difference between the reactant and transition state energy) and the reaction energy is the change in free energy of the process. All numbers used are from B3LYP calculations because this enables comparison.

As explained the in introduction to exercise 2, the endo-TS is lower in energy than the exo-TS due to secondary orbital interactions. The endo-product is also lower in energy, which is expected due to its structure.

===Key file links===

[[Media:SG 13DIOXOLE MINIMUM BETTER.LOG|1,3-dioxole B3LYP]]

[[Media:SG CYCLOHEXADIENE MINIMUM BETTER.LOG|Cyclohexadiene B3LYP]]

[[Media:SG Ex2 EXO TS BETTER.LOG|Exo transition state B3LYP]]

[[Media:SG Ex2 ENDO TS BETTER.LOG|Endo transition state B3LYP]]

[[Media:SG_EXO_PRODUCT_MINIMUM_BETTER.LOG|Exo product B3LYP]]

[[Media:SG ENDO PRODUCT MINIMUM BETTER.LOG|Endo product B3LYP]]

== Exercise 3: Diels-Alder vs Cheletropic ==
[[File:SG Ex3 Reaction scheme.PNG|thumb|Possible reactions between Xylylene and SO<sub>2</sub>]]
This experiment aimed to investigate how the trajectory of approach affected the type of reaction that proceeds. The reaction scheme for the three potential pathways can be seen on the right. The possible products are the exo-, endo- and cheletropic.

According to IUPAC, a cheletropic reaction is a “cycloaddition across the terminal atoms of a fully conjugated system with formation of two new σ-bonds to a single atom of the reagent. There is a formal loss of one π-bond in the substrate and an increase in the coordination number of the relevant atom of the reagent.”<ref><nowiki>https://goldbook.iupac.org/html/C/C01014.html (viewed: 03/11/17)</nowiki></ref>

===IRC calculations===

Below are the IRC clips for the three reactions. The exo- and endo- reactions show that the C-O bond forms before the C-S bond. In the cheletropic reaction, the both bonds to the sulfur form at the same time, this is known as a synchronous reaction.
{| class="wikitable"
|[[Image:SG Chelotropic IRC.gif|thumb|centre|Animation of the cheletropic IRC (which shows the reaction progression)]]
|[[Image:SG Ex3 Exo IRC.gif|thumb|centre|Animation of the exo approach IRC (which shows the reaction progression)]]
|[[Image:SG Ex3 Endo IRC.gif|thumb|centre|Animation of the endo approach IRC (which shows the reaction progression)]]
|}

By looking at the IRCs, it is possible to see that the 6-membered ring that is present in the starting material becomes a structure where all Cs in the ring have a 1.5 bond order. This resembles an aromatic structure (as in the final product), which is extremely stable. This causes xylylene to be very unstable as it wants to react to form its aromatic counterpart, via a Diels-Alder reaction.

=== Thermodynamics ===
{| class="wikitable"
!Reaction
!Reaction barrier / kJ mol<sup>-1</sup>
!Reaction energy / kJ mol<sup>-1</sup>
|-
|Exo
|83.01
|<nowiki>-100.60</nowiki>
|-
|Endo
|79.07
|<nowiki>-99.96</nowiki>
|-
|Cheletropic
|101.17
|<nowiki>-156.38</nowiki>
|}

From the table, the kinetic and thermodynamic products can be determined. Here, the kinetic product is the endo-product because it has the lowest reaction barrier (i.e. the transition state energy is the lowest energy), again due to the secondary orbital overlap. The thermodynamic product is the product with the lowest energy, which is the cheletropic product. The chelotropic product is most stable because it still contains two S=O double bonds, which are very strong. The transition state for this reaction is higher in energy though due to the strain of forming a 5-membered ring (less favourable than a 6-membered ring).

This can be visualised by looking at this reaction profiles below.

[[Image:SG Ex3 reaction profile.jpg|thumb|Energy profile for the exo- and endo- products of the Diels-Alder reaction]]

[[Image:SG Ex3 cheletropic reaction profile.jpg|thumb|Energy profile for the cheletropic reaction]]

===Alternative reaction pathways===

Cis-butadiene has a second diene site that can potentially react. By examining it’s reaction barrier, it is possible to explain why it does not react here. The exo- reaction gives an activation energy of 116.75 kJ mol-1 and the endo- reaction gives a value of 108.99 kJ mol-1. The reason for this is the lack of formation of the aromatic ring in the transition state structures, meaning they are less favourable and so have higher in energy. This would always produce a high energy product (relative to reaction at the other site, due to the lack of pi-delocalisation).

===Key file links===

[[Media:SG REACTANT 1 MINIMUM.LOG|Xylylene]]

[[Media:SG SO2 MINIMUM.LOG|SO2]]

[[Media:SG Ex3 EXO TS.LOG|Exo transition state]]

[[Media:SG Ex3 ENDO TS.LOG|Endo transition state]]

[[Media:SG Ex3 CHELE TS.LOG|Cheletropic transition state]]

[[Media:SG Ex3 EXO PRODUCT MINIMUM.LOG|Exo product]]

[[Media:SG Ex3 ENDO PRODUCT MINIMUM.LOG|Endo product]]

[[Media:SG CHELE PRODUCT MINIMUM.LOG|Cheletropic product]]

[[Media:SG Ex3 EXO TS IRC.LOG|Second diene site exo IRC]]

[[Media:SG Ex3 ENDO TS IRC.LOG|Second diene site endo IRC]]

==Conclusion==

==References==

File:SG Ex3 cheletropic reaction profile.jpg

2017-11-04T11:20:17Z

Sg3415:

File:SG Ex3 reaction profile.jpg

2017-11-04T11:19:52Z

Sg3415:

Rep:Mod:sg3415TS

2017-11-04T11:16:26Z

Sg3415: /* IRC calculations */

==Introduction==
Gaussian is a computational chemistry tool which enables efficient calculations of reaction coordinates across a potential energy surface. By differing the levels of approximations, Gaussian uses different quantum methods to calculate transition states to varying degrees of accuracy.

===Potential energy surface===
[[Image:SG PES image.JPG|right|thumb|Labelled potential energy surface <ref>D. S. Kaliakin, R. R. Zaari, S. A. Varganov, ''J. Chem. Educ.'', 2015, '''92''', 2106−2112.</ref>]]

A potential energy surface (PES) is a multi-dimensional plot of potential energy. The number of dimensions is the number of degrees of freedom, which are the normal modes. There are multiple possible reaction pathways to progress from reactants to products via a transition state, and deciding upon one relies on prior understanding of the approach trajectory and orientation of reactants.

=== Transition state ===
A transition state is the point of highest energy in the progression of a reaction. It is a second order saddle point with a gradient of zero, where the second derivative (corresponding to the force constant) is negative. This is relevant to computational theory because where a transition state is located, this vibrational frequency will appear negative. This is an imaginary number which resembles the vibration of the bonding atoms moving towards each other.

=== Minima ===
Potential energy surfaces contain both global and local minima. A local minimum is a low-energy geometry when compared to all surrounding points. This could be a reagent for example, which is generally a stable state of a compound, while not necessarily being the lowest energy substrate in the reaction. A global minimum point is the lowest energy point on the entire energy surface. This resembles the most stable state of any structure present in the reaction. At a minimum point, the surrounding gradient is zero and the second derivative is positive.

=== Method ===
In this experiment, three different methods were used. The first involved guessing the transition structure, which was optimised to give the transition state saddle point. Method 2 involved guessing the transition state of the reaction and freezing the atoms which form bonds in the process. This structure was then optimised to a minimum energy then used to find a transition structure by optimising. The third method began by drawing the either the reactants or products (the one with fewer atoms was chosen, here it was the product) and optimising. The bonds formed in the desired reaction were then broken and the atoms bonding were frozen. The process then continued as per method 2. For more complex structures, such as those in exercises 2 and 3, method 3 was the most successful in giving a good initial estimate for the transition state, saving time by not having to guess the structure.

Using Gaussian enables Slater functions to be replaced by a linear combination of Gaussian functions. This is similar to the method used to create molecular orbitals by linearly combining atomic orbitals. The simplification using Gaussian functions allows a relatively accurate calculation to be carried out more quickly.<ref>E. G. Lewars, ''Computational Chemistry: Introduction to the Theory And Applications of Molecular and Quantum Mechanics'', Springer, New York, 2003.</ref>

The concept of optimisation is that the computer runs a calculation to solve the Hamiltonian matrix. The geometry is then altered and the calculation is run again, until it gets closer to a solution in a step-wise fashion. PM6 is a semi-empirical method which works by using pre-set values for certain integrals. It therefore requires less calculations because it has made more approximations, making the tasks quicker to complete but less accurate. The other method used was a DFT-hybrid calculation, B3LYP, using the 6-31G(d) basis set. This is a more time-consuming but more accurate method as there are fewer approximations of integrals, which therefore produces results that are comparable to literature. DFT evaluates the Hamiltonian as a function of electron density. B3LYP uses both this and a Hartree-Fock calculation to calculate the exchange correlation contributions to the energy. It does not include correlations due to many-body interactions, but this level of complexity would make it more computationally expensive.<ref><nowiki>http://cmt.dur.ac.uk/sjc/thesis_prt/node23.html (viewed: 01/11/17)</nowiki></ref>

Exercises 1 and 3 used the PM6 method as this was sufficient for the relevant analysis. In exercise 2, where the required analysis was more quantitative (such as determining the energy of the reaction barrier), the B3LYP method was used after PM6 to further optimise the transition states to show the secondary orbital overlap more clearly. By doing these optimisations step-wise, it sped up the process. Crucially, the energies determined from either PM6 or B3LYP cannot be compared between the different levels of theory. IRC (intrinsic reaction coordinate) calculations were then used to confirm that the reaction has proceeded through the expected pathway to end up at the minimum point corresponding to the products.

== Exercise 1: Reaction of butadiene and ethene ==

Exercise 1 focussed on the reaction of butadiene with ethene in a <sub>π</sub>4<sub>s</sub> + <sub>π</sub>2<sub>s</sub> Diels-Alder cycloaddition, forming cyclohexene, as seen in the scheme below.

[[Image:SG ex1 Reaction Scheme.jpg|centre|thumb|Cycloaddition of butadiene with ethene]]

===MO analysis===
The MO diagram below shows the HOMO and LUMO MOs from each butadiene and ethene forming the four key MOs of the transition state. This was constructed by looking at the PM6 calculations of the reactants and the transition state. These are the HOMO-1, the HOMO, the LUMO and the LUMO+1. The HOMO-1 is slightly higher in energy than the HOMO of butadiene (despite being a new MO, which is usually lower in energy) because it is the transition state, which is at a maximum in terms of the reaction coordinate.

[[Image:SG4 Ex1 MO diagram.jpg|centre|thumb|MO diagram to show the formation of the transition state of the reaction between butadiene and ethene.]]

The Jmols below show that the HOMO of butadiene and the LUMO of ethene are both antisymmetric orbitals, and the LUMO of butadiene and the HOMO of ethene are both symmetric. Only orbitals of the same symmetry are “allowed” reactions because these have a non-zero orbital overlap integral. An overlap integral of zero (due to orbitals of different symmetry) means the reaction is “forbidden” and the orbitals therefore do not mix.
{| class="wikitable"
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|-
|[[File:TS HOMO-1.png|thumb|TS HOMO-1 (orbital 5)]]
|[[File:SG TS HOMO.png|thumb|TS HOMO (orbital 6)]]
|-
|[[File:TS LUMO.png|thumb|TS LUMO (orbital 7)]]
|[[File:TS LUMO+1.png|thumb|TS LUMO+1 (orbital 8)]]
|}

Normal electron demand Diels-Alder reactions occur between an electron rich diene and an electron deficient dienophile and occur between the diene HOMO and the dienophile LUMO. Inverse electron demand is the opposite.

In MO diagrams, the molecular orbital has the greater contribution from the atomic orbital that is closer in energy to the newly formed MO. This same principle applies in this situation, which is demonstrated by the MOs in the Jmols. For example, the LUMO of butadiene is further in energy from the transition state HOMO than the ethene HOMO, so it’s orbital contribution is smaller.

===Bond length analysis===
{| class="wikitable"
! rowspan="2" |C-C bond
! colspan="3" |Bond length / Å
|-
!Reactants
!TS
!Products
|-
|C<sub>1</sub>-C<sub>2</sub>
|1.333
|1.380
|1.501
|-
|C<sub>2</sub>-C<sub>3</sub>
|1.471
|1.411
|1.337
|-
|C<sub>3</sub>-C<sub>4</sub>
|1.333
|1.380
|1.501
|-
|C<sub>4</sub>-C<sub>5</sub>
|N/A
|2.116
|1.537
|-
|C<sub>5</sub>-C<sub>6</sub>
|1.327
|1.382
|1.535
|-
|C<sub>6</sub>-C<sub>1</sub>
|N/A
|2.113
|1.537
|-
| colspan="4" |Budadiene C=C bond: 1.454 Å, butadiene C-C bond: 1.338 Å, ethene C=C: 1.331 Å.<ref>N. C. Craig, P. Groner, and D. C. McKean, ''J. Phys. Chem.'', 2006, '''110''', 7461.</ref>
|}

C1-C2 and C3-C4 were previously the double bonds in butadiene, which originally started at 1.333 Å. In the product they have become sp<sup>3</sup> carbons forming single bonds, which lengthen from the reagent, through the transition state, to the product. The C5-C6 bond was the double bond of ethene and underwent the same transition. The C2-C3 bond was the single bond in butadiene, which became the double bond in the product. This was reflected by the shortening of the bond length from 1.471 A to 1.337 Å.

The Van der Waals radius of carbon is 1.7 Å.<ref>S. S. Batsanov, ''Inorganic Materials'', 2001, '''37(9),''' 871–885.</ref> For a new bond to be forming, double the Van der Waals radii should be less than the “bond” length in the transition state. For carbon, this would be 3.4 Å and the newly forming bonds have an average distance of 2.114 Å in the transition state, showing the orbitals are interacting and a bond is forming. The bond length becomes 1.537 Å, which is the typical C-C carbon single bond length.

[[Image:SG Ex1 TS negative freq vibrations.gif|right|thumb|Animation of the "negative" frequency, showing the bonding atoms moving towards each other]]

By undertaking a frequency calculation, it is possible to clarify that the transition state has been optimised is the single negative vibration, which corresponds to an imaginary number. This is due to the negative force constant when moving over the maxima of the transition state (with respect to the reaction coordinate) between the two local minima of the reactants and the products. This vibrational mode is shown below and demonstrates the bonding atoms moving towards each other at the same time, which is concordant with the concerted nature of a cycloaddition.

===Key file links===

[[Media:SG ETHYLENE MINIMUM 2.LOG|Ethene]]

[[Media:SG BUTADIENE CIS MINIMUM 2.LOG|Butadiene]]

[[Media:SG CYCLOHEXENE TS.LOG|Transition state]]

[[Media:SG_CYCLOHEXENE_PRODUCT_MINIMUM_2.LOG|Cyclohexene]]

== Exercise 2: Reaction of cyclohexadiene and 1,3-dioxole ==

Exercise two investigated the reaction of cyclohexadiene and 1,3-dioxole, as shown in the scheme below. This is another [4+2] cycloaddition, which can result in an exo or endo product, depending on the trajectory of approach of the reactants.

The endo-transition state is lower in energy due to stabilisation via both primary and secondary orbital interactions. Primary orbital interactions are those depicted by the dotted lines in the transition states in the schemes below. The secondary orbital interactions depend on the location of the oxygen atoms. Each O has a lone pair in a p-orbital, which, in the endo-TS is able to stabilise the structure by overlapping with the forming pi-bond and reducing its energy. This causes the endo-product to be the kinetic product as the reaction runs faster through this TS. The exo-product is often lower in energy due to the reduced steric clash, however in this reaction the endo-product is lower as the Hs are further away from each other, which reduces the energy.<ref>I. Flemming, ''Frontier Orbitals and Organic Chemical Reactions, ''Wiley, London, 1978, 29-109.</ref>

[[File:SG Ex2 reaction scheme.PNG|center|thumb|Reaction scheme showing the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole and the exo and endo transition states and products]]

===MO analysis===

[[File:SG4 Ex2 MO diagram.PNG|centre|thumb|MO diagram showing the formation of the endo- and exo- transition state orbitals (numbered 5-8)]]
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Exo transition state HOMO (orbital 6)

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Exo transition state LUMO (orbital 7)

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Endo transition state HOMO (orbital 5)

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</jmol></nowiki>
Endo transition state LUMO (orbital 8)
|}

Above is the MO diagram of the Diels-Alder exo- and endo- transition states and molecular orbitals for energy levels 5-8. The MO diagram has been constructed by running a single point energy calculation on the first frame of the IRC. This gives the energy levels of the orbitals of the fragments in relation to each other, enabling identification of the order of the fragment energies and so allowing confirmation that this is inverse electron demand.

The 1,3-dioxole orbitals have been raised in energy due to the oxygens donating their lone pairs to the double bond, making it a more electron rich dienophile. This causes the HOMO of the dienophile being a good energy match to interact with the LUMO of the diene, making the reaction inverse electron demand (which corresponds to the single point energy calculation). This overlap can be seen in the MO diagrams, where the LUMOs show a slightly bigger contribution from the diene LUMO, showing that this orbital is a little higher in energy than the HOMO of the dienophile.

In the Jmols of orbitals 5 and 6 (the endo- and exo-HOMOs), the secondary orbital interaction can only be seen in the endo-TS. It is between the oxygens and the pi-bond that is forming, and acts to stabilise this transition state greater than the exo-TS. The thermodynamics behind this has been explored in the section below.

===Thermodynamics analysis===
{| class="wikitable"
!Reaction
!Reaction barrier / kJ mol<sup>-1</sup>
!Reaction energy / kJ mol<sup>-1</sup>
|-
|Exo
|166.02
|<nowiki>-63.19</nowiki>
|-
|Endo
|158.26
|<nowiki>-66.75</nowiki>
|-
| colspan="3" |A table setting out the reaction barrier and the reaction energy for the formation of both the endo- and exo- products.
|}
The reaction barrier correlates to the activation energy of the process (the difference between the reactant and transition state energy) and the reaction energy is the change in free energy of the process. All numbers used are from B3LYP calculations because this enables comparison.

As explained the in introduction to exercise 2, the endo-TS is lower in energy than the exo-TS due to secondary orbital interactions. The endo-product is also lower in energy, which is expected due to its structure.

===Key file links===

[[Media:SG 13DIOXOLE MINIMUM BETTER.LOG|1,3-dioxole B3LYP]]

[[Media:SG CYCLOHEXADIENE MINIMUM BETTER.LOG|Cyclohexadiene B3LYP]]

[[Media:SG Ex2 EXO TS BETTER.LOG|Exo transition state B3LYP]]

[[Media:SG Ex2 ENDO TS BETTER.LOG|Endo transition state B3LYP]]

[[Media:SG_EXO_PRODUCT_MINIMUM_BETTER.LOG|Exo product B3LYP]]

[[Media:SG ENDO PRODUCT MINIMUM BETTER.LOG|Endo product B3LYP]]

== Exercise 3: Diels-Alder vs Cheletropic ==
[[File:SG Ex3 Reaction scheme.PNG|thumb|Possible reactions between Xylylene and SO<sub>2</sub>]]
This experiment aimed to investigate how the trajectory of approach affected the type of reaction that proceeds. The reaction scheme for the three potential pathways can be seen on the right. The possible products are the exo-, endo- and cheletropic.

According to IUPAC, a cheletropic reaction is a “cycloaddition across the terminal atoms of a fully conjugated system with formation of two new σ-bonds to a single atom of the reagent. There is a formal loss of one π-bond in the substrate and an increase in the coordination number of the relevant atom of the reagent.”<ref><nowiki>https://goldbook.iupac.org/html/C/C01014.html (viewed: 03/11/17)</nowiki></ref>

===IRC calculations===

Below are the IRC clips for the three reactions. The exo- and endo- reactions show that the C-O bond forms before the C-S bond. In the cheletropic reaction, the both bonds to the sulfur form at the same time, this is known as a synchronous reaction.
{| class="wikitable"
|[[Image:SG Chelotropic IRC.gif|thumb|centre|Animation of the cheletropic IRC (which shows the reaction progression)]]
|[[Image:SG Ex3 Exo IRC.gif|thumb|centre|Animation of the exo approach IRC (which shows the reaction progression)]]
|[[Image:SG Ex3 Endo IRC.gif|thumb|centre|Animation of the endo approach IRC (which shows the reaction progression)]]
|}

By looking at the IRCs, it is possible to see that the 6-membered ring that is present in the starting material becomes a structure where all Cs in the ring have a 1.5 bond order. This resembles an aromatic structure (as in the final product), which is extremely stable. This causes xylylene to be very unstable as it wants to react to form its aromatic counterpart, via a Diels-Alder reaction.

=== Thermodynamics ===
{| class="wikitable"
!Reaction
!Reaction barrier / kJ mol<sup>-1</sup>
!Reaction energy / kJ mol<sup>-1</sup>
|-
|Exo
|83.01
|<nowiki>-100.60</nowiki>
|-
|Endo
|79.07
|<nowiki>-99.96</nowiki>
|-
|Cheletropic
|101.17
|<nowiki>-156.38</nowiki>
|}

From the table, the kinetic and thermodynamic products can be determined. Here, the kinetic product is the endo-product because it has the lowest reaction barrier (i.e. the transition state energy is the lowest energy), again due to the secondary orbital overlap. The thermodynamic product is the product with the lowest energy, which is the cheletropic product. The chelotropic product is most stable because it still contains two S=O double bonds, which are very strong. The transition state for this reaction is higher in energy though due to the strain of forming a 5-membered ring (less favourable than a 6-membered ring).

===Alternative reaction pathways===

Cis-butadiene has a second diene site that can potentially react. By examining it’s reaction barrier, it is possible to explain why it does not react here. The exo- reaction gives an activation energy of 116.75 kJ mol-1 and the endo- reaction gives a value of 108.99 kJ mol-1. The reason for this is the lack of formation of the aromatic ring in the transition state structures, meaning they are less favourable and so have higher in energy. This would always produce a high energy product (relative to reaction at the other site, due to the lack of pi-delocalisation).

===Key file links===

[[Media:SG REACTANT 1 MINIMUM.LOG|Xylylene]]

[[Media:SG SO2 MINIMUM.LOG|SO2]]

[[Media:SG Ex3 EXO TS.LOG|Exo transition state]]

[[Media:SG Ex3 ENDO TS.LOG|Endo transition state]]

[[Media:SG Ex3 CHELE TS.LOG|Cheletropic transition state]]

[[Media:SG Ex3 EXO PRODUCT MINIMUM.LOG|Exo product]]

[[Media:SG Ex3 ENDO PRODUCT MINIMUM.LOG|Endo product]]

[[Media:SG CHELE PRODUCT MINIMUM.LOG|Cheletropic product]]

[[Media:SG Ex3 EXO TS IRC.LOG|Second diene site exo IRC]]

[[Media:SG Ex3 ENDO TS IRC.LOG|Second diene site endo IRC]]

==Conclusion==

==References==

Rep:Mod:sg3415TS

2017-11-03T18:36:28Z

Sg3415: /* Thermodynamics */

==Introduction==
Gaussian is a computational chemistry tool which enables efficient calculations of reaction coordinates across a potential energy surface. By differing the levels of approximations, Gaussian uses different quantum methods to calculate transition states to varying degrees of accuracy.

===Potential energy surface===
[[Image:SG PES image.JPG|right|thumb|Labelled potential energy surface <ref>D. S. Kaliakin, R. R. Zaari, S. A. Varganov, ''J. Chem. Educ.'', 2015, '''92''', 2106−2112.</ref>]]

A potential energy surface (PES) is a multi-dimensional plot of potential energy. The number of dimensions is the number of degrees of freedom, which are the normal modes. There are multiple possible reaction pathways to progress from reactants to products via a transition state, and deciding upon one relies on prior understanding of the approach trajectory and orientation of reactants.

=== Transition state ===
A transition state is the point of highest energy in the progression of a reaction. It is a second order saddle point with a gradient of zero, where the second derivative (corresponding to the force constant) is negative. This is relevant to computational theory because where a transition state is located, this vibrational frequency will appear negative. This is an imaginary number which resembles the vibration of the bonding atoms moving towards each other.

=== Minima ===
Potential energy surfaces contain both global and local minima. A local minimum is a low-energy geometry when compared to all surrounding points. This could be a reagent for example, which is generally a stable state of a compound, while not necessarily being the lowest energy substrate in the reaction. A global minimum point is the lowest energy point on the entire energy surface. This resembles the most stable state of any structure present in the reaction. At a minimum point, the surrounding gradient is zero and the second derivative is positive.

=== Method ===
In this experiment, three different methods were used. The first involved guessing the transition structure, which was optimised to give the transition state saddle point. Method 2 involved guessing the transition state of the reaction and freezing the atoms which form bonds in the process. This structure was then optimised to a minimum energy then used to find a transition structure by optimising. The third method began by drawing the either the reactants or products (the one with fewer atoms was chosen, here it was the product) and optimising. The bonds formed in the desired reaction were then broken and the atoms bonding were frozen. The process then continued as per method 2. For more complex structures, such as those in exercises 2 and 3, method 3 was the most successful in giving a good initial estimate for the transition state, saving time by not having to guess the structure.

Using Gaussian enables Slater functions to be replaced by a linear combination of Gaussian functions. This is similar to the method used to create molecular orbitals by linearly combining atomic orbitals. The simplification using Gaussian functions allows a relatively accurate calculation to be carried out more quickly.<ref>E. G. Lewars, ''Computational Chemistry: Introduction to the Theory And Applications of Molecular and Quantum Mechanics'', Springer, New York, 2003.</ref>

The concept of optimisation is that the computer runs a calculation to solve the Hamiltonian matrix. The geometry is then altered and the calculation is run again, until it gets closer to a solution in a step-wise fashion. PM6 is a semi-empirical method which works by using pre-set values for certain integrals. It therefore requires less calculations because it has made more approximations, making the tasks quicker to complete but less accurate. The other method used was a DFT-hybrid calculation, B3LYP, using the 6-31G(d) basis set. This is a more time-consuming but more accurate method as there are fewer approximations of integrals, which therefore produces results that are comparable to literature. DFT evaluates the Hamiltonian as a function of electron density. B3LYP uses both this and a Hartree-Fock calculation to calculate the exchange correlation contributions to the energy. It does not include correlations due to many-body interactions, but this level of complexity would make it more computationally expensive.<ref><nowiki>http://cmt.dur.ac.uk/sjc/thesis_prt/node23.html (viewed: 01/11/17)</nowiki></ref>

Exercises 1 and 3 used the PM6 method as this was sufficient for the relevant analysis. In exercise 2, where the required analysis was more quantitative (such as determining the energy of the reaction barrier), the B3LYP method was used after PM6 to further optimise the transition states to show the secondary orbital overlap more clearly. By doing these optimisations step-wise, it sped up the process. Crucially, the energies determined from either PM6 or B3LYP cannot be compared between the different levels of theory. IRC (intrinsic reaction coordinate) calculations were then used to confirm that the reaction has proceeded through the expected pathway to end up at the minimum point corresponding to the products.

== Exercise 1: Reaction of butadiene and ethene ==

Exercise 1 focussed on the reaction of butadiene with ethene in a <sub>π</sub>4<sub>s</sub> + <sub>π</sub>2<sub>s</sub> Diels-Alder cycloaddition, forming cyclohexene, as seen in the scheme below.

[[Image:SG ex1 Reaction Scheme.jpg|centre|thumb|Cycloaddition of butadiene with ethene]]

===MO analysis===
The MO diagram below shows the HOMO and LUMO MOs from each butadiene and ethene forming the four key MOs of the transition state. This was constructed by looking at the PM6 calculations of the reactants and the transition state. These are the HOMO-1, the HOMO, the LUMO and the LUMO+1. The HOMO-1 is slightly higher in energy than the HOMO of butadiene (despite being a new MO, which is usually lower in energy) because it is the transition state, which is at a maximum in terms of the reaction coordinate.

[[Image:SG4 Ex1 MO diagram.jpg|centre|thumb|MO diagram to show the formation of the transition state of the reaction between butadiene and ethene.]]

The Jmols below show that the HOMO of butadiene and the LUMO of ethene are both antisymmetric orbitals, and the LUMO of butadiene and the HOMO of ethene are both symmetric. Only orbitals of the same symmetry are “allowed” reactions because these have a non-zero orbital overlap integral. An overlap integral of zero (due to orbitals of different symmetry) means the reaction is “forbidden” and the orbitals therefore do not mix.
{| class="wikitable"
|+A table setting out the HOMO and LUMO of butadiene and ethene
|<nowiki><jmol>
<jmolApplet>
<title>Butadiene HOMO</title>
<color>white</color>
<size>200</size>
<script>frame 6; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>SG BUTADIENE CIS MINIMUM 2.LOG</uploadedFileContents></jmolApplet></jmol></nowiki>
|<nowiki><jmol>
<jmolApplet>
<title>Butadiene LUMO</title>
<color>white</color>
<size>200</size>
<script>frame 6; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>SG BUTADIENE CIS MINIMUM 2.LOG</uploadedFileContents></jmolApplet></jmol></nowiki>
|-
|<nowiki><jmol>
<jmolApplet>
<title>Ethene HOMO</title>
<color>white</color>
<size>200</size>
<script>frame 6; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>SG ETHYLENE MINIMUM 2.LOG</uploadedFileContents></jmolApplet></jmol></nowiki>
|<nowiki><jmol>
<jmolApplet>
<title>EtheneLUMO</title>
<color>white</color>
<size>200</size>
<script>frame 6; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>SG ETHYLENE MINIMUM 2.LOG</uploadedFileContents>
</jmolApplet>
</jmol></nowiki>
|-
|[[File:TS HOMO-1.png|thumb|TS HOMO-1 (orbital 5)]]
|[[File:SG TS HOMO.png|thumb|TS HOMO (orbital 6)]]
|-
|[[File:TS LUMO.png|thumb|TS LUMO (orbital 7)]]
|[[File:TS LUMO+1.png|thumb|TS LUMO+1 (orbital 8)]]
|}

Normal electron demand Diels-Alder reactions occur between an electron rich diene and an electron deficient dienophile and occur between the diene HOMO and the dienophile LUMO. Inverse electron demand is the opposite.

In MO diagrams, the molecular orbital has the greater contribution from the atomic orbital that is closer in energy to the newly formed MO. This same principle applies in this situation, which is demonstrated by the MOs in the Jmols. For example, the LUMO of butadiene is further in energy from the transition state HOMO than the ethene HOMO, so it’s orbital contribution is smaller.

===Bond length analysis===
{| class="wikitable"
! rowspan="2" |C-C bond
! colspan="3" |Bond length / Å
|-
!Reactants
!TS
!Products
|-
|C<sub>1</sub>-C<sub>2</sub>
|1.333
|1.380
|1.501
|-
|C<sub>2</sub>-C<sub>3</sub>
|1.471
|1.411
|1.337
|-
|C<sub>3</sub>-C<sub>4</sub>
|1.333
|1.380
|1.501
|-
|C<sub>4</sub>-C<sub>5</sub>
|N/A
|2.116
|1.537
|-
|C<sub>5</sub>-C<sub>6</sub>
|1.327
|1.382
|1.535
|-
|C<sub>6</sub>-C<sub>1</sub>
|N/A
|2.113
|1.537
|-
| colspan="4" |Budadiene C=C bond: 1.454 Å, butadiene C-C bond: 1.338 Å, ethene C=C: 1.331 Å.<ref>N. C. Craig, P. Groner, and D. C. McKean, ''J. Phys. Chem.'', 2006, '''110''', 7461.</ref>
|}

C1-C2 and C3-C4 were previously the double bonds in butadiene, which originally started at 1.333 Å. In the product they have become sp<sup>3</sup> carbons forming single bonds, which lengthen from the reagent, through the transition state, to the product. The C5-C6 bond was the double bond of ethene and underwent the same transition. The C2-C3 bond was the single bond in butadiene, which became the double bond in the product. This was reflected by the shortening of the bond length from 1.471 A to 1.337 Å.

The Van der Waals radius of carbon is 1.7 Å.<ref>S. S. Batsanov, ''Inorganic Materials'', 2001, '''37(9),''' 871–885.</ref> For a new bond to be forming, double the Van der Waals radii should be less than the “bond” length in the transition state. For carbon, this would be 3.4 Å and the newly forming bonds have an average distance of 2.114 Å in the transition state, showing the orbitals are interacting and a bond is forming. The bond length becomes 1.537 Å, which is the typical C-C carbon single bond length.

[[Image:SG Ex1 TS negative freq vibrations.gif|right|thumb|Animation of the "negative" frequency, showing the bonding atoms moving towards each other]]

By undertaking a frequency calculation, it is possible to clarify that the transition state has been optimised is the single negative vibration, which corresponds to an imaginary number. This is due to the negative force constant when moving over the maxima of the transition state (with respect to the reaction coordinate) between the two local minima of the reactants and the products. This vibrational mode is shown below and demonstrates the bonding atoms moving towards each other at the same time, which is concordant with the concerted nature of a cycloaddition.

===Key file links===

[[Media:SG ETHYLENE MINIMUM 2.LOG|Ethene]]

[[Media:SG BUTADIENE CIS MINIMUM 2.LOG|Butadiene]]

[[Media:SG CYCLOHEXENE TS.LOG|Transition state]]

[[Media:SG_CYCLOHEXENE_PRODUCT_MINIMUM_2.LOG|Cyclohexene]]

== Exercise 2: Reaction of cyclohexadiene and 1,3-dioxole ==

Exercise two investigated the reaction of cyclohexadiene and 1,3-dioxole, as shown in the scheme below. This is another [4+2] cycloaddition, which can result in an exo or endo product, depending on the trajectory of approach of the reactants.

The endo-transition state is lower in energy due to stabilisation via both primary and secondary orbital interactions. Primary orbital interactions are those depicted by the dotted lines in the transition states in the schemes below. The secondary orbital interactions depend on the location of the oxygen atoms. Each O has a lone pair in a p-orbital, which, in the endo-TS is able to stabilise the structure by overlapping with the forming pi-bond and reducing its energy. This causes the endo-product to be the kinetic product as the reaction runs faster through this TS. The exo-product is often lower in energy due to the reduced steric clash, however in this reaction the endo-product is lower as the Hs are further away from each other, which reduces the energy.<ref>I. Flemming, ''Frontier Orbitals and Organic Chemical Reactions, ''Wiley, London, 1978, 29-109.</ref>

[[File:SG Ex2 reaction scheme.PNG|center|thumb|Reaction scheme showing the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole and the exo and endo transition states and products]]

===MO analysis===

[[File:SG4 Ex2 MO diagram.PNG|centre|thumb|MO diagram showing the formation of the endo- and exo- transition state orbitals (numbered 5-8)]]
{| class="wikitable"
|
<nowiki><jmol>
<jmolApplet>
<color>white</color>
<size>500</size>
<script>frame 20; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>SG_Ex2_EXO_TS_BETTER.LOG</uploadedFileContents></jmolApplet></jmol></nowiki>
|<nowiki><jmol></jmolApplet>
</jmol>
Exo transition state HOMO (orbital 6)

<jmol>
<jmolApplet>
<color>white</color>
<size>500</size>
<script>frame 20; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>SG_Ex2_EXO_TS_BETTER.LOG</uploadedFileContents>
</jmol></nowiki>
|-
|<nowiki><jmol></jmolApplet>
</jmol>
Exo transition state LUMO (orbital 7)

<jmol>
<jmolApplet>
<color>white</color>
<size>500</size>
<script>frame 20; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>SG_Ex2_ENDO_TS_BETTER.LOG</uploadedFileContents>
</jmol></nowiki>
|<nowiki><jmol></jmolApplet>
</jmol>
Endo transition state HOMO (orbital 5)

<jmol>
<jmolApplet>
<color>white</color>
<size>500</size>
<script>frame 20; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>SG_Ex2_ENDO_TS_BETTER.LOG</uploadedFileContents>
</jmolApplet>
</jmol></nowiki>
Endo transition state LUMO (orbital 8)
|}

Above is the MO diagram of the Diels-Alder exo- and endo- transition states and molecular orbitals for energy levels 5-8. The MO diagram has been constructed by running a single point energy calculation on the first frame of the IRC. This gives the energy levels of the orbitals of the fragments in relation to each other, enabling identification of the order of the fragment energies and so allowing confirmation that this is inverse electron demand.

The 1,3-dioxole orbitals have been raised in energy due to the oxygens donating their lone pairs to the double bond, making it a more electron rich dienophile. This causes the HOMO of the dienophile being a good energy match to interact with the LUMO of the diene, making the reaction inverse electron demand (which corresponds to the single point energy calculation). This overlap can be seen in the MO diagrams, where the LUMOs show a slightly bigger contribution from the diene LUMO, showing that this orbital is a little higher in energy than the HOMO of the dienophile.

In the Jmols of orbitals 5 and 6 (the endo- and exo-HOMOs), the secondary orbital interaction can only be seen in the endo-TS. It is between the oxygens and the pi-bond that is forming, and acts to stabilise this transition state greater than the exo-TS. The thermodynamics behind this has been explored in the section below.

===Thermodynamics analysis===
{| class="wikitable"
!Reaction
!Reaction barrier / kJ mol<sup>-1</sup>
!Reaction energy / kJ mol<sup>-1</sup>
|-
|Exo
|166.02
|<nowiki>-63.19</nowiki>
|-
|Endo
|158.26
|<nowiki>-66.75</nowiki>
|-
| colspan="3" |A table setting out the reaction barrier and the reaction energy for the formation of both the endo- and exo- products.
|}
The reaction barrier correlates to the activation energy of the process (the difference between the reactant and transition state energy) and the reaction energy is the change in free energy of the process. All numbers used are from B3LYP calculations because this enables comparison.

As explained the in introduction to exercise 2, the endo-TS is lower in energy than the exo-TS due to secondary orbital interactions. The endo-product is also lower in energy, which is expected due to its structure.

===Key file links===

[[Media:SG 13DIOXOLE MINIMUM BETTER.LOG|1,3-dioxole B3LYP]]

[[Media:SG CYCLOHEXADIENE MINIMUM BETTER.LOG|Cyclohexadiene B3LYP]]

[[Media:SG Ex2 EXO TS BETTER.LOG|Exo transition state B3LYP]]

[[Media:SG Ex2 ENDO TS BETTER.LOG|Endo transition state B3LYP]]

[[Media:SG_EXO_PRODUCT_MINIMUM_BETTER.LOG|Exo product B3LYP]]

[[Media:SG ENDO PRODUCT MINIMUM BETTER.LOG|Endo product B3LYP]]

== Exercise 3: Diels-Alder vs Cheletropic ==
[[File:SG Ex3 Reaction scheme.PNG|thumb|Possible reactions between Xylylene and SO<sub>2</sub>]]
This experiment aimed to investigate how the trajectory of approach affected the type of reaction that proceeds. The reaction scheme for the three potential pathways can be seen on the right. The possible products are the exo-, endo- and cheletropic.

According to IUPAC, a cheletropic reaction is a “cycloaddition across the terminal atoms of a fully conjugated system with formation of two new σ-bonds to a single atom of the reagent. There is a formal loss of one π-bond in the substrate and an increase in the coordination number of the relevant atom of the reagent.”<ref><nowiki>https://goldbook.iupac.org/html/C/C01014.html (viewed: 03/11/17)</nowiki></ref>

===IRC calculations===

Below are the IRC clips for the three reactions. The exo- and endo- reactions show that the C-O bond forms before the C-S bond. In the cheletropic reaction, the both bonds to the sulfur form at the same time, this is known as a synchronous reaction.
{| class="wikitable"
|[[Image:SG Chelotropic IRC.gif|thumb|centre|Animation of the cheletropic IRC (which shows the reaction progression)]]
|[[Image:SG Ex3 Exo IRC.gif|thumb|centre|Animation of the exo approach IRC (which shows the reaction progression)]]
|[[Image:SG Ex3 Endo IRC.gif|thumb|centre|Animation of the endo approach IRC (which shows the reaction progression)]]
|}

By looking at the IRCs, it is possible to see that the 6-membered ring that is present in the starting material becomes a structure where all Cs in the ring have a 1.5 bond order. This resembles an aromatic structure (as in the final product), which is extremely stable. This causes xylylene to be very unstable as it wants to react to form its aromatic counterpart.

=== Thermodynamics ===
{| class="wikitable"
!Reaction
!Reaction barrier / kJ mol<sup>-1</sup>
!Reaction energy / kJ mol<sup>-1</sup>
|-
|Exo
|83.01
|<nowiki>-100.60</nowiki>
|-
|Endo
|79.07
|<nowiki>-99.96</nowiki>
|-
|Cheletropic
|101.17
|<nowiki>-156.38</nowiki>
|}

From the table, the kinetic and thermodynamic products can be determined. Here, the kinetic product is the endo-product because it has the lowest reaction barrier (i.e. the transition state energy is the lowest energy), again due to the secondary orbital overlap. The thermodynamic product is the product with the lowest energy, which is the cheletropic product. The chelotropic product is most stable because it still contains two S=O double bonds, which are very strong. The transition state for this reaction is higher in energy though due to the strain of forming a 5-membered ring (less favourable than a 6-membered ring).

===Alternative reaction pathways===

Cis-butadiene has a second diene site that can potentially react. By examining it’s reaction barrier, it is possible to explain why it does not react here. The exo- reaction gives an activation energy of 116.75 kJ mol-1 and the endo- reaction gives a value of 108.99 kJ mol-1. The reason for this is the lack of formation of the aromatic ring in the transition state structures, meaning they are less favourable and so have higher in energy. This would always produce a high energy product (relative to reaction at the other site, due to the lack of pi-delocalisation).

===Key file links===

[[Media:SG REACTANT 1 MINIMUM.LOG|Xylylene]]

[[Media:SG SO2 MINIMUM.LOG|SO2]]

[[Media:SG Ex3 EXO TS.LOG|Exo transition state]]

[[Media:SG Ex3 ENDO TS.LOG|Endo transition state]]

[[Media:SG Ex3 CHELE TS.LOG|Cheletropic transition state]]

[[Media:SG Ex3 EXO PRODUCT MINIMUM.LOG|Exo product]]

[[Media:SG Ex3 ENDO PRODUCT MINIMUM.LOG|Endo product]]

[[Media:SG CHELE PRODUCT MINIMUM.LOG|Cheletropic product]]

[[Media:SG Ex3 EXO TS IRC.LOG|Second diene site exo IRC]]

[[Media:SG Ex3 ENDO TS IRC.LOG|Second diene site endo IRC]]

==Conclusion==

==References==

Rep:Mod:sg3415TS

2017-11-03T18:35:34Z

2017-11-03T18:00:49Z

Sg3415: /* IRC calculations */

==Introduction==
Gaussian is a computational chemistry tool which enables efficient calculations of reaction coordinates across a potential energy surface. By differing the levels of approximations, Gaussian uses different quantum methods to calculate transition states to varying degrees of accuracy.

===Potential energy surface===
[[Image:SG PES image.JPG|right|thumb|Labelled potential energy surface <ref>D. S. Kaliakin, R. R. Zaari, S. A. Varganov, ''J. Chem. Educ.'', 2015, '''92''', 2106−2112.</ref>]]

A potential energy surface (PES) is a multi-dimensional plot of potential energy. The number of dimensions is the number of degrees of freedom, which are the normal modes. There are multiple possible reaction pathways to progress from reactants to products via a transition state, and deciding upon one relies on prior understanding of the approach trajectory and orientation of reactants.

=== Transition state ===
A transition state is the point of highest energy in the progression of a reaction. It is a second order saddle point with a gradient of zero, where the second derivative (corresponding to the force constant) is negative. This is relevant to computational theory because where a transition state is located, this vibrational frequency will appear negative. This is an imaginary number which resembles the vibration of the bonding atoms moving towards each other.

=== Minima ===
Potential energy surfaces contain both global and local minima. A local minimum is a low-energy geometry when compared to all surrounding points. This could be a reagent for example, which is generally a stable state of a compound, while not necessarily being the lowest energy substrate in the reaction. A global minimum point is the lowest energy point on the entire energy surface. This resembles the most stable state of any structure present in the reaction. At a minimum point, the surrounding gradient is zero and the second derivative is positive.

=== Method ===
In this experiment, three different methods were used. The first involved guessing the transition structure, which was optimised to give the transition state saddle point. Method 2 involved guessing the transition state of the reaction and freezing the atoms which form bonds in the process. This structure was then optimised to a minimum energy then used to find a transition structure by optimising. The third method began by drawing the either the reactants or products (the one with fewer atoms was chosen, here it was the product) and optimising. The bonds formed in the desired reaction were then broken and the atoms bonding were frozen. The process then continued as per method 2. For more complex structures, such as those in exercises 2 and 3, method 3 was the most successful in giving a good initial estimate for the transition state, saving time by not having to guess the structure.

Using Gaussian enables Slater functions to be replaced by a linear combination of Gaussian functions. This is similar to the method used to create molecular orbitals by linearly combining atomic orbitals. The simplification using Gaussian functions allows a relatively accurate calculation to be carried out more quickly.<ref>E. G. Lewars, ''Computational Chemistry: Introduction to the Theory And Applications of Molecular and Quantum Mechanics'', Springer, New York, 2003.</ref>

The concept of optimisation is that the computer runs a calculation to solve the Hamiltonian matrix. The geometry is then altered and the calculation is run again, until it gets closer to a solution in a step-wise fashion. PM6 is a semi-empirical method which works by using pre-set values for certain integrals. It therefore requires less calculations because it has made more approximations, making the tasks quicker to complete but less accurate. The other method used was a DFT-hybrid calculation, B3LYP, using the 6-31G(d) basis set. This is a more time-consuming but more accurate method as there are fewer approximations of integrals, which therefore produces results that are comparable to literature. DFT evaluates the Hamiltonian as a function of electron density. B3LYP uses both this and a Hartree-Fock calculation to calculate the exchange correlation contributions to the energy. It does not include correlations due to many-body interactions, but this level of complexity would make it more computationally expensive.<ref><nowiki>http://cmt.dur.ac.uk/sjc/thesis_prt/node23.html (viewed: 01/11/17)</nowiki></ref>

Exercises 1 and 3 used the PM6 method as this was sufficient for the relevant analysis. In exercise 2, where the required analysis was more quantitative (such as determining the energy of the reaction barrier), the B3LYP method was used after PM6 to further optimise the transition states to show the secondary orbital overlap more clearly. By doing these optimisations step-wise, it sped up the process. Crucially, the energies determined from either PM6 or B3LYP cannot be compared between the different levels of theory. IRC (intrinsic reaction coordinate) calculations were then used to confirm that the reaction has proceeded through the expected pathway to end up at the minimum point corresponding to the products.

== Exercise 1: Reaction of butadiene and ethene ==

Exercise 1 focussed on the reaction of butadiene with ethene in a <sub>π</sub>4<sub>s</sub> + <sub>π</sub>2<sub>s</sub> Diels-Alder cycloaddition, forming cyclohexene, as seen in the scheme below.

[[Image:SG ex1 Reaction Scheme.jpg|centre|thumb|Cycloaddition of butadiene with ethene]]

===MO analysis===
The MO diagram below shows the HOMO and LUMO MOs from each butadiene and ethene forming the four key MOs of the transition state. This was constructed by looking at the PM6 calculations of the reactants and the transition state. These are the HOMO-1, the HOMO, the LUMO and the LUMO+1. The HOMO-1 is slightly higher in energy than the HOMO of butadiene (despite being a new MO, which is usually lower in energy) because it is the transition state, which is at a maximum in terms of the reaction coordinate.

[[Image:SG4 Ex1 MO diagram.jpg|centre|thumb|MO diagram to show the formation of the transition state of the reaction between butadiene and ethene.]]

The Jmols below show that the HOMO of butadiene and the LUMO of ethene are both antisymmetric orbitals, and the LUMO of butadiene and the HOMO of ethene are both symmetric. Only orbitals of the same symmetry are “allowed” reactions because these have a non-zero orbital overlap integral. An overlap integral of zero (due to orbitals of different symmetry) means the reaction is “forbidden” and the orbitals therefore do not mix.
{| class="wikitable"
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|<nowiki><jmol>
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|<nowiki><jmol>
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|-
|<nowiki><jmol>
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<uploadedFileContents>SG ETHYLENE MINIMUM 2.LOG</uploadedFileContents></jmolApplet></jmol></nowiki>
|<nowiki><jmol>
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<uploadedFileContents>SG ETHYLENE MINIMUM 2.LOG</uploadedFileContents>
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|-
|[[File:TS HOMO-1.png|thumb|TS HOMO-1 (orbital 5)]]
|[[File:SG TS HOMO.png|thumb|TS HOMO (orbital 6)]]
|-
|[[File:TS LUMO.png|thumb|TS LUMO (orbital 7)]]
|[[File:TS LUMO+1.png|thumb|TS LUMO+1 (orbital 8)]]
|}

Normal electron demand Diels-Alder reactions occur between an electron rich diene and an electron deficient dienophile and occur between the diene HOMO and the dienophile LUMO. Inverse electron demand is the opposite.

In MO diagrams, the molecular orbital has the greater contribution from the atomic orbital that is closer in energy to the newly formed MO. This same principle applies in this situation, which is demonstrated by the MOs in the Jmols. For example, the LUMO of butadiene is further in energy from the transition state HOMO than the ethene HOMO, so it’s orbital contribution is smaller.

===Bond length analysis===
{| class="wikitable"
! rowspan="2" |C-C bond
! colspan="3" |Bond length / Å
|-
!Reactants
!TS
!Products
|-
|C<sub>1</sub>-C<sub>2</sub>
|1.333
|1.380
|1.501
|-
|C<sub>2</sub>-C<sub>3</sub>
|1.471
|1.411
|1.337
|-
|C<sub>3</sub>-C<sub>4</sub>
|1.333
|1.380
|1.501
|-
|C<sub>4</sub>-C<sub>5</sub>
|N/A
|2.116
|1.537
|-
|C<sub>5</sub>-C<sub>6</sub>
|1.327
|1.382
|1.535
|-
|C<sub>6</sub>-C<sub>1</sub>
|N/A
|2.113
|1.537
|-
| colspan="4" |Budadiene C=C bond: 1.454 Å, butadiene C-C bond: 1.338 Å, ethene C=C: 1.331 Å.<ref>N. C. Craig, P. Groner, and D. C. McKean, ''J. Phys. Chem.'', 2006, '''110''', 7461.</ref>
|}

C1-C2 and C3-C4 were previously the double bonds in butadiene, which originally started at 1.333 Å. In the product they have become sp<sup>3</sup> carbons forming single bonds, which lengthen from the reagent, through the transition state, to the product. The C5-C6 bond was the double bond of ethene and underwent the same transition. The C2-C3 bond was the single bond in butadiene, which became the double bond in the product. This was reflected by the shortening of the bond length from 1.471 A to 1.337 Å.

The Van der Waals radius of carbon is 1.7 Å.<ref>S. S. Batsanov, ''Inorganic Materials'', 2001, '''37(9),''' 871–885.</ref> For a new bond to be forming, double the Van der Waals radii should be less than the “bond” length in the transition state. For carbon, this would be 3.4 Å and the newly forming bonds have an average distance of 2.114 Å in the transition state, showing the orbitals are interacting and a bond is forming. The bond length becomes 1.537 Å, which is the typical C-C carbon single bond length.

[[Image:SG Ex1 TS negative freq vibrations.gif|right|thumb|Animation of the "negative" frequency, showing the bonding atoms moving towards each other]]

By undertaking a frequency calculation, it is possible to clarify that the transition state has been optimised is the single negative vibration, which corresponds to an imaginary number. This is due to the negative force constant when moving over the maxima of the transition state (with respect to the reaction coordinate) between the two local minima of the reactants and the products. This vibrational mode is shown below and demonstrates the bonding atoms moving towards each other at the same time, which is concordant with the concerted nature of a cycloaddition.

===Key file links===

[[Media:SG ETHYLENE MINIMUM 2.LOG|Ethene]]

[[Media:SG BUTADIENE CIS MINIMUM 2.LOG|Butadiene]]

[[Media:SG CYCLOHEXENE TS.LOG|Transition state]]

[[Media:SG_CYCLOHEXENE_PRODUCT_MINIMUM_2.LOG|Cyclohexene]]

== Exercise 2: Reaction of cyclohexadiene and 1,3-dioxole ==

Exercise two investigated the reaction of cyclohexadiene and 1,3-dioxole, as shown in the scheme below. This is another [4+2] cycloaddition, which can result in an exo or endo product, depending on the trajectory of approach of the reactants.

The endo-transition state is lower in energy due to stabilisation via both primary and secondary orbital interactions. Primary orbital interactions are those depicted by the dotted lines in the transition states in the schemes below. The secondary orbital interactions depend on the location of the oxygen atoms. Each O has a lone pair in a p-orbital, which, in the endo-TS is able to stabilise the structure by overlapping with the forming pi-bond and reducing its energy. This causes the endo-product to be the kinetic product as the reaction runs faster through this TS. The exo-product is often lower in energy due to the reduced steric clash, however in this reaction the endo-product is lower as the Hs are further away from each other, which reduces the energy.<ref>I. Flemming, ''Frontier Orbitals and Organic Chemical Reactions, ''Wiley, London, 1978, 29-109.</ref>

[[File:SG Ex2 reaction scheme.PNG|center|thumb|Reaction scheme showing the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole and the exo and endo transition states and products]]

===MO analysis===

[[File:SG4 Ex2 MO diagram.PNG|centre|thumb|MO diagram showing the formation of the endo- and exo- transition state orbitals (numbered 5-8)]]
{| class="wikitable"
|
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|<nowiki><jmol></jmolApplet>
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Exo transition state HOMO (orbital 6)

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Exo transition state LUMO (orbital 7)

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Endo transition state HOMO (orbital 5)

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Endo transition state LUMO (orbital 8)
|}

Above is the MO diagram of the Diels-Alder exo- and endo- transition states and molecular orbitals for energy levels 5-8. The MO diagram has been constructed by running a single point energy calculation on the first frame of the IRC. This gives the energy levels of the orbitals of the fragments in relation to each other, enabling identification of the order of the fragment energies and so allowing confirmation that this is inverse electron demand.

The 1,3-dioxole orbitals have been raised in energy due to the oxygens donating their lone pairs to the double bond, making it a more electron rich dienophile. This causes the HOMO of the dienophile being a good energy match to interact with the LUMO of the diene, making the reaction inverse electron demand (which corresponds to the single point energy calculation). This overlap can be seen in the MO diagrams, where the LUMOs show a slightly bigger contribution from the diene LUMO, showing that this orbital is a little higher in energy than the HOMO of the dienophile.

In the Jmols of orbitals 5 and 6 (the endo- and exo-HOMOs), the secondary orbital interaction can only be seen in the endo-TS. It is between the oxygens and the pi-bond that is forming, and acts to stabilise this transition state greater than the exo-TS. The thermodynamics behind this has been explored in the section below.

===Thermodynamics analysis===
{| class="wikitable"
!Reaction
!Reaction barrier / kJ mol<sup>-1</sup>
!Reaction energy / kJ mol<sup>-1</sup>
|-
|Exo
|166.02
|<nowiki>-63.19</nowiki>
|-
|Endo
|158.26
|<nowiki>-66.75</nowiki>
|-
| colspan="3" |A table setting out the reaction barrier and the reaction energy for the formation of both the endo- and exo- products.
|}
The reaction barrier correlates to the activation energy of the process (the difference between the reactant and transition state energy) and the reaction energy is the change in free energy of the process. All numbers used are from B3LYP calculations because this enables comparison.

As explained the in introduction to exercise 2, the endo-TS is lower in energy than the exo-TS due to secondary orbital interactions. The endo-product is also lower in energy, which is expected due to its structure.

===Key file links===

[[Media:SG 13DIOXOLE MINIMUM BETTER.LOG|1,3-dioxole B3LYP]]

[[Media:SG CYCLOHEXADIENE MINIMUM BETTER.LOG|Cyclohexadiene B3LYP]]

[[Media:SG Ex2 EXO TS BETTER.LOG|Exo transition state B3LYP]]

[[Media:SG Ex2 ENDO TS BETTER.LOG|Endo transition state B3LYP]]

[[Media:SG_EXO_PRODUCT_MINIMUM_BETTER.LOG|Exo product B3LYP]]

[[Media:SG ENDO PRODUCT MINIMUM BETTER.LOG|Endo product B3LYP]]

== Exercise 3: Diels-Alder vs Cheletropic ==
[[File:SG Ex3 Reaction scheme.PNG|thumb|Possible reactions between Xylylene and SO<sub>2</sub>]]
This experiment aimed to investigate how the trajectory of approach affected the type of reaction that proceeds. The reaction scheme for the three potential pathways can be seen on the right. The possible products are the exo-, endo- and cheletropic.

According to IUPAC, a cheletropic reaction is a “cycloaddition across the terminal atoms of a fully conjugated system with formation of two new σ-bonds to a single atom of the reagent. There is a formal loss of one π-bond in the substrate and an increase in the coordination number of the relevant atom of the reagent.”<ref><nowiki>https://goldbook.iupac.org/html/C/C01014.html (viewed: 03/11/17)</nowiki></ref>

===IRC calculations===

Below are the IRC clips for the three reactions. The exo- and endo- reactions show that the C-O bond forms before the C-S bond. In the cheletropic reaction, the both bonds to the sulfur form at the same time, this is known as a synchronous reaction.

[[Image:SG Chelotropic IRC.gif|thumb|Animation of the cheletropic IRC (which shows the reaction progression)]]
[[Image:SG Ex3 Exo IRC.gif|thumb|Animation of the exo approach IRC (which shows the reaction progression)]]
[[Image:SG Ex3 Endo IRC.gif|thumb|Animation of the endo approach IRC (which shows the reaction progression)]]

By looking at the IRCs, it is possible to see that the 6-membered ring that is present in the starting material becomes a structure where all Cs in the ring have a 1.5 bond order. This resembles an aromatic structure (as in the final product), which is extremely stable. This causes xylylene to be very unstable as it wants to react to form its aromatic counterpart.

=== Thermodynamics ===
{| class="wikitable"
!Reaction
!Reaction barrier / kJ mol<sup>-1</sup>
!Reaction energy / kJ mol<sup>-1</sup>
|-
|Exo
|83.01
|<nowiki>-100.60</nowiki>
|-
|Endo
|79.07
|<nowiki>-99.96</nowiki>
|-
|Cheletropic
|101.17
|<nowiki>-156.38</nowiki>
|}

From the table, the kinetic and thermodynamic products can be determined. Here, the kinetic product is the endo-product because it has the lowest reaction barrier (i.e. the transition state energy is the lowest energy), again due to the secondary orbital overlap. The thermodynamic product is the product with the lowest energy, which is the cheletropic product. The chelotropic product is most stable because it still contains two S=O double bonds, which are very strong. The transition state for this reaction is higher in energy though due to the strain of forming a 5-membered ring (less favourable than a 6-membered ring).

==Conclusion==

==References==

File:SG Ex3 Exo IRC.gif

2017-11-03T18:00:36Z

Sg3415:

File:SG Ex3 Endo IRC.gif

2017-11-03T17:59:46Z

Sg3415:

Rep:Mod:sg3415TS

2017-11-03T17:58:48Z

Sg3415: /* IRC calculations */

==Introduction==
Gaussian is a computational chemistry tool which enables efficient calculations of reaction coordinates across a potential energy surface. By differing the levels of approximations, Gaussian uses different quantum methods to calculate transition states to varying degrees of accuracy.

===Potential energy surface===
[[Image:SG PES image.JPG|right|thumb|Labelled potential energy surface <ref>D. S. Kaliakin, R. R. Zaari, S. A. Varganov, ''J. Chem. Educ.'', 2015, '''92''', 2106−2112.</ref>]]

A potential energy surface (PES) is a multi-dimensional plot of potential energy. The number of dimensions is the number of degrees of freedom, which are the normal modes. There are multiple possible reaction pathways to progress from reactants to products via a transition state, and deciding upon one relies on prior understanding of the approach trajectory and orientation of reactants.

=== Transition state ===
A transition state is the point of highest energy in the progression of a reaction. It is a second order saddle point with a gradient of zero, where the second derivative (corresponding to the force constant) is negative. This is relevant to computational theory because where a transition state is located, this vibrational frequency will appear negative. This is an imaginary number which resembles the vibration of the bonding atoms moving towards each other.

=== Minima ===
Potential energy surfaces contain both global and local minima. A local minimum is a low-energy geometry when compared to all surrounding points. This could be a reagent for example, which is generally a stable state of a compound, while not necessarily being the lowest energy substrate in the reaction. A global minimum point is the lowest energy point on the entire energy surface. This resembles the most stable state of any structure present in the reaction. At a minimum point, the surrounding gradient is zero and the second derivative is positive.

=== Method ===
In this experiment, three different methods were used. The first involved guessing the transition structure, which was optimised to give the transition state saddle point. Method 2 involved guessing the transition state of the reaction and freezing the atoms which form bonds in the process. This structure was then optimised to a minimum energy then used to find a transition structure by optimising. The third method began by drawing the either the reactants or products (the one with fewer atoms was chosen, here it was the product) and optimising. The bonds formed in the desired reaction were then broken and the atoms bonding were frozen. The process then continued as per method 2. For more complex structures, such as those in exercises 2 and 3, method 3 was the most successful in giving a good initial estimate for the transition state, saving time by not having to guess the structure.

Using Gaussian enables Slater functions to be replaced by a linear combination of Gaussian functions. This is similar to the method used to create molecular orbitals by linearly combining atomic orbitals. The simplification using Gaussian functions allows a relatively accurate calculation to be carried out more quickly.<ref>E. G. Lewars, ''Computational Chemistry: Introduction to the Theory And Applications of Molecular and Quantum Mechanics'', Springer, New York, 2003.</ref>

The concept of optimisation is that the computer runs a calculation to solve the Hamiltonian matrix. The geometry is then altered and the calculation is run again, until it gets closer to a solution in a step-wise fashion. PM6 is a semi-empirical method which works by using pre-set values for certain integrals. It therefore requires less calculations because it has made more approximations, making the tasks quicker to complete but less accurate. The other method used was a DFT-hybrid calculation, B3LYP, using the 6-31G(d) basis set. This is a more time-consuming but more accurate method as there are fewer approximations of integrals, which therefore produces results that are comparable to literature. DFT evaluates the Hamiltonian as a function of electron density. B3LYP uses both this and a Hartree-Fock calculation to calculate the exchange correlation contributions to the energy. It does not include correlations due to many-body interactions, but this level of complexity would make it more computationally expensive.<ref><nowiki>http://cmt.dur.ac.uk/sjc/thesis_prt/node23.html (viewed: 01/11/17)</nowiki></ref>

Exercises 1 and 3 used the PM6 method as this was sufficient for the relevant analysis. In exercise 2, where the required analysis was more quantitative (such as determining the energy of the reaction barrier), the B3LYP method was used after PM6 to further optimise the transition states to show the secondary orbital overlap more clearly. By doing these optimisations step-wise, it sped up the process. Crucially, the energies determined from either PM6 or B3LYP cannot be compared between the different levels of theory. IRC (intrinsic reaction coordinate) calculations were then used to confirm that the reaction has proceeded through the expected pathway to end up at the minimum point corresponding to the products.

== Exercise 1: Reaction of butadiene and ethene ==

Exercise 1 focussed on the reaction of butadiene with ethene in a <sub>π</sub>4<sub>s</sub> + <sub>π</sub>2<sub>s</sub> Diels-Alder cycloaddition, forming cyclohexene, as seen in the scheme below.

[[Image:SG ex1 Reaction Scheme.jpg|centre|thumb|Cycloaddition of butadiene with ethene]]

===MO analysis===
The MO diagram below shows the HOMO and LUMO MOs from each butadiene and ethene forming the four key MOs of the transition state. This was constructed by looking at the PM6 calculations of the reactants and the transition state. These are the HOMO-1, the HOMO, the LUMO and the LUMO+1. The HOMO-1 is slightly higher in energy than the HOMO of butadiene (despite being a new MO, which is usually lower in energy) because it is the transition state, which is at a maximum in terms of the reaction coordinate.

[[Image:SG4 Ex1 MO diagram.jpg|centre|thumb|MO diagram to show the formation of the transition state of the reaction between butadiene and ethene.]]

The Jmols below show that the HOMO of butadiene and the LUMO of ethene are both antisymmetric orbitals, and the LUMO of butadiene and the HOMO of ethene are both symmetric. Only orbitals of the same symmetry are “allowed” reactions because these have a non-zero orbital overlap integral. An overlap integral of zero (due to orbitals of different symmetry) means the reaction is “forbidden” and the orbitals therefore do not mix.
{| class="wikitable"
|+A table setting out the HOMO and LUMO of butadiene and ethene
|<nowiki><jmol>
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|<nowiki><jmol>
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|-
|<nowiki><jmol>
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|-
|[[File:TS HOMO-1.png|thumb|TS HOMO-1 (orbital 5)]]
|[[File:SG TS HOMO.png|thumb|TS HOMO (orbital 6)]]
|-
|[[File:TS LUMO.png|thumb|TS LUMO (orbital 7)]]
|[[File:TS LUMO+1.png|thumb|TS LUMO+1 (orbital 8)]]
|}

Normal electron demand Diels-Alder reactions occur between an electron rich diene and an electron deficient dienophile and occur between the diene HOMO and the dienophile LUMO. Inverse electron demand is the opposite.

In MO diagrams, the molecular orbital has the greater contribution from the atomic orbital that is closer in energy to the newly formed MO. This same principle applies in this situation, which is demonstrated by the MOs in the Jmols. For example, the LUMO of butadiene is further in energy from the transition state HOMO than the ethene HOMO, so it’s orbital contribution is smaller.

===Bond length analysis===
{| class="wikitable"
! rowspan="2" |C-C bond
! colspan="3" |Bond length / Å
|-
!Reactants
!TS
!Products
|-
|C<sub>1</sub>-C<sub>2</sub>
|1.333
|1.380
|1.501
|-
|C<sub>2</sub>-C<sub>3</sub>
|1.471
|1.411
|1.337
|-
|C<sub>3</sub>-C<sub>4</sub>
|1.333
|1.380
|1.501
|-
|C<sub>4</sub>-C<sub>5</sub>
|N/A
|2.116
|1.537
|-
|C<sub>5</sub>-C<sub>6</sub>
|1.327
|1.382
|1.535
|-
|C<sub>6</sub>-C<sub>1</sub>
|N/A
|2.113
|1.537
|-
| colspan="4" |Budadiene C=C bond: 1.454 Å, butadiene C-C bond: 1.338 Å, ethene C=C: 1.331 Å.<ref>N. C. Craig, P. Groner, and D. C. McKean, ''J. Phys. Chem.'', 2006, '''110''', 7461.</ref>
|}

C1-C2 and C3-C4 were previously the double bonds in butadiene, which originally started at 1.333 Å. In the product they have become sp<sup>3</sup> carbons forming single bonds, which lengthen from the reagent, through the transition state, to the product. The C5-C6 bond was the double bond of ethene and underwent the same transition. The C2-C3 bond was the single bond in butadiene, which became the double bond in the product. This was reflected by the shortening of the bond length from 1.471 A to 1.337 Å.

The Van der Waals radius of carbon is 1.7 Å.<ref>S. S. Batsanov, ''Inorganic Materials'', 2001, '''37(9),''' 871–885.</ref> For a new bond to be forming, double the Van der Waals radii should be less than the “bond” length in the transition state. For carbon, this would be 3.4 Å and the newly forming bonds have an average distance of 2.114 Å in the transition state, showing the orbitals are interacting and a bond is forming. The bond length becomes 1.537 Å, which is the typical C-C carbon single bond length.

[[Image:SG Ex1 TS negative freq vibrations.gif|right|thumb|Animation of the "negative" frequency, showing the bonding atoms moving towards each other]]

By undertaking a frequency calculation, it is possible to clarify that the transition state has been optimised is the single negative vibration, which corresponds to an imaginary number. This is due to the negative force constant when moving over the maxima of the transition state (with respect to the reaction coordinate) between the two local minima of the reactants and the products. This vibrational mode is shown below and demonstrates the bonding atoms moving towards each other at the same time, which is concordant with the concerted nature of a cycloaddition.

===Key file links===

[[Media:SG ETHYLENE MINIMUM 2.LOG|Ethene]]

[[Media:SG BUTADIENE CIS MINIMUM 2.LOG|Butadiene]]

[[Media:SG CYCLOHEXENE TS.LOG|Transition state]]

[[Media:SG_CYCLOHEXENE_PRODUCT_MINIMUM_2.LOG|Cyclohexene]]

== Exercise 2: Reaction of cyclohexadiene and 1,3-dioxole ==

Exercise two investigated the reaction of cyclohexadiene and 1,3-dioxole, as shown in the scheme below. This is another [4+2] cycloaddition, which can result in an exo or endo product, depending on the trajectory of approach of the reactants.

The endo-transition state is lower in energy due to stabilisation via both primary and secondary orbital interactions. Primary orbital interactions are those depicted by the dotted lines in the transition states in the schemes below. The secondary orbital interactions depend on the location of the oxygen atoms. Each O has a lone pair in a p-orbital, which, in the endo-TS is able to stabilise the structure by overlapping with the forming pi-bond and reducing its energy. This causes the endo-product to be the kinetic product as the reaction runs faster through this TS. The exo-product is often lower in energy due to the reduced steric clash, however in this reaction the endo-product is lower as the Hs are further away from each other, which reduces the energy.<ref>I. Flemming, ''Frontier Orbitals and Organic Chemical Reactions, ''Wiley, London, 1978, 29-109.</ref>

[[File:SG Ex2 reaction scheme.PNG|center|thumb|Reaction scheme showing the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole and the exo and endo transition states and products]]

===MO analysis===

[[File:SG4 Ex2 MO diagram.PNG|centre|thumb|MO diagram showing the formation of the endo- and exo- transition state orbitals (numbered 5-8)]]
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Exo transition state HOMO (orbital 6)

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Exo transition state LUMO (orbital 7)

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Endo transition state HOMO (orbital 5)

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Endo transition state LUMO (orbital 8)
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Above is the MO diagram of the Diels-Alder exo- and endo- transition states and molecular orbitals for energy levels 5-8. The MO diagram has been constructed by running a single point energy calculation on the first frame of the IRC. This gives the energy levels of the orbitals of the fragments in relation to each other, enabling identification of the order of the fragment energies and so allowing confirmation that this is inverse electron demand.

The 1,3-dioxole orbitals have been raised in energy due to the oxygens donating their lone pairs to the double bond, making it a more electron rich dienophile. This causes the HOMO of the dienophile being a good energy match to interact with the LUMO of the diene, making the reaction inverse electron demand (which corresponds to the single point energy calculation). This overlap can be seen in the MO diagrams, where the LUMOs show a slightly bigger contribution from the diene LUMO, showing that this orbital is a little higher in energy than the HOMO of the dienophile.

In the Jmols of orbitals 5 and 6 (the endo- and exo-HOMOs), the secondary orbital interaction can only be seen in the endo-TS. It is between the oxygens and the pi-bond that is forming, and acts to stabilise this transition state greater than the exo-TS. The thermodynamics behind this has been explored in the section below.

===Thermodynamics analysis===
{| class="wikitable"
!Reaction
!Reaction barrier / kJ mol<sup>-1</sup>
!Reaction energy / kJ mol<sup>-1</sup>
|-
|Exo
|166.02
|<nowiki>-63.19</nowiki>
|-
|Endo
|158.26
|<nowiki>-66.75</nowiki>
|-
| colspan="3" |A table setting out the reaction barrier and the reaction energy for the formation of both the endo- and exo- products.
|}
The reaction barrier correlates to the activation energy of the process (the difference between the reactant and transition state energy) and the reaction energy is the change in free energy of the process. All numbers used are from B3LYP calculations because this enables comparison.

As explained the in introduction to exercise 2, the endo-TS is lower in energy than the exo-TS due to secondary orbital interactions. The endo-product is also lower in energy, which is expected due to its structure.

===Key file links===

[[Media:SG 13DIOXOLE MINIMUM BETTER.LOG|1,3-dioxole B3LYP]]

[[Media:SG CYCLOHEXADIENE MINIMUM BETTER.LOG|Cyclohexadiene B3LYP]]

[[Media:SG Ex2 EXO TS BETTER.LOG|Exo transition state B3LYP]]

[[Media:SG Ex2 ENDO TS BETTER.LOG|Endo transition state B3LYP]]

[[Media:SG_EXO_PRODUCT_MINIMUM_BETTER.LOG|Exo product B3LYP]]

[[Media:SG ENDO PRODUCT MINIMUM BETTER.LOG|Endo product B3LYP]]

== Exercise 3: Diels-Alder vs Cheletropic ==
[[File:SG Ex3 Reaction scheme.PNG|thumb|Possible reactions between Xylylene and SO<sub>2</sub>]]
This experiment aimed to investigate how the trajectory of approach affected the type of reaction that proceeds. The reaction scheme for the three potential pathways can be seen on the right. The possible products are the exo-, endo- and cheletropic.

According to IUPAC, a cheletropic reaction is a “cycloaddition across the terminal atoms of a fully conjugated system with formation of two new σ-bonds to a single atom of the reagent. There is a formal loss of one π-bond in the substrate and an increase in the coordination number of the relevant atom of the reagent.”<ref><nowiki>https://goldbook.iupac.org/html/C/C01014.html (viewed: 03/11/17)</nowiki></ref>

===IRC calculations===

Below are the IRC clips for the three reactions. The exo- and endo- reactions show that the C-O bond forms before the C-S bond. In the cheletropic reaction, the both bonds to the sulfur form at the same time, this is known as a synchronous reaction.

[[Image:SG Chelotropic IRC.gif|right|thumb|Animation of the cheletropic IRC (which shows the reaction progression)]]

<nowiki>*</nowiki>insert gif IRCs*

By looking at the IRCs, it is possible to see that the 6-membered ring that is present in the starting material becomes a structure where all Cs in the ring have a 1.5 bond order. This resembles an aromatic structure (as in the final product), which is extremely stable. This causes xylylene to be very unstable as it wants to react to form its aromatic counterpart.

=== Thermodynamics ===
{| class="wikitable"
!Reaction
!Reaction barrier / kJ mol<sup>-1</sup>
!Reaction energy / kJ mol<sup>-1</sup>
|-
|Exo
|83.01
|<nowiki>-100.60</nowiki>
|-
|Endo
|79.07
|<nowiki>-99.96</nowiki>
|-
|Cheletropic
|101.17
|<nowiki>-156.38</nowiki>
|}

From the table, the kinetic and thermodynamic products can be determined. Here, the kinetic product is the endo-product because it has the lowest reaction barrier (i.e. the transition state energy is the lowest energy), again due to the secondary orbital overlap. The thermodynamic product is the product with the lowest energy, which is the cheletropic product. The chelotropic product is most stable because it still contains two S=O double bonds, which are very strong. The transition state for this reaction is higher in energy though due to the strain of forming a 5-membered ring (less favourable than a 6-membered ring).

==Conclusion==

==References==

File:SG Chelotropic IRC.gif

2017-11-03T17:57:38Z

Sg3415:

Rep:Mod:sg3415TS

2017-11-03T17:55:48Z

Sg3415: /* Exercise 3: Diels-Alder vs Cheletropic */

==Introduction==
Gaussian is a computational chemistry tool which enables efficient calculations of reaction coordinates across a potential energy surface. By differing the levels of approximations, Gaussian uses different quantum methods to calculate transition states to varying degrees of accuracy.

===Potential energy surface===
[[Image:SG PES image.JPG|right|thumb|Labelled potential energy surface <ref>D. S. Kaliakin, R. R. Zaari, S. A. Varganov, ''J. Chem. Educ.'', 2015, '''92''', 2106−2112.</ref>]]

A potential energy surface (PES) is a multi-dimensional plot of potential energy. The number of dimensions is the number of degrees of freedom, which are the normal modes. There are multiple possible reaction pathways to progress from reactants to products via a transition state, and deciding upon one relies on prior understanding of the approach trajectory and orientation of reactants.

=== Transition state ===
A transition state is the point of highest energy in the progression of a reaction. It is a second order saddle point with a gradient of zero, where the second derivative (corresponding to the force constant) is negative. This is relevant to computational theory because where a transition state is located, this vibrational frequency will appear negative. This is an imaginary number which resembles the vibration of the bonding atoms moving towards each other.

=== Minima ===
Potential energy surfaces contain both global and local minima. A local minimum is a low-energy geometry when compared to all surrounding points. This could be a reagent for example, which is generally a stable state of a compound, while not necessarily being the lowest energy substrate in the reaction. A global minimum point is the lowest energy point on the entire energy surface. This resembles the most stable state of any structure present in the reaction. At a minimum point, the surrounding gradient is zero and the second derivative is positive.

=== Method ===
In this experiment, three different methods were used. The first involved guessing the transition structure, which was optimised to give the transition state saddle point. Method 2 involved guessing the transition state of the reaction and freezing the atoms which form bonds in the process. This structure was then optimised to a minimum energy then used to find a transition structure by optimising. The third method began by drawing the either the reactants or products (the one with fewer atoms was chosen, here it was the product) and optimising. The bonds formed in the desired reaction were then broken and the atoms bonding were frozen. The process then continued as per method 2. For more complex structures, such as those in exercises 2 and 3, method 3 was the most successful in giving a good initial estimate for the transition state, saving time by not having to guess the structure.

Using Gaussian enables Slater functions to be replaced by a linear combination of Gaussian functions. This is similar to the method used to create molecular orbitals by linearly combining atomic orbitals. The simplification using Gaussian functions allows a relatively accurate calculation to be carried out more quickly.<ref>E. G. Lewars, ''Computational Chemistry: Introduction to the Theory And Applications of Molecular and Quantum Mechanics'', Springer, New York, 2003.</ref>

The concept of optimisation is that the computer runs a calculation to solve the Hamiltonian matrix. The geometry is then altered and the calculation is run again, until it gets closer to a solution in a step-wise fashion. PM6 is a semi-empirical method which works by using pre-set values for certain integrals. It therefore requires less calculations because it has made more approximations, making the tasks quicker to complete but less accurate. The other method used was a DFT-hybrid calculation, B3LYP, using the 6-31G(d) basis set. This is a more time-consuming but more accurate method as there are fewer approximations of integrals, which therefore produces results that are comparable to literature. DFT evaluates the Hamiltonian as a function of electron density. B3LYP uses both this and a Hartree-Fock calculation to calculate the exchange correlation contributions to the energy. It does not include correlations due to many-body interactions, but this level of complexity would make it more computationally expensive.<ref><nowiki>http://cmt.dur.ac.uk/sjc/thesis_prt/node23.html (viewed: 01/11/17)</nowiki></ref>

Exercises 1 and 3 used the PM6 method as this was sufficient for the relevant analysis. In exercise 2, where the required analysis was more quantitative (such as determining the energy of the reaction barrier), the B3LYP method was used after PM6 to further optimise the transition states to show the secondary orbital overlap more clearly. By doing these optimisations step-wise, it sped up the process. Crucially, the energies determined from either PM6 or B3LYP cannot be compared between the different levels of theory. IRC (intrinsic reaction coordinate) calculations were then used to confirm that the reaction has proceeded through the expected pathway to end up at the minimum point corresponding to the products.

== Exercise 1: Reaction of butadiene and ethene ==

Exercise 1 focussed on the reaction of butadiene with ethene in a <sub>π</sub>4<sub>s</sub> + <sub>π</sub>2<sub>s</sub> Diels-Alder cycloaddition, forming cyclohexene, as seen in the scheme below.

[[Image:SG ex1 Reaction Scheme.jpg|centre|thumb|Cycloaddition of butadiene with ethene]]

===MO analysis===
The MO diagram below shows the HOMO and LUMO MOs from each butadiene and ethene forming the four key MOs of the transition state. This was constructed by looking at the PM6 calculations of the reactants and the transition state. These are the HOMO-1, the HOMO, the LUMO and the LUMO+1. The HOMO-1 is slightly higher in energy than the HOMO of butadiene (despite being a new MO, which is usually lower in energy) because it is the transition state, which is at a maximum in terms of the reaction coordinate.

[[Image:SG4 Ex1 MO diagram.jpg|centre|thumb|MO diagram to show the formation of the transition state of the reaction between butadiene and ethene.]]

The Jmols below show that the HOMO of butadiene and the LUMO of ethene are both antisymmetric orbitals, and the LUMO of butadiene and the HOMO of ethene are both symmetric. Only orbitals of the same symmetry are “allowed” reactions because these have a non-zero orbital overlap integral. An overlap integral of zero (due to orbitals of different symmetry) means the reaction is “forbidden” and the orbitals therefore do not mix.
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|-
|[[File:TS HOMO-1.png|thumb|TS HOMO-1 (orbital 5)]]
|[[File:SG TS HOMO.png|thumb|TS HOMO (orbital 6)]]
|-
|[[File:TS LUMO.png|thumb|TS LUMO (orbital 7)]]
|[[File:TS LUMO+1.png|thumb|TS LUMO+1 (orbital 8)]]
|}

Normal electron demand Diels-Alder reactions occur between an electron rich diene and an electron deficient dienophile and occur between the diene HOMO and the dienophile LUMO. Inverse electron demand is the opposite.

In MO diagrams, the molecular orbital has the greater contribution from the atomic orbital that is closer in energy to the newly formed MO. This same principle applies in this situation, which is demonstrated by the MOs in the Jmols. For example, the LUMO of butadiene is further in energy from the transition state HOMO than the ethene HOMO, so it’s orbital contribution is smaller.

===Bond length analysis===
{| class="wikitable"
! rowspan="2" |C-C bond
! colspan="3" |Bond length / Å
|-
!Reactants
!TS
!Products
|-
|C<sub>1</sub>-C<sub>2</sub>
|1.333
|1.380
|1.501
|-
|C<sub>2</sub>-C<sub>3</sub>
|1.471
|1.411
|1.337
|-
|C<sub>3</sub>-C<sub>4</sub>
|1.333
|1.380
|1.501
|-
|C<sub>4</sub>-C<sub>5</sub>
|N/A
|2.116
|1.537
|-
|C<sub>5</sub>-C<sub>6</sub>
|1.327
|1.382
|1.535
|-
|C<sub>6</sub>-C<sub>1</sub>
|N/A
|2.113
|1.537
|-
| colspan="4" |Budadiene C=C bond: 1.454 Å, butadiene C-C bond: 1.338 Å, ethene C=C: 1.331 Å.<ref>N. C. Craig, P. Groner, and D. C. McKean, ''J. Phys. Chem.'', 2006, '''110''', 7461.</ref>
|}

C1-C2 and C3-C4 were previously the double bonds in butadiene, which originally started at 1.333 Å. In the product they have become sp<sup>3</sup> carbons forming single bonds, which lengthen from the reagent, through the transition state, to the product. The C5-C6 bond was the double bond of ethene and underwent the same transition. The C2-C3 bond was the single bond in butadiene, which became the double bond in the product. This was reflected by the shortening of the bond length from 1.471 A to 1.337 Å.

The Van der Waals radius of carbon is 1.7 Å.<ref>S. S. Batsanov, ''Inorganic Materials'', 2001, '''37(9),''' 871–885.</ref> For a new bond to be forming, double the Van der Waals radii should be less than the “bond” length in the transition state. For carbon, this would be 3.4 Å and the newly forming bonds have an average distance of 2.114 Å in the transition state, showing the orbitals are interacting and a bond is forming. The bond length becomes 1.537 Å, which is the typical C-C carbon single bond length.

[[Image:SG Ex1 TS negative freq vibrations.gif|right|thumb|Animation of the "negative" frequency, showing the bonding atoms moving towards each other]]

By undertaking a frequency calculation, it is possible to clarify that the transition state has been optimised is the single negative vibration, which corresponds to an imaginary number. This is due to the negative force constant when moving over the maxima of the transition state (with respect to the reaction coordinate) between the two local minima of the reactants and the products. This vibrational mode is shown below and demonstrates the bonding atoms moving towards each other at the same time, which is concordant with the concerted nature of a cycloaddition.

===Key file links===

[[Media:SG ETHYLENE MINIMUM 2.LOG|Ethene]]

[[Media:SG BUTADIENE CIS MINIMUM 2.LOG|Butadiene]]

[[Media:SG CYCLOHEXENE TS.LOG|Transition state]]

[[Media:SG_CYCLOHEXENE_PRODUCT_MINIMUM_2.LOG|Cyclohexene]]

== Exercise 2: Reaction of cyclohexadiene and 1,3-dioxole ==

Exercise two investigated the reaction of cyclohexadiene and 1,3-dioxole, as shown in the scheme below. This is another [4+2] cycloaddition, which can result in an exo or endo product, depending on the trajectory of approach of the reactants.

The endo-transition state is lower in energy due to stabilisation via both primary and secondary orbital interactions. Primary orbital interactions are those depicted by the dotted lines in the transition states in the schemes below. The secondary orbital interactions depend on the location of the oxygen atoms. Each O has a lone pair in a p-orbital, which, in the endo-TS is able to stabilise the structure by overlapping with the forming pi-bond and reducing its energy. This causes the endo-product to be the kinetic product as the reaction runs faster through this TS. The exo-product is often lower in energy due to the reduced steric clash, however in this reaction the endo-product is lower as the Hs are further away from each other, which reduces the energy.<ref>I. Flemming, ''Frontier Orbitals and Organic Chemical Reactions, ''Wiley, London, 1978, 29-109.</ref>

[[File:SG Ex2 reaction scheme.PNG|center|thumb|Reaction scheme showing the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole and the exo and endo transition states and products]]

===MO analysis===

[[File:SG4 Ex2 MO diagram.PNG|centre|thumb|MO diagram showing the formation of the endo- and exo- transition state orbitals (numbered 5-8)]]
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Exo transition state HOMO (orbital 6)

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Exo transition state LUMO (orbital 7)

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Endo transition state HOMO (orbital 5)

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Endo transition state LUMO (orbital 8)
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Above is the MO diagram of the Diels-Alder exo- and endo- transition states and molecular orbitals for energy levels 5-8. The MO diagram has been constructed by running a single point energy calculation on the first frame of the IRC. This gives the energy levels of the orbitals of the fragments in relation to each other, enabling identification of the order of the fragment energies and so allowing confirmation that this is inverse electron demand.

The 1,3-dioxole orbitals have been raised in energy due to the oxygens donating their lone pairs to the double bond, making it a more electron rich dienophile. This causes the HOMO of the dienophile being a good energy match to interact with the LUMO of the diene, making the reaction inverse electron demand (which corresponds to the single point energy calculation). This overlap can be seen in the MO diagrams, where the LUMOs show a slightly bigger contribution from the diene LUMO, showing that this orbital is a little higher in energy than the HOMO of the dienophile.

In the Jmols of orbitals 5 and 6 (the endo- and exo-HOMOs), the secondary orbital interaction can only be seen in the endo-TS. It is between the oxygens and the pi-bond that is forming, and acts to stabilise this transition state greater than the exo-TS. The thermodynamics behind this has been explored in the section below.

===Thermodynamics analysis===
{| class="wikitable"
!Reaction
!Reaction barrier / kJ mol<sup>-1</sup>
!Reaction energy / kJ mol<sup>-1</sup>
|-
|Exo
|166.02
|<nowiki>-63.19</nowiki>
|-
|Endo
|158.26
|<nowiki>-66.75</nowiki>
|-
| colspan="3" |A table setting out the reaction barrier and the reaction energy for the formation of both the endo- and exo- products.
|}
The reaction barrier correlates to the activation energy of the process (the difference between the reactant and transition state energy) and the reaction energy is the change in free energy of the process. All numbers used are from B3LYP calculations because this enables comparison.

As explained the in introduction to exercise 2, the endo-TS is lower in energy than the exo-TS due to secondary orbital interactions. The endo-product is also lower in energy, which is expected due to its structure.

===Key file links===

[[Media:SG 13DIOXOLE MINIMUM BETTER.LOG|1,3-dioxole B3LYP]]

[[Media:SG CYCLOHEXADIENE MINIMUM BETTER.LOG|Cyclohexadiene B3LYP]]

[[Media:SG Ex2 EXO TS BETTER.LOG|Exo transition state B3LYP]]

[[Media:SG Ex2 ENDO TS BETTER.LOG|Endo transition state B3LYP]]

[[Media:SG_EXO_PRODUCT_MINIMUM_BETTER.LOG|Exo product B3LYP]]

[[Media:SG ENDO PRODUCT MINIMUM BETTER.LOG|Endo product B3LYP]]

== Exercise 3: Diels-Alder vs Cheletropic ==
[[File:SG Ex3 Reaction scheme.PNG|thumb|Possible reactions between Xylylene and SO<sub>2</sub>]]
This experiment aimed to investigate how the trajectory of approach affected the type of reaction that proceeds. The reaction scheme for the three potential pathways can be seen on the right. The possible products are the exo-, endo- and cheletropic.

According to IUPAC, a cheletropic reaction is a “cycloaddition across the terminal atoms of a fully conjugated system with formation of two new σ-bonds to a single atom of the reagent. There is a formal loss of one π-bond in the substrate and an increase in the coordination number of the relevant atom of the reagent.”<ref><nowiki>https://goldbook.iupac.org/html/C/C01014.html (viewed: 03/11/17)</nowiki></ref>

===IRC calculations===

Below are the IRC clips for the three reactions. The exo- and endo- reactions show that the C-O bond forms before the C-S bond. In the cheletropic reaction, the both bonds to the sulfur form at the same time, this is known as a synchronous reaction.

<nowiki>*</nowiki>insert gif IRCs*

By looking at the IRCs, it is possible to see that the 6-membered ring that is present in the starting material becomes a structure where all Cs in the ring have a 1.5 bond order. This resembles an aromatic structure (as in the final product), which is extremely stable. This causes xylylene to be very unstable as it wants to react to form its aromatic counterpart.

=== Thermodynamics ===
{| class="wikitable"
!Reaction
!Reaction barrier / kJ mol<sup>-1</sup>
!Reaction energy / kJ mol<sup>-1</sup>
|-
|Exo
|83.01
|<nowiki>-100.60</nowiki>
|-
|Endo
|79.07
|<nowiki>-99.96</nowiki>
|-
|Cheletropic
|101.17
|<nowiki>-156.38</nowiki>
|}

From the table, the kinetic and thermodynamic products can be determined. Here, the kinetic product is the endo-product because it has the lowest reaction barrier (i.e. the transition state energy is the lowest energy), again due to the secondary orbital overlap. The thermodynamic product is the product with the lowest energy, which is the cheletropic product. The chelotropic product is most stable because it still contains two S=O double bonds, which are very strong. The transition state for this reaction is higher in energy though due to the strain of forming a 5-membered ring (less favourable than a 6-membered ring).

==Conclusion==

==References==

Rep:Mod:sg3415TS

2017-11-03T17:54:36Z

Sg3415: /* Key file links */

==Introduction==
Gaussian is a computational chemistry tool which enables efficient calculations of reaction coordinates across a potential energy surface. By differing the levels of approximations, Gaussian uses different quantum methods to calculate transition states to varying degrees of accuracy.

===Potential energy surface===
[[Image:SG PES image.JPG|right|thumb|Labelled potential energy surface <ref>D. S. Kaliakin, R. R. Zaari, S. A. Varganov, ''J. Chem. Educ.'', 2015, '''92''', 2106−2112.</ref>]]

A potential energy surface (PES) is a multi-dimensional plot of potential energy. The number of dimensions is the number of degrees of freedom, which are the normal modes. There are multiple possible reaction pathways to progress from reactants to products via a transition state, and deciding upon one relies on prior understanding of the approach trajectory and orientation of reactants.

=== Transition state ===
A transition state is the point of highest energy in the progression of a reaction. It is a second order saddle point with a gradient of zero, where the second derivative (corresponding to the force constant) is negative. This is relevant to computational theory because where a transition state is located, this vibrational frequency will appear negative. This is an imaginary number which resembles the vibration of the bonding atoms moving towards each other.

=== Minima ===
Potential energy surfaces contain both global and local minima. A local minimum is a low-energy geometry when compared to all surrounding points. This could be a reagent for example, which is generally a stable state of a compound, while not necessarily being the lowest energy substrate in the reaction. A global minimum point is the lowest energy point on the entire energy surface. This resembles the most stable state of any structure present in the reaction. At a minimum point, the surrounding gradient is zero and the second derivative is positive.

=== Method ===
In this experiment, three different methods were used. The first involved guessing the transition structure, which was optimised to give the transition state saddle point. Method 2 involved guessing the transition state of the reaction and freezing the atoms which form bonds in the process. This structure was then optimised to a minimum energy then used to find a transition structure by optimising. The third method began by drawing the either the reactants or products (the one with fewer atoms was chosen, here it was the product) and optimising. The bonds formed in the desired reaction were then broken and the atoms bonding were frozen. The process then continued as per method 2. For more complex structures, such as those in exercises 2 and 3, method 3 was the most successful in giving a good initial estimate for the transition state, saving time by not having to guess the structure.

Using Gaussian enables Slater functions to be replaced by a linear combination of Gaussian functions. This is similar to the method used to create molecular orbitals by linearly combining atomic orbitals. The simplification using Gaussian functions allows a relatively accurate calculation to be carried out more quickly.<ref>E. G. Lewars, ''Computational Chemistry: Introduction to the Theory And Applications of Molecular and Quantum Mechanics'', Springer, New York, 2003.</ref>

The concept of optimisation is that the computer runs a calculation to solve the Hamiltonian matrix. The geometry is then altered and the calculation is run again, until it gets closer to a solution in a step-wise fashion. PM6 is a semi-empirical method which works by using pre-set values for certain integrals. It therefore requires less calculations because it has made more approximations, making the tasks quicker to complete but less accurate. The other method used was a DFT-hybrid calculation, B3LYP, using the 6-31G(d) basis set. This is a more time-consuming but more accurate method as there are fewer approximations of integrals, which therefore produces results that are comparable to literature. DFT evaluates the Hamiltonian as a function of electron density. B3LYP uses both this and a Hartree-Fock calculation to calculate the exchange correlation contributions to the energy. It does not include correlations due to many-body interactions, but this level of complexity would make it more computationally expensive.<ref><nowiki>http://cmt.dur.ac.uk/sjc/thesis_prt/node23.html (viewed: 01/11/17)</nowiki></ref>

Exercises 1 and 3 used the PM6 method as this was sufficient for the relevant analysis. In exercise 2, where the required analysis was more quantitative (such as determining the energy of the reaction barrier), the B3LYP method was used after PM6 to further optimise the transition states to show the secondary orbital overlap more clearly. By doing these optimisations step-wise, it sped up the process. Crucially, the energies determined from either PM6 or B3LYP cannot be compared between the different levels of theory. IRC (intrinsic reaction coordinate) calculations were then used to confirm that the reaction has proceeded through the expected pathway to end up at the minimum point corresponding to the products.

== Exercise 1: Reaction of butadiene and ethene ==

Exercise 1 focussed on the reaction of butadiene with ethene in a <sub>π</sub>4<sub>s</sub> + <sub>π</sub>2<sub>s</sub> Diels-Alder cycloaddition, forming cyclohexene, as seen in the scheme below.

[[Image:SG ex1 Reaction Scheme.jpg|centre|thumb|Cycloaddition of butadiene with ethene]]

===MO analysis===
The MO diagram below shows the HOMO and LUMO MOs from each butadiene and ethene forming the four key MOs of the transition state. This was constructed by looking at the PM6 calculations of the reactants and the transition state. These are the HOMO-1, the HOMO, the LUMO and the LUMO+1. The HOMO-1 is slightly higher in energy than the HOMO of butadiene (despite being a new MO, which is usually lower in energy) because it is the transition state, which is at a maximum in terms of the reaction coordinate.

[[Image:SG4 Ex1 MO diagram.jpg|centre|thumb|MO diagram to show the formation of the transition state of the reaction between butadiene and ethene.]]

The Jmols below show that the HOMO of butadiene and the LUMO of ethene are both antisymmetric orbitals, and the LUMO of butadiene and the HOMO of ethene are both symmetric. Only orbitals of the same symmetry are “allowed” reactions because these have a non-zero orbital overlap integral. An overlap integral of zero (due to orbitals of different symmetry) means the reaction is “forbidden” and the orbitals therefore do not mix.
{| class="wikitable"
|+A table setting out the HOMO and LUMO of butadiene and ethene
|<nowiki><jmol>
<jmolApplet>
<title>Butadiene HOMO</title>
<color>white</color>
<size>200</size>
<script>frame 6; mo 11; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>SG BUTADIENE CIS MINIMUM 2.LOG</uploadedFileContents></jmolApplet></jmol></nowiki>
|<nowiki><jmol>
<jmolApplet>
<title>Butadiene LUMO</title>
<color>white</color>
<size>200</size>
<script>frame 6; mo 12; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>SG BUTADIENE CIS MINIMUM 2.LOG</uploadedFileContents></jmolApplet></jmol></nowiki>
|-
|<nowiki><jmol>
<jmolApplet>
<title>Ethene HOMO</title>
<color>white</color>
<size>200</size>
<script>frame 6; mo 6; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>SG ETHYLENE MINIMUM 2.LOG</uploadedFileContents></jmolApplet></jmol></nowiki>
|<nowiki><jmol>
<jmolApplet>
<title>EtheneLUMO</title>
<color>white</color>
<size>200</size>
<script>frame 6; mo 7; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>SG ETHYLENE MINIMUM 2.LOG</uploadedFileContents>
</jmolApplet>
</jmol></nowiki>
|-
|[[File:TS HOMO-1.png|thumb|TS HOMO-1 (orbital 5)]]
|[[File:SG TS HOMO.png|thumb|TS HOMO (orbital 6)]]
|-
|[[File:TS LUMO.png|thumb|TS LUMO (orbital 7)]]
|[[File:TS LUMO+1.png|thumb|TS LUMO+1 (orbital 8)]]
|}

Normal electron demand Diels-Alder reactions occur between an electron rich diene and an electron deficient dienophile and occur between the diene HOMO and the dienophile LUMO. Inverse electron demand is the opposite.

In MO diagrams, the molecular orbital has the greater contribution from the atomic orbital that is closer in energy to the newly formed MO. This same principle applies in this situation, which is demonstrated by the MOs in the Jmols. For example, the LUMO of butadiene is further in energy from the transition state HOMO than the ethene HOMO, so it’s orbital contribution is smaller.

===Bond length analysis===
{| class="wikitable"
! rowspan="2" |C-C bond
! colspan="3" |Bond length / Å
|-
!Reactants
!TS
!Products
|-
|C<sub>1</sub>-C<sub>2</sub>
|1.333
|1.380
|1.501
|-
|C<sub>2</sub>-C<sub>3</sub>
|1.471
|1.411
|1.337
|-
|C<sub>3</sub>-C<sub>4</sub>
|1.333
|1.380
|1.501
|-
|C<sub>4</sub>-C<sub>5</sub>
|N/A
|2.116
|1.537
|-
|C<sub>5</sub>-C<sub>6</sub>
|1.327
|1.382
|1.535
|-
|C<sub>6</sub>-C<sub>1</sub>
|N/A
|2.113
|1.537
|-
| colspan="4" |Budadiene C=C bond: 1.454 Å, butadiene C-C bond: 1.338 Å, ethene C=C: 1.331 Å.<ref>N. C. Craig, P. Groner, and D. C. McKean, ''J. Phys. Chem.'', 2006, '''110''', 7461.</ref>
|}

C1-C2 and C3-C4 were previously the double bonds in butadiene, which originally started at 1.333 Å. In the product they have become sp<sup>3</sup> carbons forming single bonds, which lengthen from the reagent, through the transition state, to the product. The C5-C6 bond was the double bond of ethene and underwent the same transition. The C2-C3 bond was the single bond in butadiene, which became the double bond in the product. This was reflected by the shortening of the bond length from 1.471 A to 1.337 Å.

The Van der Waals radius of carbon is 1.7 Å.<ref>S. S. Batsanov, ''Inorganic Materials'', 2001, '''37(9),''' 871–885.</ref> For a new bond to be forming, double the Van der Waals radii should be less than the “bond” length in the transition state. For carbon, this would be 3.4 Å and the newly forming bonds have an average distance of 2.114 Å in the transition state, showing the orbitals are interacting and a bond is forming. The bond length becomes 1.537 Å, which is the typical C-C carbon single bond length.

[[Image:SG Ex1 TS negative freq vibrations.gif|right|thumb|Animation of the "negative" frequency, showing the bonding atoms moving towards each other]]

By undertaking a frequency calculation, it is possible to clarify that the transition state has been optimised is the single negative vibration, which corresponds to an imaginary number. This is due to the negative force constant when moving over the maxima of the transition state (with respect to the reaction coordinate) between the two local minima of the reactants and the products. This vibrational mode is shown below and demonstrates the bonding atoms moving towards each other at the same time, which is concordant with the concerted nature of a cycloaddition.

===Key file links===

[[Media:SG ETHYLENE MINIMUM 2.LOG|Ethene]]

[[Media:SG BUTADIENE CIS MINIMUM 2.LOG|Butadiene]]

[[Media:SG CYCLOHEXENE TS.LOG|Transition state]]

[[Media:SG_CYCLOHEXENE_PRODUCT_MINIMUM_2.LOG|Cyclohexene]]

== Exercise 2: Reaction of cyclohexadiene and 1,3-dioxole ==

Exercise two investigated the reaction of cyclohexadiene and 1,3-dioxole, as shown in the scheme below. This is another [4+2] cycloaddition, which can result in an exo or endo product, depending on the trajectory of approach of the reactants.

The endo-transition state is lower in energy due to stabilisation via both primary and secondary orbital interactions. Primary orbital interactions are those depicted by the dotted lines in the transition states in the schemes below. The secondary orbital interactions depend on the location of the oxygen atoms. Each O has a lone pair in a p-orbital, which, in the endo-TS is able to stabilise the structure by overlapping with the forming pi-bond and reducing its energy. This causes the endo-product to be the kinetic product as the reaction runs faster through this TS. The exo-product is often lower in energy due to the reduced steric clash, however in this reaction the endo-product is lower as the Hs are further away from each other, which reduces the energy.<ref>I. Flemming, ''Frontier Orbitals and Organic Chemical Reactions, ''Wiley, London, 1978, 29-109.</ref>

[[File:SG Ex2 reaction scheme.PNG|center|thumb|Reaction scheme showing the Diels-Alder reaction between cyclohexadiene and 1,3-dioxole and the exo and endo transition states and products]]

===MO analysis===

[[File:SG4 Ex2 MO diagram.PNG|centre|thumb|MO diagram showing the formation of the endo- and exo- transition state orbitals (numbered 5-8)]]
{| class="wikitable"
|
<nowiki><jmol>
<jmolApplet>
<color>white</color>
<size>500</size>
<script>frame 20; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>SG_Ex2_EXO_TS_BETTER.LOG</uploadedFileContents></jmolApplet></jmol></nowiki>
|<nowiki><jmol></jmolApplet>
</jmol>
Exo transition state HOMO (orbital 6)

<jmol>
<jmolApplet>
<color>white</color>
<size>500</size>
<script>frame 20; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>SG_Ex2_EXO_TS_BETTER.LOG</uploadedFileContents>
</jmol></nowiki>
|-
|<nowiki><jmol></jmolApplet>
</jmol>
Exo transition state LUMO (orbital 7)

<jmol>
<jmolApplet>
<color>white</color>
<size>500</size>
<script>frame 20; mo 41; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>SG_Ex2_ENDO_TS_BETTER.LOG</uploadedFileContents>
</jmol></nowiki>
|<nowiki><jmol></jmolApplet>
</jmol>
Endo transition state HOMO (orbital 5)

<jmol>
<jmolApplet>
<color>white</color>
<size>500</size>
<script>frame 20; mo 42; mo nodots nomesh fill translucent; mo titleformat ""; set antialiasdisplay on</script>
<uploadedFileContents>SG_Ex2_ENDO_TS_BETTER.LOG</uploadedFileContents>
</jmolApplet>
</jmol></nowiki>
Endo transition state LUMO (orbital 8)
|}

Above is the MO diagram of the Diels-Alder exo- and endo- transition states and molecular orbitals for energy levels 5-8. The MO diagram has been constructed by running a single point energy calculation on the first frame of the IRC. This gives the energy levels of the orbitals of the fragments in relation to each other, enabling identification of the order of the fragment energies and so allowing confirmation that this is inverse electron demand.

The 1,3-dioxole orbitals have been raised in energy due to the oxygens donating their lone pairs to the double bond, making it a more electron rich dienophile. This causes the HOMO of the dienophile being a good energy match to interact with the LUMO of the diene, making the reaction inverse electron demand (which corresponds to the single point energy calculation). This overlap can be seen in the MO diagrams, where the LUMOs show a slightly bigger contribution from the diene LUMO, showing that this orbital is a little higher in energy than the HOMO of the dienophile.

In the Jmols of orbitals 5 and 6 (the endo- and exo-HOMOs), the secondary orbital interaction can only be seen in the endo-TS. It is between the oxygens and the pi-bond that is forming, and acts to stabilise this transition state greater than the exo-TS. The thermodynamics behind this has been explored in the section below.

===Thermodynamics analysis===
{| class="wikitable"
!Reaction
!Reaction barrier / kJ mol<sup>-1</sup>
!Reaction energy / kJ mol<sup>-1</sup>
|-
|Exo
|166.02
|<nowiki>-63.19</nowiki>
|-
|Endo
|158.26
|<nowiki>-66.75</nowiki>
|-
| colspan="3" |A table setting out the reaction barrier and the reaction energy for the formation of both the endo- and exo- products.
|}
The reaction barrier correlates to the activation energy of the process (the difference between the reactant and transition state energy) and the reaction energy is the change in free energy of the process. All numbers used are from B3LYP calculations because this enables comparison.

As explained the in introduction to exercise 2, the endo-TS is lower in energy than the exo-TS due to secondary orbital interactions. The endo-product is also lower in energy, which is expected due to its structure.

===Key file links===

[[Media:SG 13DIOXOLE MINIMUM BETTER.LOG|1,3-dioxole B3LYP]]

[[Media:SG CYCLOHEXADIENE MINIMUM BETTER.LOG|Cyclohexadiene B3LYP]]

[[Media:SG Ex2 EXO TS BETTER.LOG|Exo transition state B3LYP]]

[[Media:SG Ex2 ENDO TS BETTER.LOG|Endo transition state B3LYP]]

[[Media:SG_EXO_PRODUCT_MINIMUM_BETTER.LOG|Exo product B3LYP]]

[[Media:SG ENDO PRODUCT MINIMUM BETTER.LOG|Endo product B3LYP]]

== Exercise 3: Diels-Alder vs Cheletropic ==
[[File:SG Ex3 Reaction scheme.PNG|thumb|Possible reactions between Xylylene and SO<sub>2</sub>]]
This experiment aimed to investigate how the trajectory of approach affected the type of reaction that proceeds. The reaction scheme for the three potential pathways can be seen on the right. The possible products are the exo-, endo- and cheletropic.

According to IUPAC, a cheletropic reaction is a “cycloaddition across the terminal atoms of a fully conjugated system with formation of two new σ-bonds to a single atom of the reagent. There is a formal loss of one π-bond in the substrate and an increase in the coordination number of the relevant atom of the reagent.”

===IRC calculations===

Below are the IRC clips for the three reactions. The exo- and endo- reactions show that the C-O bond forms before the C-S bond. In the cheletropic reaction, the both bonds to the sulfur form at the same time, this is known as a synchronous reaction.

<nowiki>*</nowiki>insert gif IRCs*

By looking at the IRCs, it is possible to see that the 6-membered ring that is present in the starting material becomes a structure where all Cs in the ring have a 1.5 bond order. This resembles an aromatic structure (as in the final product), which is extremely stable. This causes xylylene to be very unstable as it wants to react to form its aromatic counterpart.

=== Thermodynamics ===
{| class="wikitable"
!Reaction
!Reaction barrier / kJ mol<sup>-1</sup>
!Reaction energy / kJ mol<sup>-1</sup>
|-
|Exo
|83.01
|<nowiki>-100.60</nowiki>
|-
|Endo
|79.07
|<nowiki>-99.96</nowiki>
|-
|Cheletropic
|101.17
|<nowiki>-156.38</nowiki>
|}

From the table, the kinetic and thermodynamic products can be determined. Here, the kinetic product is the endo-product because it has the lowest reaction barrier (i.e. the transition state energy is the lowest energy), again due to the secondary orbital overlap. The thermodynamic product is the product with the lowest energy, which is the cheletropic product. The chelotropic product is most stable because it still contains two S=O double bonds, which are very strong. The transition state for this reaction is higher in energy though due to the strain of forming a 5-membered ring (less favourable than a 6-membered ring).

==Conclusion==

==References==