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Rep:Mod:parkbom

2009-11-14T16:49:30Z

Byt07:

=Module 3 - The Computation of the Transition State=

==Introduction==

The '''Transition State''' is an energetic maximum within a given reaction coordinate and represents the point at which bond breaking and bond forming is occurring simultaneously. This is represented by dotted lines in formal notation and depicts the transient nature of the bond(s) in question in the transition state.

The analysis of potential energy surfaces are effective in finding transition states; the transition states can be described graphically as "dams" that represent energy metastability, that the reactants must "cross" in order to form the products.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Potential Energy Surface Reaction Coordinate'''<ref>http://www.ch.ic.ac.uk/motm/porphyrins/TSdiscovery.html</ref>||'''Cope Rearrangement'''<ref>http://www.ch.ic.ac.uk/local/organic/pericyclic/</ref>
|-
|[[Image:PES.gif]]||[[Image:Cope Arr.gif|350px]]
|}

From the point of view of computational chemistry, the methods of molecular mechanical structure determination that employed imaginary force fields are ineffective in the calculations concerning the transition state in larger molecules, as they do not define the dynamics of bond making and bond forming that are apparent in the transient nature of the transition state.

As such, in combination with molecular orbital based calculations, the following of the potential energy surface of a molecule can be used to locate and analyse the transition state.

This computational exercise concerns the rearrangement of ''1,5-Hexadiene'' in the '''Cope Rearrangment'''(above).

The Cope Rearrangement proceeds via a [3,3]-Sigmatropic shift involving the motion of 6 electrons and is thermally-driven.

In terms of transition states, the cope rearrangement can reportedly proceed via two checkpoints, the '''boat''' or the '''chair''':-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Chair'''||'''Boat'''
|-
|[[Image:pic2a.jpg]]||[[Image:pic2b.jpg]]
|}

The purpose of this exercise is to ascertain which reaction pathway is preferred by computation of each transition state in the elucidation of geometry, repective energies and other thermochemical data.

==''Exercise 1'' - Optimising the Reactants and Products==

In this section, the "''reactant''" '''1,5-hexadiene''' and the "''product''" '''1,5-hexadiene''' are modelled in correspondence to the cope rearrangement.

The flexible nature of 1,5-hexadiene means that there is an initial dilemma of likely conformation at rest.

Several starting conformations were thus taken and their geometries optimised to find the lowest energy conformer.

Initially, 1,5-hexadiene was taken, ensuring a roughly "'''anti'''" linkage in the centre of the molecule, so that the central C atoms '''3''' and '''4''' are approximately '''antiperiplanar''' to one another. This molecule was cleaned in the gaussview builder interface and then taken and optimised using the '''Hartree-Fock''' method with basis set '''3-21G''' through Gaussian.

'''b)''' similarly, a synclinal, or "gauche" version of 1,5-hexadiene was taken and optimised at the HF 3-21G level.

The energies of both conformers and also their symmetry were noted.

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|'''"''Anti''" 1,5-Hexadiene'''||'''"''Gauche''" 1,5-Hexadiene'''
|-align="center"
|[[Image:Bo Anti.jpg|420px]]||[[Image:Bo Gauche.jpg|400px]]
|}

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|'''Confomer'''||'''''Anti''-1,5-Hexadiene'''||'''''Gauche''-1,5-Hexadiene'''
|-align="center"
|'''Total Energy'''(a.u.)||-231.69254||-231.69266
|-align="center"
|'''Point Group'''||ci||None
|}

From a purely steric-orientated argument, the ''anti'' configuration should have a lower energy than the ''gauche'', as there is less steric hindrance from the two ends of the carbon chain 180 degrees from one another than next to each other in the gauche conformation.

As it so happens, the energy for the ''anti'' conformer is reportedly lower than that of the ''gauche'' conformer, but by a minute amount of roughly a ten-thousandth of a hartree ('''1 hartree = 4.3597482E-18 J''').

Even so, it was appreciated that the central C-C-C-C linkage of 1,5-hexadiene lends itself to quite a high degree of rotational freedom, and thus, further examples of conformational isomerism. This is reinforced by the fact of the two remarkably different conformers having remarkably similar energies, and points to the fact that 1,5-hexadiene experiences conformational ambiguity often<ref>Cope Rearrangement of 1,5-Hexadiene: Full Geometry Optimizations Using Analytic MR-CISD and MR-AQCC Gradient Methods, doi:10.1021/jp0259014</ref>.

As such, 2 other conformers were taken and optimised:-

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|colspan=4|'''<big>Different Conformers of 1,5-Hexadiene</big>'''
|-align="center"
|(''g120,a,g120'')-'''Anti1'''||(''g120,a,g-120'')-'''Anti2'''||(''g120,g-60,g120'')-'''Gauche1'''||(''g120,g60,g-120'')-'''Gauche3'''
|-align="center"
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Bo Anti1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Bo Anti2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Gauche 1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Gauche 3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-align="center"
|colspan=4|'''Energy'''(a.u.)
|-align="center"
| -231.69260213||-231.69253506||-231.69166702||-231.69266120
|-align="center"
|colspan=4|'''Point Group'''
|-align="center"
|C2||Ci||C2||C1
|}

The energies here from the HF 3-21G are similarly close, but contrary to what was expected, it was found that ''gauche3'' was actually the lowest in energy.

Beneath this, the ''anti'' conformers are found to be lower in energy than the other ''gauche'' conformer. This may be rationalised as the minimalisation of steric hindrance lowering the energy of the overall molecule, as the antiperiplanar groups maintain the least steric strain.

However, the '''''cis'''''-''gauche'' conformer is lowest in energy due to stereoelectronic arguments.

The '''π'''-bonds of the C=C bonds are lying criss-crossed over each other in this conformer. This promotes effectual π-molecular orbital overlap and hence stabilises the molecule.

Next, the anti2 conformer with ci symmetry and the gauche1 conformer was taken and further optimised under the DFT B3LYP method with a more sophisticated basis set, 6-31G(d).

The resulting optimised molecule was then compared with the earlier attempt in terms of energy.

A further frequency analysis was conducted in order to check that the optimisation has proceeded to a minimum, being the case when all frequencies were real and positive.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=5|'''Comparison of the HF 3-21G and DFT B3LYP 6-31G(d) Optimisations'''
|-align="center"
|'''Method'''||colspan=2|'''HF 3-21G'''||colspan=2|'''DFT B3LYP 6-31G(d)'''
|-align="center"
|'''Conformer'''||'''anti2'''||'''gauche1'''||'''anti2'''||'''gauche1'''
|-align="center"
|'''Total Energy'''(a.u.)||-231.69253506||-231.69166702||-231.61170616||-231.61068821
|-align="center"
|1-2 C=C Bond Length (A)||1.3162||1.3156||1.3335||1.3331
|-align="center"
|2-3 C-C Bond Length (A)||1.5091||1.5041||1.5083||1.5044
|-align="center"
|3-4 C-C Bond Length (A)||1.5527||1.5481||1.5508||1.5482
|-align="center"
|4-5 C-C Bond Length (A)||1.5091||1.5041||1.5083||1.5044
|-align="center"
|5-6 C=C Bond Length (A)||1.3163||1.3335||1.3157||1.3331
|-align="center"
|1-2-3-4 Dihedral (o)||114.66||118.49||123.93||124.24
|-align="center"
|2-3-4-5 Dihedral (o)||180.00||180.00||-64.18||-65.01
|-align="center"
|3-4-5-6 Dihedral (o)||-114.66||-118.49||123.93||124.24
|}

As can be seen here, the energies returned differ in a noticeable manner. However, comparison of energies computed under different basis sets is unbalanced.

As such, comparisons of the geometry are notably more reliable in discerning the computed differences.

The bond lengths differ by negligible amounts. However, it can be seen that the various dihedral angles have been "tightened" more to their ideal figures (closer to 120o for the anticlinal dihedrals.

As such, using a more sophisticated basis set ensures a more accurate approximation to reality.

The log file yielded the following thermochemical data:-

'''1)Sum of electronic and zero-point Energies= -234.469195'''

'''2)Sum of electronic and thermal Energies= -234.461847'''

'''3)Sum of electronic and thermal Enthalpies= -234.460903'''

'''4)Sum of electronic and thermal Free Energies= -234.500782'''

Theses values concern the 1)Potential Energy at 0K that includes the zero-point vibrational energy term 2)Potential Energy at 298.15K and 1atm and adds contributions from translational, rotational and vibrational energy 3)Energy including a correction for RT(H=E+RT) for dissociation reactions and 4)The Entropic contribution.

==''Exercise 2'' - Optimising the "Chair" and "Boat" Transition Structures==

The Cope Rearrangement has been found to proceed via one of two possible transition states with differing conformation, as mentioned in the introduction:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=2|'''<big>Cope Rearrangement Transition State Conformations</big>'''
|-align="center"
|''Chair''||''Boat''
|-align="center"
|[[Image:pic2a.jpg]]||[[Image:pic2b.jpg]]
|}

When the rearrangment is thought as one that is concerted and proceeds with synchronicity of bond breaking and formation, these two transition states come to mind.

There is indeed a third type of transition state for this reaction, dubbed the '''''Dewar''''' transition state, that reportedly proceeds via a diradical tight-chair intermediate species that was discovered by Dupuis et al.<ref>The Cope Rearrangement Revisited, DOI:10.1021/ja00026a007</ref>, and is thus NOT concerted.

===The Chair Transition State===

The chair transition state was initally synthesised using two "resonant" '''allyl''' ('''C3H5''') fragments in gaussview, which were optimised under the 3-21G basis set, and subsequently placed in appropriate positions relative to one another to be optimised into the transition state.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=2|'''Allyl Fragment('''C3H5''')'''
|-align="center"
|colspan=2|[[Image:Allyl Frag.jpg|275px]]
|-align="center"
|'''sp3''' Centre Bond Angle (o)||109.5
|-align="center"
|'''Allyl''' C-C Bond Angle (o)||124.3
|-align="center"
|'''C-C''' Bond Length (A)||1.54
|-align="center"
|'''C=C''' Bond Length (A)||1.34
|-align="center"
|'''Allyl C-C''' Bond Length (A)||1.39
|}

The optimised allyl C-C bond angle is greater than the normal sp3 bond angle of '''109.5o''' due to resonance.

More importantly, the allyl C-C bond length is between that of a single bond and a double bond.

This optimised allyl fragment was then taken, duplicated and placed, facing opposite directions, with roughly a 2.2A distance between the terminal carbons of each fragment to mimic the shape of the chair transition state.

Thus, a frequency + optimisation was run that directed gaussian to reach a '''TS(Berny)''', with the elucidation of force constants.

The optimisation was then repeated but this time using the '''"frozen" coordinate method''', effectively fixing the two pairs of ends at a distance of 2.2A.

Clearly, this method requires that the general structure of the transition state in question be known. However, by fixing the positions in such a way, a more accurate optimisation can be done that avoids divergence.

Finally, the optimisation was performed a third time, using a normal guess '''Hessian''' method, which involves calculation of the force constant matrix.

After the three optimisations were complete, the following data were obtained:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=3|'''<big>Optimisation of Chair Transition State</big>'''
|-align="center"
|colspan=3|'''Structure'''
|-align="center"
|'''TS(Berny)'''||'''Frozen Coordinate'''||'''Hessian'''
|-align="center"
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Berny.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Frozen.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>red</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Hessian.mol</uploadedFileContents>
</jmolApplet></jmol>
|-align="center"
|colspan=3|'''Energies''' (a.u.)
|-align="center"
| -231.61932||-231.61932||-231.69167
|}

The structures returned from the normal force-constant calculation optimisation and the frozen coordinate method are indistinguishable from first glance, and their energies are virtually identical. However, the structure obtained from the Hessian is different, and so is its energy value.

'''FREQUENCY ANALYSIS'''

Vibration-wise, there was one negative, and thus "imaginary" frequency found with a wavenumber of '''-817.91cm-1'''.

When the stretch was visualised, the following was observed:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Imaginary Stretch at -817.91cm-1'''
|-
|[[Image:Imag Stretch.jpg|300px]]
|}

The appearance of this imaginary frequency resembles the motion that would ensue under the Cope Rearrangment - the action of bond breaking of the C-C σ bond in the middle of the 1,5-hexadiene with the subsequent formation of a new σ bond on across the terminals on the other side.

The fact that there is only '''one''' imaginary frequency value points to the notion that the cope rearrangment through this transition state is ''concerted'', and thus occurs in one ''synchronous'' step.

The number of imaginary frequencies produced from a frequency analysis for the midpoint of a reaction points towards the ''synchronicity'' of the reaction<ref>Cope Rearrangement Revisited, DOI:10.1021/ja00026a007</ref>.

===The Boat Transition State===

For the boat transition state, a different method was utilised, namely the '''QST2 method''', whereby reactant and product are defined, and the transition state is converged from these.

This required the previously optimised '''1,5-hexadiene''' molecule to be taken and numbered along its carbon skeleton.

The molecule was then duplicated and juxtaposed to the intial "reactant" molecule and renumbered in such a way to reflect the reacted, rearragement "product" as shown below:-

{|style="margin:1em auto 1em auto" border="1"
|-
|[[Image:Boat Method.JPG|400px]]
|}

The first instance of optimisation failed to produce an output, and thus the molecule was changed in the following way:-

The dihedral angle of both reactant and product were changed to 0 degrees, such that the two carbon groups either side of C3 and C4 are eclipsed, and the C2-C3-C4 and C3-C4-C5 bond angles were both changed to 100 degrees.

The modified structures were once again optimised under the QST2 method to yield the following structure:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''''QST2'' Optimised Boat Transition State'''
|-
|[[Image:FINAL Boat.jpg|400px]]
|}

'''DISCUSSION'''

Here, it is illustrated that the QST2 method is automated, and is thus very efficient, but is only effective if the shape provided of the reactants and products cloely mirror that of the transition state, otherwise convergence is difficult.

The '''QST3''' method was also attempted <ref>QST Calculation, http://hdl.handle.net/10042/to-2902</ref>, whereby an addition guess of the transition state geometry was implemented. If the guess of the transition state is sound, then the QST3 is by far more accurate than the QST2 method, and does not rely so much on the shape of the reactants and products.

initally, it would seem that in order for the cope rearrangement to occur, to two carbon terminals would be required to be close to one another. This is not the case with the "''anti''" conformers of the molecule, as the terminals are situated as far apart as they may possibly be, '''antiperplanar''' to one another.

Once free rotation moves the carbon groups closer together, into the gauche configuration, where the terminals are more poised to react, cope rearrangement would proceed.

The relative orientations of the vinyl groups would ultimately dictate which transition state would be traversed. For instance, if the vectors of the vinyl groups cross, such as the case for the gauche1 conformer, then as the terminals align into a bonding orientaton, the ''chair'' conformation would be adopted by the transition state.

Conversely, if the two vinyl groups lie in a pseudo-parallel fashion, where their vectors do NOT cross, such as is the case for gauche3, then the transition state would be of ''boat'' topography.

'''IRC'''

The Intrinsic Reaction Coordinate looks more cloesly at the potential surface of a reaction and works to pinpoint the highest point in a local area of the potential surface by creeping up the steepest slopes in the immediate area.

This calculation was attempted with 50 points, and did not converge within the maximum number of points.

Thus, this was carried out again whilst calculating the force constant every single point.

{|style="margin:1em auto 1em auto" border="1"
|-
|[[Image:BO_IRC.JPG]]
|}

Here, upon closer inspection of the RMS Gradient of the energy, the gradient falls to zero, thus indicating that a minimum has been reached.

'''Energy Calculation'''

The two transition states were optimised a final time under DFT B3LYP 6-31G(d) conditions.

The geometries that were outputted were very similar to the previous examples, however, the energies received for both transition states are notably more varied:-

'''Final Energy of Chair Conformation''' (a.u.) = -234.61071
'''Final Energy of Boat Conformation''' (a.u.) = -234.54309

Here, it can be conclusively said that the ''Boat'' conformation transition state is indeed lower in energy than the ''Chair'' conformation.

This corresponds to better orbital overlap of the adjacent carbons on the framework, which are exactly eclipsed and not staggered like in the chair formation.

==The Diels-Alder Cycloaddition==

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Diels-Alder Reaction of Butadiene and Ethylene'''
|-align="center"
|[[Image:Mb_da3.jpg|400px]]
|}

The Diels-Alder Reaction is an example of a pericyclic cycloaddition, that usually involves the reaction between an electron rich diene and an electron poor dienophile.

The process is described as a [4s + 2s], 6 electron process that forms 2 new σ bonds.

When viewing the molecular orbital picture of butadiene and ethylene, the molecular orbitals are classified as either '''''s'''''(''symmetric'') or '''''a'''''(''antisymmetric'') in relation to the bisecting plane as shown below.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Bisecting Plane for Diels-alder'''
|-
|[[Image:Mb_da2.jpg|200px]]
|}

===Investigation of Molecular Orbital Symmetries of Butadiene and Ethene===

Butadiene and Ethylene were optimised and their molecular orbitals visualised:-

'''''Butadiene'''''

The HOMO and LUMO are considered with reference to the bisecting plane to determine their symmetry:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=4|'''HOMO and LUMO of Butadiene'''
|-align="center"
|width=250px|'''Molecular Orbital'''||'''2D Representation'''||'''3D Representation'''||width=250px|'''s/a'''
|-align="center"
|'''HOMO'''||[[Image:BDE HOMO.png|200px]]||[[Image:3D BDE HOMO.jpg|300px]]||a
|-align="center"
|'''LUMO'''||[[Image:BDE LUMO.png|200px]]||[[Image:3D BDE LUMO.jpg|300px]]||s
|}

'''''Ethene'''''

Similarly, the HOMO and LUMO of ethylene were characterised:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=4|'''HOMO and LUMO of Ethene'''
|-align="center"
|width=250px|'''Molecular Orbital'''||'''2D Representation'''||'''3D Representation'''||width=250px|'''s/a'''
|-align="center"
|'''HOMO'''||[[Image:ET HOMO.jpg|200px]]||[[Image:3D ET HOMO.jpg|300px]]||s
|-align="center"
|'''LUMO'''||[[Image:ET LUMO.jpg|200px]]||[[Image:3D ET LUMO.jpg|300px]]||a
|}

===Optimisation of Diels-Alder Transition State===

The '''Transition State''' whereby the concerted cycloaddition of butadiene and ethylene was modelled to a guess initially, before being optimised under HF 3-21G basis set conditions to a TS(Berny).

The initial guess was modelled from the bicyclo-system of cyclohexene and subsequently removing one of the -CH2CH2- arches to arrive at an "''envelope''"-shaped 6-membered ring.

The new σ bonds were stripped before the optimisation.

A frequency analysis was first concocted via optimisation to a transition state.

There were 3 imaginary frequencies reported by the analysis, detailed below:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=4|'''Imaginary Frequency Vibrations of Transition State'''
|-align="center"
|'''Stretches'''||[[Image:CHEX VIB 1.jpg|250px]]|||[[Image:CHEX VIB 2.jpg|250px]]|||[[Image:CHEX VIB 3.jpg|250px]]
|-align="center"
|'''Frequency''' (cm-1)||-716.19||-557.32||-198.58
|}

The first imaginary frequency at -716.19cm-1 did not correspond to any particular action within the transition state.

However, it can be seen that the 2nd and 3rd vibrations tie into the formation of the two conformations of cyclohexene, i.e. the ''chair'' and the ''boat''.

The presence of 3 imaginary frequencies in the analysis points to the notion that the formation of the two bonds is asynchronous.

The average of the inter-C-terminal distances was then calculated and used in a separate ''frozen-coordinate'' optimisation to a minimum, to get a more accurate representation of the geometry:-

{|style="margin=1em auto 1em auto" border="1"
|-align="center"
|colspan=3|'''Comparisons of the Optimisations of transition state'''
|-align="center"
|'''Method'''||'''Ts(Berny)'''||'''Frozen-Coordinate'''
|-align="center"
|'''Structure'''||[[Image:CHEX BERNY.jpg|300px]]||[[Image:CHEX FROZEN.jpg|300px]]
|-align="center"
|'''Energy'''(a.u.)||-230.30655||-231.67157
|}

The length of the partly formed σ C-C bonds were optimised to a value of '''1.5075A''', which is notably in between the lengths of sp2 and sp3 bond lengths of '''1.34A''' and '''1.54A''' and is a result of the point of breaking of the double bonds in the diene and formation of new σ bonds between the butadiene and ethene.

The Van-der-Waals radius of ''carbon'' is '''1.70A'''.

By freezing the terminals, the energy of the optimised structure is markedly higher.

'''AM1 Molecular Orbital Analysis'''

The AM1 semi-empirical method was used to visualise the highest-occupied molecular orbital of the transition state.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''''HOMO'''''||'''''LUMO'''''
|-align="center"
|[[Image:CHEX TS HOMO.jpg|300px]]||[[Image:CHEX TS LUMO.jpg|300px]]
|-align="center"
|colspan=2|'''Symmetry'''
|-align="center"
|s||a
|-align="center"
|}

With reference to the bisecting plane as shown above, the HOMO of the transition state can be seen to be symmetric when reflected.

The individual MOs of butadiene and ethene thus must both be symmetric with respect to the plane in order to react in such a fashion.

This is consistent with the idea that only molecular orbitals with the same symmetry are allowed to react.

Given this, the pertinent orbitals that are symmetric are the '''LUMO''' of the butadiene and the '''HOMO''' of the ethene.

As such, electron density is donated into the empty π* LUMO of the butadiene.

The LUMO in this case is antisymmetric with respect to the bisecting plane and corresponds to the '''HOMO''' of the butadiene and the '''LUMO''' of the ethene.

===Diels-Alder Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride===

The Diels-Alder reaction of '''Cyclohexa-1,3-diene and Maleic Anhydride''' is also an example of a 6-electron [4+2] Diels-Alder Cycloaddition intrinsically.

However, it differs from the reaction between butadiene and ethene in that there is now a question of regioselectivity; the bigger anhydride ring can either react to adopt an '''EXO''' or an '''ENDO''' topography:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|[[Image:EXO ENDO.jpg|400px]]
|}

The vinyl goup on maleic anhydride, in comparison to ethene, is electron-deficient, thanks to the highly electron-withdrawing nature of the anhydride group.

==References==

<references />

Rep:Mod:parkbom

2009-11-14T16:48:48Z

Byt07:

=Module 3 - The Computation of the Transition State=

==Introduction==

The '''Transition State''' is an energetic maximum within a given reaction coordinate and represents the point at which bond breaking and bond forming is occurring simultaneously. This is represented by dotted lines in formal notation and depicts the transient nature of the bond(s) in question in the transition state.

The analysis of potential energy surfaces are effective in finding transition states; the transition states can be described graphically as "dams" that represent energy metastability, that the reactants must "cross" in order to form the products.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Potential Energy Surface Reaction Coordinate'''<ref>http://www.ch.ic.ac.uk/motm/porphyrins/TSdiscovery.html</ref>||'''Cope Rearrangement'''<ref>http://www.ch.ic.ac.uk/local/organic/pericyclic/</ref>
|-
|[[Image:PES.gif]]||[[Image:Cope Arr.gif|350px]]
|}

From the point of view of computational chemistry, the methods of molecular mechanical structure determination that employed imaginary force fields are ineffective in the calculations concerning the transition state in larger molecules, as they do not define the dynamics of bond making and bond forming that are apparent in the transient nature of the transition state.

As such, in combination with molecular orbital based calculations, the following of the potential energy surface of a molecule can be used to locate and analyse the transition state.

This computational exercise concerns the rearrangement of ''1,5-Hexadiene'' in the '''Cope Rearrangment'''(above).

The Cope Rearrangement proceeds via a [3,3]-Sigmatropic shift involving the motion of 6 electrons and is thermally-driven.

In terms of transition states, the cope rearrangement can reportedly proceed via two checkpoints, the '''boat''' or the '''chair''':-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Chair'''||'''Boat'''
|-
|[[Image:pic2a.jpg]]||[[Image:pic2b.jpg]]
|}

The purpose of this exercise is to ascertain which reaction pathway is preferred by computation of each transition state in the elucidation of geometry, repective energies and other thermochemical data.

==''Exercise 1'' - Optimising the Reactants and Products==

In this section, the "''reactant''" '''1,5-hexadiene''' and the "''product''" '''1,5-hexadiene''' are modelled in correspondence to the cope rearrangement.

The flexible nature of 1,5-hexadiene means that there is an initial dilemma of likely conformation at rest.

Several starting conformations were thus taken and their geometries optimised to find the lowest energy conformer.

Initially, 1,5-hexadiene was taken, ensuring a roughly "'''anti'''" linkage in the centre of the molecule, so that the central C atoms '''3''' and '''4''' are approximately '''antiperiplanar''' to one another. This molecule was cleaned in the gaussview builder interface and then taken and optimised using the '''Hartree-Fock''' method with basis set '''3-21G''' through Gaussian.

'''b)''' similarly, a synclinal, or "gauche" version of 1,5-hexadiene was taken and optimised at the HF 3-21G level.

The energies of both conformers and also their symmetry were noted.

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|'''"''Anti''" 1,5-Hexadiene'''||'''"''Gauche''" 1,5-Hexadiene'''
|-align="center"
|[[Image:Bo Anti.jpg|420px]]||[[Image:Bo Gauche.jpg|400px]]
|}

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|'''Confomer'''||'''''Anti''-1,5-Hexadiene'''||'''''Gauche''-1,5-Hexadiene'''
|-align="center"
|'''Total Energy'''(a.u.)||-231.69254||-231.69266
|-align="center"
|'''Point Group'''||ci||None
|}

From a purely steric-orientated argument, the ''anti'' configuration should have a lower energy than the ''gauche'', as there is less steric hindrance from the two ends of the carbon chain 180 degrees from one another than next to each other in the gauche conformation.

As it so happens, the energy for the ''anti'' conformer is reportedly lower than that of the ''gauche'' conformer, but by a minute amount of roughly a ten-thousandth of a hartree ('''1 hartree = 4.3597482E-18 J''').

Even so, it was appreciated that the central C-C-C-C linkage of 1,5-hexadiene lends itself to quite a high degree of rotational freedom, and thus, further examples of conformational isomerism. This is reinforced by the fact of the two remarkably different conformers having remarkably similar energies, and points to the fact that 1,5-hexadiene experiences conformational ambiguity often<ref>Cope Rearrangement of 1,5-Hexadiene: Full Geometry Optimizations Using Analytic MR-CISD and MR-AQCC Gradient Methods, doi:10.1021/jp0259014</ref>.

As such, 2 other conformers were taken and optimised:-

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|colspan=4|'''<big>Different Conformers of 1,5-Hexadiene</big>'''
|-align="center"
|(''g120,a,g120'')-'''Anti1'''||(''g120,a,g-120'')-'''Anti2'''||(''g120,g-60,g120'')-'''Gauche1'''||(''g120,g60,g-120'')-'''Gauche3'''
|-align="center"
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Bo Anti1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Bo Anti2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Gauche 1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Gauche 3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-align="center"
|colspan=4|'''Energy'''(a.u.)
|-align="center"
| -231.69260213||-231.69253506||-231.69166702||-231.69266120
|-align="center"
|colspan=4|'''Point Group'''
|-align="center"
|C2||Ci||C2||C1
|}

The energies here from the HF 3-21G are similarly close, but contrary to what was expected, it was found that ''gauche3'' was actually the lowest in energy.

Beneath this, the ''anti'' conformers are found to be lower in energy than the other ''gauche'' conformer. This may be rationalised as the minimalisation of steric hindrance lowering the energy of the overall molecule, as the antiperiplanar groups maintain the least steric strain.

However, the '''''cis'''''-''gauche'' conformer is lowest in energy due to stereoelectronic arguments.

The '''π'''-bonds of the C=C bonds are lying criss-crossed over each other in this conformer. This promotes effectual π-molecular orbital overlap and hence stabilises the molecule.

Next, the anti2 conformer with ci symmetry and the gauche1 conformer was taken and further optimised under the DFT B3LYP method with a more sophisticated basis set, 6-31G(d).

The resulting optimised molecule was then compared with the earlier attempt in terms of energy.

A further frequency analysis was conducted in order to check that the optimisation has proceeded to a minimum, being the case when all frequencies were real and positive.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=5|'''Comparison of the HF 3-21G and DFT B3LYP 6-31G(d) Optimisations'''
|-align="center"
|'''Method'''||colspan=2|'''HF 3-21G'''||colspan=2|'''DFT B3LYP 6-31G(d)'''
|-align="center"
|'''Conformer'''||'''anti2'''||'''gauche1'''||'''anti2'''||'''gauche1'''
|-align="center"
|'''Total Energy'''(a.u.)||-231.69253506||-231.69166702||-231.61170616||-231.61068821
|-align="center"
|1-2 C=C Bond Length (A)||1.3162||1.3156||1.3335||1.3331
|-align="center"
|2-3 C-C Bond Length (A)||1.5091||1.5041||1.5083||1.5044
|-align="center"
|3-4 C-C Bond Length (A)||1.5527||1.5481||1.5508||1.5482
|-align="center"
|4-5 C-C Bond Length (A)||1.5091||1.5041||1.5083||1.5044
|-align="center"
|5-6 C=C Bond Length (A)||1.3163||1.3335||1.3157||1.3331
|-align="center"
|1-2-3-4 Dihedral (o)||114.66||118.49||123.93||124.24
|-align="center"
|2-3-4-5 Dihedral (o)||180.00||180.00||-64.18||-65.01
|-align="center"
|3-4-5-6 Dihedral (o)||-114.66||-118.49||123.93||124.24
|}

As can be seen here, the energies returned differ in a noticeable manner. However, comparison of energies computed under different basis sets is unbalanced.

As such, comparisons of the geometry are notably more reliable in discerning the computed differences.

The bond lengths differ by negligible amounts. However, it can be seen that the various dihedral angles have been "tightened" more to their ideal figures (closer to 120o for the anticlinal dihedrals.

As such, using a more sophisticated basis set ensures a more accurate approximation to reality.

The log file yielded the following thermochemical data:-

'''1)Sum of electronic and zero-point Energies= -234.469195'''

'''2)Sum of electronic and thermal Energies= -234.461847'''

'''3)Sum of electronic and thermal Enthalpies= -234.460903'''

'''4)Sum of electronic and thermal Free Energies= -234.500782'''

Theses values concern the 1)Potential Energy at 0K that includes the zero-point vibrational energy term 2)Potential Energy at 298.15K and 1atm and adds contributions from translational, rotational and vibrational energy 3)Energy including a correction for RT(H=E+RT) for dissociation reactions and 4)The Entropic contribution.

==''Exercise 2'' - Optimising the "Chair" and "Boat" Transition Structures==

The Cope Rearrangement has been found to proceed via one of two possible transition states with differing conformation, as mentioned in the introduction:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=2|'''<big>Cope Rearrangement Transition State Conformations</big>'''
|-align="center"
|''Chair''||''Boat''
|-align="center"
|[[Image:pic2a.jpg]]||[[Image:pic2b.jpg]]
|}

When the rearrangment is thought as one that is concerted and proceeds with synchronicity of bond breaking and formation, these two transition states come to mind.

There is indeed a third type of transition state for this reaction, dubbed the '''''Dewar''''' transition state, that reportedly proceeds via a diradical tight-chair intermediate species that was discovered by Dupuis et al.<ref>The Cope Rearrangement Revisited, DOI:10.1021/ja00026a007</ref>, and is thus NOT concerted.

===The Chair Transition State===

The chair transition state was initally synthesised using two "resonant" '''allyl''' ('''C3H5''') fragments in gaussview, which were optimised under the 3-21G basis set, and subsequently placed in appropriate positions relative to one another to be optimised into the transition state.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=2|'''Allyl Fragment('''C3H5''')'''
|-align="center"
|colspan=2|[[Image:Allyl Frag.jpg|275px]]
|-align="center"
|'''sp3''' Centre Bond Angle (o)||109.5
|-align="center"
|'''Allyl''' C-C Bond Angle (o)||124.3
|-align="center"
|'''C-C''' Bond Length (A)||1.54
|-align="center"
|'''C=C''' Bond Length (A)||1.34
|-align="center"
|'''Allyl C-C''' Bond Length (A)||1.39
|}

The optimised allyl C-C bond angle is greater than the normal sp3 bond angle of '''109.5o''' due to resonance.

More importantly, the allyl C-C bond length is between that of a single bond and a double bond.

This optimised allyl fragment was then taken, duplicated and placed, facing opposite directions, with roughly a 2.2A distance between the terminal carbons of each fragment to mimic the shape of the chair transition state.

Thus, a frequency + optimisation was run that directed gaussian to reach a '''TS(Berny)''', with the elucidation of force constants.

The optimisation was then repeated but this time using the '''"frozen" coordinate method''', effectively fixing the two pairs of ends at a distance of 2.2A.

Clearly, this method requires that the general structure of the transition state in question be known. However, by fixing the positions in such a way, a more accurate optimisation can be done that avoids divergence.

Finally, the optimisation was performed a third time, using a normal guess '''Hessian''' method, which involves calculation of the force constant matrix.

After the three optimisations were complete, the following data were obtained:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=3|'''<big>Optimisation of Chair Transition State</big>'''
|-align="center"
|colspan=3|'''Structure'''
|-align="center"
|'''TS(Berny)'''||'''Frozen Coordinate'''||'''Hessian'''
|-align="center"
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Berny.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Frozen.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>red</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Hessian.mol</uploadedFileContents>
</jmolApplet></jmol>
|-align="center"
|colspan=3|'''Energies''' (a.u.)
|-align="center"
| -231.61932||-231.61932||-231.69167
|}

The structures returned from the normal force-constant calculation optimisation and the frozen coordinate method are indistinguishable from first glance, and their energies are virtually identical. However, the structure obtained from the Hessian is different, and so is its energy value.

'''FREQUENCY ANALYSIS'''

Vibration-wise, there was one negative, and thus "imaginary" frequency found with a wavenumber of '''-817.91cm-1'''.

When the stretch was visualised, the following was observed:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Imaginary Stretch at -817.91cm-1'''
|-
|[[Image:Imag Stretch.jpg|300px]]
|}

The appearance of this imaginary frequency resembles the motion that would ensue under the Cope Rearrangment - the action of bond breaking of the C-C σ bond in the middle of the 1,5-hexadiene with the subsequent formation of a new σ bond on across the terminals on the other side.

The fact that there is only '''one''' imaginary frequency value points to the notion that the cope rearrangment through this transition state is ''concerted'', and thus occurs in one ''synchronous'' step.

The number of imaginary frequencies produced from a frequency analysis for the midpoint of a reaction points towards the ''synchronicity'' of the reaction<ref>Cope Rearrangement Revisited, DOI:10.1021/ja00026a007</ref>.

===The Boat Transition State===

For the boat transition state, a different method was utilised, namely the '''QST2 method''', whereby reactant and product are defined, and the transition state is converged from these.

This required the previously optimised '''1,5-hexadiene''' molecule to be taken and numbered along its carbon skeleton.

The molecule was then duplicated and juxtaposed to the intial "reactant" molecule and renumbered in such a way to reflect the reacted, rearragement "product" as shown below:-

{|style="margin:1em auto 1em auto" border="1"
|-
|[[Image:Boat Method.JPG|400px]]
|}

The first instance of optimisation failed to produce an output, and thus the molecule was changed in the following way:-

The dihedral angle of both reactant and product were changed to 0 degrees, such that the two carbon groups either side of C3 and C4 are eclipsed, and the C2-C3-C4 and C3-C4-C5 bond angles were both changed to 100 degrees.

The modified structures were once again optimised under the QST2 method to yield the following structure:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''''QST2'' Optimised Boat Transition State'''
|-
|[[Image:FINAL Boat.jpg|400px]]
|}

'''DISCUSSION'''

Here, it is illustrated that the QST2 method is automated, and is thus very efficient, but is only effective if the shape provided of the reactants and products cloely mirror that of the transition state, otherwise convergence is difficult.

The '''QST3''' method was also attempted <ref>QST Calculation, http://hdl.handle.net/10042/to-2902</ref>, whereby an addition guess of the transition state geometry was implemented. If the guess of the transition state is sound, then the QST3 is by far more accurate than the QST2 method, and does not rely so much on the shape of the reactants and products.

initally, it would seem that in order for the cope rearrangement to occur, to two carbon terminals would be required to be close to one another. This is not the case with the "''anti''" conformers of the molecule, as the terminals are situated as far apart as they may possibly be, '''antiperplanar''' to one another.

Once free rotation moves the carbon groups closer together, into the gauche configuration, where the terminals are more poised to react, cope rearrangement would proceed.

The relative orientations of the vinyl groups would ultimately dictate which transition state would be traversed. For instance, if the vectors of the vinyl groups cross, such as the case for the gauche1 conformer, then as the terminals align into a bonding orientaton, the ''chair'' conformation would be adopted by the transition state.

Conversely, if the two vinyl groups lie in a pseudo-parallel fashion, where their vectors do NOT cross, such as is the case for gauche3, then the transition state would be of ''boat'' topography.

'''IRC'''

The Intrinsic Reaction Coordinate looks more cloesly at the potential surface of a reaction and works to pinpoint the highest point in a local area of the potential surface by creeping up the steepest slopes in the immediate area.

This calculation was attempted with 50 points, and did not converge within the maximum number of points.

Thus, this was carried out again whilst calculating the force constant every single point.

{|style="margin:1em auto 1em auto" border="1"
|-
|[[Image:BO_IRC.JPG]]
|}

Here, upon closer inspection of the RMS Gradient of the energy, the gradient falls to zero, thus indicating that a minimum has been reached.

'''Energy Calculation'''

The two transition states were optimised a final time under DFT B3LYP 6-31G(d) conditions.

The geometries that were outputted were very similar to the previous examples, however, the energies received for both transition states are notably more varied:-

'''Final Energy of Chair Conformation''' (a.u.) = -234.61071
'''Final Energy of Boat Conformation''' (a.u.) = -234.54309

Here, it can be conclusively said that the ''Boat'' conformation transition state is indeed lower in energy than the ''Chair'' conformation.

This corresponds to better orbital overlap of the adjacent carbons on the framework, which are exactly eclipsed and not staggered like in the chair formation.

==The Diels-Alder Cycloaddition==

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Diels-Alder Reaction of Butadiene and Ethylene'''
|-align="center"
|[[Image:Mb_da3.jpg|400px]]
|}

The Diels-Alder Reaction is an example of a pericyclic cycloaddition, that usually involves the reaction between an electron rich diene and an electron poor dienophile.

The process is described as a [4s + 2s], 6 electron process that forms 2 new σ bonds.

When viewing the molecular orbital picture of butadiene and ethylene, the molecular orbitals are classified as either '''''s'''''(''symmetric'') or '''''a'''''(''antisymmetric'') in relation to the bisecting plane as shown below.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Bisecting Plane for Diels-alder'''
|-
|[[Image:Mb_da2.jpg|200px]]
|}

===Investigation of Molecular Orbital Symmetries of Butadiene and Ethene===

Butadiene and Ethylene were optimised and their molecular orbitals visualised:-

'''''Butadiene'''''

The HOMO and LUMO are considered with reference to the bisecting plane to determine their symmetry:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=4|'''HOMO and LUMO of Butadiene'''
|-align="center"
|width=250px|'''Molecular Orbital'''||'''2D Representation'''||'''3D Representation'''||width=250px|'''s/a'''
|-align="center"
|'''HOMO'''||[[Image:BDE HOMO.png|200px]]||[[Image:3D BDE HOMO.jpg|300px]]||a
|-align="center"
|'''LUMO'''||[[Image:BDE LUMO.png|200px]]||[[Image:3D BDE LUMO.jpg|300px]]||s
|}

'''''Ethene'''''

Similarly, the HOMO and LUMO of ethylene were characterised:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=4|'''HOMO and LUMO of Ethene'''
|-align="center"
|width=250px|'''Molecular Orbital'''||'''2D Representation'''||'''3D Representation'''||width=250px|'''s/a'''
|-align="center"
|'''HOMO'''||[[Image:ET HOMO.jpg|200px]]||[[Image:3D ET HOMO.jpg|300px]]||s
|-align="center"
|'''LUMO'''||[[Image:ET LUMO.jpg|200px]]||[[Image:3D ET LUMO.jpg|300px]]||a
|}

===Optimisation of Diels-Alder Transition State===

The '''Transition State''' whereby the concerted cycloaddition of butadiene and ethylene was modelled to a guess initially, before being optimised under HF 3-21G basis set conditions to a TS(Berny).

The initial guess was modelled from the bicyclo-system of cyclohexene and subsequently removing one of the -CH2CH2- arches to arrive at an "''envelope''"-shaped 6-membered ring.

The new σ bonds were stripped before the optimisation.

A frequency analysis was first concocted via optimisation to a transition state.

There were 3 imaginary frequencies reported by the analysis, detailed below:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=4|'''Imaginary Frequency Vibrations of Transition State'''
|-align="center"
|'''Stretches'''||[[Image:CHEX VIB 1.jpg|250px]]|||[[Image:CHEX VIB 2.jpg|250px]]|||[[Image:CHEX VIB 3.jpg|250px]]
|-align="center"
|'''Frequency''' (cm-1)||-716.19||-557.32||-198.58
|}

The first imaginary frequency at -716.19cm-1 did not correspond to any particular action within the transition state.

However, it can be seen that the 2nd and 3rd vibrations tie into the formation of the two conformations of cyclohexene, i.e. the ''chair'' and the ''boat''.

The presence of 3 imaginary frequencies in the analysis points to the notion that the formation of the two bonds is asynchronous.

The average of the inter-C-terminal distances was then calculated and used in a separate ''frozen-coordinate'' optimisation to a minimum, to get a more accurate representation of the geometry:-

{|style="margin=1em auto 1em auto" border="1"
|-align="center"
|colspan=3|'''Comparisons of the Optimisations of transition state'''
|-align="center"
|'''Method'''||'''Ts(Berny)'''||'''Frozen-Coordinate'''
|-align="center"
|'''Structure'''||[[Image:CHEX BERNY.jpg|300px]]||[[Image:CHEX FROZEN.jpg|300px]]
|-align="center"
|'''Energy'''(a.u.)||-230.30655||-231.67157
|}

The length of the partly formed σ C-C bonds were optimised to a value of '''1.5075A''', which is notably in between the lengths of sp2 and sp3 bond lengths of '''1.34A''' and '''1.54A''' and is a result of the point of breaking of the double bonds in the diene and formation of new σ bonds between the butadiene and ethene.

The Van-der-Waals radius of ''carbon'' is '''1.70A'''.

By freezing the terminals, the energy of the optimised structure is markedly higher.

'''AM1 Molecular Orbital Analysis'''

The AM1 semi-empirical method was used to visualise the highest-occupied molecular orbital of the transition state.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''''HOMO'''''||'''''LUMO'''''
|-align="center"
|[[Image:CHEX TS HOMO.jpg|300px]]||[[Image:CHEX TS LUMO.jpg|300px]]
|-align="center"
|colspan=2|'''Symmetry'''
|-align="center"
|s||a
|-align="center"
|}

With reference to the bisecting plane as shown above, the HOMO of the transition state can be seen to be symmetric when reflected.

The individual MOs of butadiene and ethene thus must both be symmetric with respect to the plane in order to react in such a fashion.

This is consistent with the idea that only molecular orbitals with the same symmetry are allowed to react.

Given this, the pertinent orbitals that are symmetric are the '''LUMO''' of the butadiene and the '''HOMO''' of the ethene.

As such, electron density is donated into the empty π* LUMO of the butadiene.

The LUMO in this case is antisymmetric with respect to the bisecting plane and corresponds to the '''HOMO''' of the butadiene and the '''LUMO''' of the ethene.

===Diels-Alder Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride===

The Diels-Alder reaction of '''Cyclohexa-1,3-diene and Maleic Anhydride''' is also an example of a 6-electron [4+2] Diels-Alder Cycloaddition intrinsically.

However, it differs from the reaction between butadiene and ethene in that there is now a question of regioselectivity; the bigger anhydride ring can either react to adopt an '''EXO''' or an '''ENDO''' topography:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|[[Image:EXO ENDO.jpg|400px]]
|}

The vinyl goup on maleic anhydride, in comparison to ethene, is electron-deficient, thanks to the highly electron-withdrawing nature of the anhydride group.

==References==

<references />

Rep:Mod:parkbom

2009-11-14T16:31:26Z

Byt07:

=Module 3 - The Computation of the Transition State=

==Introduction==

The '''Transition State''' is an energetic maximum within a given reaction coordinate and represents the point at which bond breaking and bond forming is occurring simultaneously. This is represented by dotted lines in formal notation and depicts the transient nature of the bond(s) in question in the transition state.

The analysis of potential energy surfaces are effective in finding transition states; the transition states can be described graphically as "dams" that represent energy metastability, that the reactants must "cross" in order to form the products.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Potential Energy Surface Reaction Coordinate'''<ref>http://www.ch.ic.ac.uk/motm/porphyrins/TSdiscovery.html</ref>||'''Cope Rearrangement'''<ref>http://www.ch.ic.ac.uk/local/organic/pericyclic/</ref>
|-
|[[Image:PES.gif]]||[[Image:Cope Arr.gif|350px]]
|}

From the point of view of computational chemistry, the methods of molecular mechanical structure determination that employed imaginary force fields are ineffective in the calculations concerning the transition state in larger molecules, as they do not define the dynamics of bond making and bond forming that are apparent in the transient nature of the transition state.

As such, in combination with molecular orbital based calculations, the following of the potential energy surface of a molecule can be used to locate and analyse the transition state.

This computational exercise concerns the rearrangement of ''1,5-Hexadiene'' in the '''Cope Rearrangment'''(above).

The Cope Rearrangement proceeds via a [3,3]-Sigmatropic shift involving the motion of 6 electrons and is thermally-driven.

In terms of transition states, the cope rearrangement can reportedly proceed via two checkpoints, the '''boat''' or the '''chair''':-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Chair'''||'''Boat'''
|-
|[[Image:pic2a.jpg]]||[[Image:pic2b.jpg]]
|}

The purpose of this exercise is to ascertain which reaction pathway is preferred by computation of each transition state in the elucidation of geometry, repective energies and other thermochemical data.

==''Exercise 1'' - Optimising the Reactants and Products==

In this section, the "''reactant''" '''1,5-hexadiene''' and the "''product''" '''1,5-hexadiene''' are modelled in correspondence to the cope rearrangement.

The flexible nature of 1,5-hexadiene means that there is an initial dilemma of likely conformation at rest.

Several starting conformations were thus taken and their geometries optimised to find the lowest energy conformer.

Initially, 1,5-hexadiene was taken, ensuring a roughly "'''anti'''" linkage in the centre of the molecule, so that the central C atoms '''3''' and '''4''' are approximately '''antiperiplanar''' to one another. This molecule was cleaned in the gaussview builder interface and then taken and optimised using the '''Hartree-Fock''' method with basis set '''3-21G''' through Gaussian.

'''b)''' similarly, a synclinal, or "gauche" version of 1,5-hexadiene was taken and optimised at the HF 3-21G level.

The energies of both conformers and also their symmetry were noted.

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|'''"''Anti''" 1,5-Hexadiene'''||'''"''Gauche''" 1,5-Hexadiene'''
|-align="center"
|[[Image:Bo Anti.jpg|420px]]||[[Image:Bo Gauche.jpg|400px]]
|}

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|'''Confomer'''||'''''Anti''-1,5-Hexadiene'''||'''''Gauche''-1,5-Hexadiene'''
|-align="center"
|'''Total Energy'''(a.u.)||-231.69254||-231.69266
|-align="center"
|'''Point Group'''||ci||None
|}

From a purely steric-orientated argument, the ''anti'' configuration should have a lower energy than the ''gauche'', as there is less steric hindrance from the two ends of the carbon chain 180 degrees from one another than next to each other in the gauche conformation.

As it so happens, the energy for the ''anti'' conformer is reportedly lower than that of the ''gauche'' conformer, but by a minute amount of roughly a ten-thousandth of a hartree ('''1 hartree = 4.3597482E-18 J''').

Even so, it was appreciated that the central C-C-C-C linkage of 1,5-hexadiene lends itself to quite a high degree of rotational freedom, and thus, further examples of conformational isomerism. This is reinforced by the fact of the two remarkably different conformers having remarkably similar energies, and points to the fact that 1,5-hexadiene experiences conformational ambiguity often<ref>Cope Rearrangement of 1,5-Hexadiene: Full Geometry Optimizations Using Analytic MR-CISD and MR-AQCC Gradient Methods, doi:10.1021/jp0259014</ref>.

As such, 2 other conformers were taken and optimised:-

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|colspan=4|'''<big>Different Conformers of 1,5-Hexadiene</big>'''
|-align="center"
|(''g120,a,g120'')-'''Anti1'''||(''g120,a,g-120'')-'''Anti2'''||(''g120,g-60,g120'')-'''Gauche1'''||(''g120,g60,g-120'')-'''Gauche3'''
|-align="center"
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Bo Anti1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Bo Anti2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Gauche 1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Gauche 3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-align="center"
|colspan=4|'''Energy'''(a.u.)
|-align="center"
| -231.69260213||-231.69253506||-231.69166702||-231.69266120
|-align="center"
|colspan=4|'''Point Group'''
|-align="center"
|C2||Ci||C2||C1
|}

The energies here from the HF 3-21G are similarly close, but contrary to what was expected, it was found that ''gauche3'' was actually the lowest in energy.

Beneath this, the ''anti'' conformers are found to be lower in energy than the other ''gauche'' conformer. This may be rationalised as the minimalisation of steric hindrance lowering the energy of the overall molecule, as the antiperiplanar groups maintain the least steric strain.

However, the '''''cis'''''-''gauche'' conformer is lowest in energy due to stereoelectronic arguments.

The '''π'''-bonds of the C=C bonds are lying criss-crossed over each other in this conformer. This promotes effectual π-molecular orbital overlap and hence stabilises the molecule.

Next, the anti2 conformer with ci symmetry and the gauche1 conformer was taken and further optimised under the DFT B3LYP method with a more sophisticated basis set, 6-31G(d).

The resulting optimised molecule was then compared with the earlier attempt in terms of energy.

A further frequency analysis was conducted in order to check that the optimisation has proceeded to a minimum, being the case when all frequencies were real and positive.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=5|'''Comparison of the HF 3-21G and DFT B3LYP 6-31G(d) Optimisations'''
|-align="center"
|'''Method'''||colspan=2|'''HF 3-21G'''||colspan=2|'''DFT B3LYP 6-31G(d)'''
|-align="center"
|'''Conformer'''||'''anti2'''||'''gauche1'''||'''anti2'''||'''gauche1'''
|-align="center"
|'''Total Energy'''(a.u.)||-231.69253506||-231.69166702||-231.61170616||-231.61068821
|-align="center"
|1-2 C=C Bond Length (A)||1.3162||1.3156||1.3335||1.3331
|-align="center"
|2-3 C-C Bond Length (A)||1.5091||1.5041||1.5083||1.5044
|-align="center"
|3-4 C-C Bond Length (A)||1.5527||1.5481||1.5508||1.5482
|-align="center"
|4-5 C-C Bond Length (A)||1.5091||1.5041||1.5083||1.5044
|-align="center"
|5-6 C=C Bond Length (A)||1.3163||1.3335||1.3157||1.3331
|-align="center"
|1-2-3-4 Dihedral (o)||114.66||118.49||123.93||124.24
|-align="center"
|2-3-4-5 Dihedral (o)||180.00||180.00||-64.18||-65.01
|-align="center"
|3-4-5-6 Dihedral (o)||-114.66||-118.49||123.93||124.24
|}

As can be seen here, the energies returned differ in a noticeable manner. However, comparison of energies computed under different basis sets is unbalanced.

As such, comparisons of the geometry are notably more reliable in discerning the computed differences.

The bond lengths differ by negligible amounts. However, it can be seen that the various dihedral angles have been "tightened" more to their ideal figures (closer to 120o for the anticlinal dihedrals.

As such, using a more sophisticated basis set ensures a more accurate approximation to reality.

The log file yielded the following thermochemical data:-

'''1)Sum of electronic and zero-point Energies= -234.469195'''

'''2)Sum of electronic and thermal Energies= -234.461847'''

'''3)Sum of electronic and thermal Enthalpies= -234.460903'''

'''4)Sum of electronic and thermal Free Energies= -234.500782'''

Theses values concern the 1)Potential Energy at 0K that includes the zero-point vibrational energy term 2)Potential Energy at 298.15K and 1atm and adds contributions from translational, rotational and vibrational energy 3)Energy including a correction for RT(H=E+RT) for dissociation reactions and 4)The Entropic contribution.

==''Exercise 2'' - Optimising the "Chair" and "Boat" Transition Structures==

The Cope Rearrangement has been found to proceed via one of two possible transition states with differing conformation, as mentioned in the introduction:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=2|'''<big>Cope Rearrangement Transition State Conformations</big>'''
|-align="center"
|''Chair''||''Boat''
|-align="center"
|[[Image:pic2a.jpg]]||[[Image:pic2b.jpg]]
|}

When the rearrangment is thought as one that is concerted and proceeds with synchronicity of bond breaking and formation, these two transition states come to mind.

There is indeed a third type of transition state for this reaction, dubbed the '''''Dewar''''' transition state, that reportedly proceeds via a diradical tight-chair intermediate species that was discovered by Dupuis et al.<ref>The Cope Rearrangement Revisited, DOI:10.1021/ja00026a007</ref>, and is thus NOT concerted.

===The Chair Transition State===

The chair transition state was initally synthesised using two "resonant" '''allyl''' ('''C3H5''') fragments in gaussview, which were optimised under the 3-21G basis set, and subsequently placed in appropriate positions relative to one another to be optimised into the transition state.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=2|'''Allyl Fragment('''C3H5''')'''
|-align="center"
|colspan=2|[[Image:Allyl Frag.jpg|275px]]
|-align="center"
|'''sp3''' Centre Bond Angle (o)||109.5
|-align="center"
|'''Allyl''' C-C Bond Angle (o)||124.3
|-align="center"
|'''C-C''' Bond Length (A)||1.54
|-align="center"
|'''C=C''' Bond Length (A)||1.34
|-align="center"
|'''Allyl C-C''' Bond Length (A)||1.39
|}

The optimised allyl C-C bond angle is greater than the normal sp3 bond angle of '''109.5o''' due to resonance.

More importantly, the allyl C-C bond length is between that of a single bond and a double bond.

This optimised allyl fragment was then taken, duplicated and placed, facing opposite directions, with roughly a 2.2A distance between the terminal carbons of each fragment to mimic the shape of the chair transition state.

Thus, a frequency + optimisation was run that directed gaussian to reach a '''TS(Berny)''', with the elucidation of force constants.

The optimisation was then repeated but this time using the '''"frozen" coordinate method''', effectively fixing the two pairs of ends at a distance of 2.2A.

Clearly, this method requires that the general structure of the transition state in question be known. However, by fixing the positions in such a way, a more accurate optimisation can be done that avoids divergence.

Finally, the optimisation was performed a third time, using a normal guess '''Hessian''' method, which involves calculation of the force constant matrix.

After the three optimisations were complete, the following data were obtained:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=3|'''<big>Optimisation of Chair Transition State</big>'''
|-align="center"
|colspan=3|'''Structure'''
|-align="center"
|'''TS(Berny)'''||'''Frozen Coordinate'''||'''Hessian'''
|-align="center"
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Berny.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Frozen.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>red</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Hessian.mol</uploadedFileContents>
</jmolApplet></jmol>
|-align="center"
|colspan=3|'''Energies''' (a.u.)
|-align="center"
| -231.61932||-231.61932||-231.69167
|}

The structures returned from the normal force-constant calculation optimisation and the frozen coordinate method are indistinguishable from first glance, and their energies are virtually identical. However, the structure obtained from the Hessian is different, and so is its energy value.

'''FREQUENCY ANALYSIS'''

Vibration-wise, there was one negative, and thus "imaginary" frequency found with a wavenumber of '''-817.91cm-1'''.

When the stretch was visualised, the following was observed:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Imaginary Stretch at -817.91cm-1'''
|-
|[[Image:Imag Stretch.jpg|300px]]
|}

The appearance of this imaginary frequency resembles the motion that would ensue under the Cope Rearrangment - the action of bond breaking of the C-C σ bond in the middle of the 1,5-hexadiene with the subsequent formation of a new σ bond on across the terminals on the other side.

The fact that there is only '''one''' imaginary frequency value points to the notion that the cope rearrangment through this transition state is ''concerted'', and thus occurs in one ''synchronous'' step.

The number of imaginary frequencies produced from a frequency analysis for the midpoint of a reaction points towards the ''synchronicity'' of the reaction<ref>Cope Rearrangement Revisited, DOI:10.1021/ja00026a007</ref>.

===The Boat Transition State===

For the boat transition state, a different method was utilised, namely the '''QST2 method''', whereby reactant and product are defined, and the transition state is converged from these.

This required the previously optimised '''1,5-hexadiene''' molecule to be taken and numbered along its carbon skeleton.

The molecule was then duplicated and juxtaposed to the intial "reactant" molecule and renumbered in such a way to reflect the reacted, rearragement "product" as shown below:-

{|style="margin:1em auto 1em auto" border="1"
|-
|[[Image:Boat Method.JPG|400px]]
|}

The first instance of optimisation failed to produce an output, and thus the molecule was changed in the following way:-

The dihedral angle of both reactant and product were changed to 0 degrees, such that the two carbon groups either side of C3 and C4 are eclipsed, and the C2-C3-C4 and C3-C4-C5 bond angles were both changed to 100 degrees.

The modified structures were once again optimised under the QST2 method to yield the following structure:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''''QST2'' Optimised Boat Transition State'''
|-
|[[Image:FINAL Boat.jpg|400px]]
|}

'''DISCUSSION'''

Here, it is illustrated that the QST2 method is automated, and is thus very efficient, but is only effective if the shape provided of the reactants and products cloely mirror that of the transition state, otherwise convergence is difficult.

The '''QST3''' method was also attempted <ref>QST Calculation, http://hdl.handle.net/10042/to-2902</ref>, whereby an addition guess of the transition state geometry was implemented. If the guess of the transition state is sound, then the QST3 is by far more accurate than the QST2 method, and does not rely so much on the shape of the reactants and products.

initally, it would seem that in order for the cope rearrangement to occur, to two carbon terminals would be required to be close to one another. This is not the case with the "''anti''" conformers of the molecule, as the terminals are situated as far apart as they may possibly be, '''antiperplanar''' to one another.

Once free rotation moves the carbon groups closer together, into the gauche configuration, where the terminals are more poised to react, cope rearrangement would proceed.

The relative orientations of the vinyl groups would ultimately dictate which transition state would be traversed. For instance, if the vectors of the vinyl groups cross, such as the case for the gauche1 conformer, then as the terminals align into a bonding orientaton, the ''chair'' conformation would be adopted by the transition state.

Conversely, if the two vinyl groups lie in a pseudo-parallel fashion, where their vectors do NOT cross, such as is the case for gauche3, then the transition state would be of ''boat'' topography.

'''IRC'''

The Intrinsic Reaction Coordinate looks more cloesly at the potential surface of a reaction and works to pinpoint the highest point in a local area of the potential surface by creeping up the steepest slopes in the immediate area.

This calculation was attempted with 50 points, and did not converge within the maximum number of points.

Thus, this was carried out again whilst calculating the force constant every single point.

{|style="margin:1em auto 1em auto" border="1"
|-
|[[Image:BO_IRC.JPG]]
|}

Here, upon closer inspection of the RMS Gradient of the energy, the gradient falls to zero, thus indicating that a minimum has been reached.

'''Energy Calculation'''

The two transition states were optimised a final time under DFT B3LYP 6-31G(d) conditions.

The geometries that were outputted were very similar to the previous examples, however, the energies received for both transition states are notably more varied:-

'''Final Energy of Chair Conformation''' (a.u.) = -234.61071
'''Final Energy of Boat Conformation''' (a.u.) = -234.54309

Here, it can be conclusively said that the ''Boat'' conformation transition state is indeed lower in energy than the ''Chair'' conformation.

This corresponds to better orbital overlap of the adjacent carbons on the framework, which are exactly eclipsed and not staggered like in the chair formation.

==The Diels-Alder Cycloaddition==

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Diels-Alder Reaction of Butadiene and Ethylene'''
|-align="center"
|[[Image:Mb_da3.jpg|400px]]
|}

The Diels-Alder Reaction is an example of a pericyclic cycloaddition, that usually involves the reaction between an electron rich diene and an electron poor dienophile.

The process is described as a [4s + 2s], 6 electron process that forms 2 new σ bonds.

When viewing the molecular orbital picture of butadiene and ethylene, the molecular orbitals are classified as either '''''s'''''(''symmetric'') or '''''a'''''(''antisymmetric'') in relation to the bisecting plane as shown below.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Bisecting Plane for Diels-alder'''
|-
|[[Image:Mb_da2.jpg|200px]]
|}

===Investigation of Molecular Orbital Symmetries of Butadiene and Ethene===

Butadiene and Ethylene were optimised and their molecular orbitals visualised:-

'''''Butadiene'''''

The HOMO and LUMO are considered with reference to the bisecting plane to determine their symmetry:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=4|'''HOMO and LUMO of Butadiene'''
|-align="center"
|width=250px|'''Molecular Orbital'''||'''2D Representation'''||'''3D Representation'''||width=250px|'''s/a'''
|-align="center"
|'''HOMO'''||[[Image:BDE HOMO.png|200px]]||[[Image:3D BDE HOMO.jpg|300px]]||a
|-align="center"
|'''LUMO'''||[[Image:BDE LUMO.png|200px]]||[[Image:3D BDE LUMO.jpg|300px]]||s
|}

'''''Ethene'''''

Similarly, the HOMO and LUMO of ethylene were characterised:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=4|'''HOMO and LUMO of Ethene'''
|-align="center"
|width=250px|'''Molecular Orbital'''||'''2D Representation'''||'''3D Representation'''||width=250px|'''s/a'''
|-align="center"
|'''HOMO'''||[[Image:ET HOMO.jpg|200px]]||[[Image:3D ET HOMO.jpg|300px]]||s
|-align="center"
|'''LUMO'''||[[Image:ET LUMO.jpg|200px]]||[[Image:3D ET LUMO.jpg|300px]]||a
|}

===Optimisation of Diels-Alder Transition State===

The '''Transition State''' whereby the concerted cycloaddition of butadiene and ethylene was modelled to a guess initially, before being optimised under HF 3-21G basis set conditions to a TS(Berny).

The initial guess was modelled from the bicyclo-system of cyclohexene and subsequently removing one of the -CH2CH2- arches to arrive at an "''envelope''"-shaped 6-membered ring.

The new σ bonds were stripped before the optimisation.

A frequency analysis was first concocted via optimisation to a transition state.

There were 3 imaginary frequencies reported by the analysis, detailed below:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=4|'''Imaginary Frequency Vibrations of Transition State'''
|-align="center"
|'''Stretches'''||[[Image:CHEX VIB 1.jpg|250px]]|||[[Image:CHEX VIB 2.jpg|250px]]|||[[Image:CHEX VIB 3.jpg|250px]]
|-align="center"
|'''Frequency''' (cm-1)||-716.19||-557.32||-198.58
|}

The first imaginary frequency at -716.19cm-1 did not correspond to any particular action within the transition state.

However, it can be seen that the 2nd and 3rd vibrations tie into the formation of the two conformations of cyclohexene, i.e. the ''chair'' and the ''boat''.

The average of the inter-C-terminal distances was then calculated and used in a separate ''frozen-coordinate'' optimisation to a minimum, to get a more accurate representation of the geometry:-

{|style="margin=1em auto 1em auto" border="1"
|-align="center"
|colspan=3|'''Comparisons of the Optimisations of transition state'''
|-align="center"
|'''Method'''||'''Ts(Berny)'''||'''Frozen-Coordinate'''
|-align="center"
|'''Structure'''||[[Image:CHEX BERNY.jpg|300px]]||[[Image:CHEX FROZEN.jpg|300px]]
|-align="center"
|'''Energy'''(a.u.)||-230.30655||-231.67157
|}

The length of the partly formed σ C-C bonds were optimised to a value of 1.5075A, which is notably in between the lengths of sp2 and sp3 bond lengths of 1.34A and 1.54A and is a result of the point of breaking of the double bonds in the diene and formation of new σ bonds between the butadiene and ethene.

By freezing the terminals, the energy of the optimised structure is markedly higher.

'''AM1 Molecular Orbital Analysis'''

The AM1 semi-empirical method was used to visualise the highest-occupied molecular orbital of the transition state.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''''HOMO'''''||'''''LUMO'''''
|-align="center"
|[[Image:CHEX TS HOMO.jpg|300px]]||[[Image:CHEX TS LUMO.jpg|300px]]
|-align="center"
|colspan=2|'''Symmetry'''
|-align="center"
|s||a
|-align="center"
|}

With reference to the bisecting plane as shown above, the HOMO of the transition state can be seen to be symmetric when reflected.

The individual MOs of butadiene and ethene thus must both be symmetric with respect to the plane in order to react in such a fashion.

This is consistent with the idea that only molecular orbitals with the same symmetry are allowed to react.

Given this, the pertinent orbitals that are symmetric are the '''LUMO''' of the butadiene and the '''HOMO''' of the ethene.

As such, electron density is donated into the empty π* LUMO of the butadiene.

The LUMO in this case is antisymmetric with respect to the bisecting plane and corresponds to the '''HOMO''' of the butadiene and the '''LUMO''' of the ethene.

===Diels-Alder Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride===

The Diels-Alder reaction of '''Cyclohexa-1,3-diene and Maleic Anhydride''' is also an example of a 6-electron [4+2] Diels-Alder Cycloaddition intrinsically.

However, it differs from the reaction between butadiene and ethene in that there is now a question of regioselectivity; the bigger anhydride ring can either react to adopt an '''EXO''' or an '''ENDO''' topography:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|[[Image:EXO ENDO.jpg|400px]]
|}

The vinyl goup on maleic anhydride, in comparison to ethene, is electron-deficient, thanks to the highly electron-withdrawing nature of the anhydride group.

==References==

<references />

Rep:Mod:parkbom

2009-11-14T16:01:52Z

Byt07:

=Module 3 - The Computation of the Transition State=

==Introduction==

The '''Transition State''' is an energetic maximum within a given reaction coordinate and represents the point at which bond breaking and bond forming is occurring simultaneously. This is represented by dotted lines in formal notation and depicts the transient nature of the bond(s) in question in the transition state.

The analysis of potential energy surfaces are effective in finding transition states; the transition states can be described graphically as "dams" that represent energy metastability, that the reactants must "cross" in order to form the products.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Potential Energy Surface Reaction Coordinate'''<ref>http://www.ch.ic.ac.uk/motm/porphyrins/TSdiscovery.html</ref>||'''Cope Rearrangement'''<ref>http://www.ch.ic.ac.uk/local/organic/pericyclic/</ref>
|-
|[[Image:PES.gif]]||[[Image:Cope Arr.gif|350px]]
|}

From the point of view of computational chemistry, the methods of molecular mechanical structure determination that employed imaginary force fields are ineffective in the calculations concerning the transition state in larger molecules, as they do not define the dynamics of bond making and bond forming that are apparent in the transient nature of the transition state.

As such, in combination with molecular orbital based calculations, the following of the potential energy surface of a molecule can be used to locate and analyse the transition state.

This computational exercise concerns the rearrangement of ''1,5-Hexadiene'' in the '''Cope Rearrangment'''(above).

The Cope Rearrangement proceeds via a [3,3]-Sigmatropic shift involving the motion of 6 electrons and is thermally-driven.

In terms of transition states, the cope rearrangement can reportedly proceed via two checkpoints, the '''boat''' or the '''chair''':-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Chair'''||'''Boat'''
|-
|[[Image:pic2a.jpg]]||[[Image:pic2b.jpg]]
|}

The purpose of this exercise is to ascertain which reaction pathway is preferred by computation of each transition state in the elucidation of geometry, repective energies and other thermochemical data.

==''Exercise 1'' - Optimising the Reactants and Products==

In this section, the "''reactant''" '''1,5-hexadiene''' and the "''product''" '''1,5-hexadiene''' are modelled in correspondence to the cope rearrangement.

The flexible nature of 1,5-hexadiene means that there is an initial dilemma of likely conformation at rest.

Several starting conformations were thus taken and their geometries optimised to find the lowest energy conformer.

Initially, 1,5-hexadiene was taken, ensuring a roughly "'''anti'''" linkage in the centre of the molecule, so that the central C atoms '''3''' and '''4''' are approximately '''antiperiplanar''' to one another. This molecule was cleaned in the gaussview builder interface and then taken and optimised using the '''Hartree-Fock''' method with basis set '''3-21G''' through Gaussian.

'''b)''' similarly, a synclinal, or "gauche" version of 1,5-hexadiene was taken and optimised at the HF 3-21G level.

The energies of both conformers and also their symmetry were noted.

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|'''"''Anti''" 1,5-Hexadiene'''||'''"''Gauche''" 1,5-Hexadiene'''
|-align="center"
|[[Image:Bo Anti.jpg|420px]]||[[Image:Bo Gauche.jpg|400px]]
|}

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|'''Confomer'''||'''''Anti''-1,5-Hexadiene'''||'''''Gauche''-1,5-Hexadiene'''
|-align="center"
|'''Total Energy'''(a.u.)||-231.69254||-231.69266
|-align="center"
|'''Point Group'''||ci||None
|}

From a purely steric-orientated argument, the ''anti'' configuration should have a lower energy than the ''gauche'', as there is less steric hindrance from the two ends of the carbon chain 180 degrees from one another than next to each other in the gauche conformation.

As it so happens, the energy for the ''anti'' conformer is reportedly lower than that of the ''gauche'' conformer, but by a minute amount of roughly a ten-thousandth of a hartree ('''1 hartree = 4.3597482E-18 J''').

Even so, it was appreciated that the central C-C-C-C linkage of 1,5-hexadiene lends itself to quite a high degree of rotational freedom, and thus, further examples of conformational isomerism. This is reinforced by the fact of the two remarkably different conformers having remarkably similar energies, and points to the fact that 1,5-hexadiene experiences conformational ambiguity often<ref>Cope Rearrangement of 1,5-Hexadiene: Full Geometry Optimizations Using Analytic MR-CISD and MR-AQCC Gradient Methods, doi:10.1021/jp0259014</ref>.

As such, 2 other conformers were taken and optimised:-

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|colspan=4|'''<big>Different Conformers of 1,5-Hexadiene</big>'''
|-align="center"
|(''g120,a,g120'')-'''Anti1'''||(''g120,a,g-120'')-'''Anti2'''||(''g120,g-60,g120'')-'''Gauche1'''||(''g120,g60,g-120'')-'''Gauche3'''
|-align="center"
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Bo Anti1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Bo Anti2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Gauche 1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Gauche 3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-align="center"
|colspan=4|'''Energy'''(a.u.)
|-align="center"
| -231.69260213||-231.69253506||-231.69166702||-231.69266120
|-align="center"
|colspan=4|'''Point Group'''
|-align="center"
|C2||Ci||C2||C1
|}

The energies here from the HF 3-21G are similarly close, but contrary to what was expected, it was found that ''gauche3'' was actually the lowest in energy.

Beneath this, the ''anti'' conformers are found to be lower in energy than the other ''gauche'' conformer. This may be rationalised as the minimalisation of steric hindrance lowering the energy of the overall molecule, as the antiperiplanar groups maintain the least steric strain.

However, the '''''cis'''''-''gauche'' conformer is lowest in energy due to stereoelectronic arguments.

The '''π'''-bonds of the C=C bonds are lying criss-crossed over each other in this conformer. This promotes effectual π-molecular orbital overlap and hence stabilises the molecule.

Next, the anti2 conformer with ci symmetry and the gauche1 conformer was taken and further optimised under the DFT B3LYP method with a more sophisticated basis set, 6-31G(d).

The resulting optimised molecule was then compared with the earlier attempt in terms of energy.

A further frequency analysis was conducted in order to check that the optimisation has proceeded to a minimum, being the case when all frequencies were real and positive.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=5|'''Comparison of the HF 3-21G and DFT B3LYP 6-31G(d) Optimisations'''
|-align="center"
|'''Method'''||colspan=2|'''HF 3-21G'''||colspan=2|'''DFT B3LYP 6-31G(d)'''
|-align="center"
|'''Conformer'''||'''anti2'''||'''gauche1'''||'''anti2'''||'''gauche1'''
|-align="center"
|'''Total Energy'''(a.u.)||-231.69253506||-231.69166702||-231.61170616||-231.61068821
|-align="center"
|1-2 C=C Bond Length (A)||1.3162||1.3156||1.3335||1.3331
|-align="center"
|2-3 C-C Bond Length (A)||1.5091||1.5041||1.5083||1.5044
|-align="center"
|3-4 C-C Bond Length (A)||1.5527||1.5481||1.5508||1.5482
|-align="center"
|4-5 C-C Bond Length (A)||1.5091||1.5041||1.5083||1.5044
|-align="center"
|5-6 C=C Bond Length (A)||1.3163||1.3335||1.3157||1.3331
|-align="center"
|1-2-3-4 Dihedral (o)||114.66||118.49||123.93||124.24
|-align="center"
|2-3-4-5 Dihedral (o)||180.00||180.00||-64.18||-65.01
|-align="center"
|3-4-5-6 Dihedral (o)||-114.66||-118.49||123.93||124.24
|}

As can be seen here, the energies returned differ in a noticeable manner. However, comparison of energies computed under different basis sets is unbalanced.

As such, comparisons of the geometry are notably more reliable in discerning the computed differences.

The bond lengths differ by negligible amounts. However, it can be seen that the various dihedral angles have been "tightened" more to their ideal figures (closer to 120o for the anticlinal dihedrals.

As such, using a more sophisticated basis set ensures a more accurate approximation to reality.

The log file yielded the following thermochemical data:-

'''1)Sum of electronic and zero-point Energies= -234.469195'''

'''2)Sum of electronic and thermal Energies= -234.461847'''

'''3)Sum of electronic and thermal Enthalpies= -234.460903'''

'''4)Sum of electronic and thermal Free Energies= -234.500782'''

Theses values concern the 1)Potential Energy at 0K that includes the zero-point vibrational energy term 2)Potential Energy at 298.15K and 1atm and adds contributions from translational, rotational and vibrational energy 3)Energy including a correction for RT(H=E+RT) for dissociation reactions and 4)The Entropic contribution.

==''Exercise 2'' - Optimising the "Chair" and "Boat" Transition Structures==

The Cope Rearrangement has been found to proceed via one of two possible transition states with differing conformation, as mentioned in the introduction:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=2|'''<big>Cope Rearrangement Transition State Conformations</big>'''
|-align="center"
|''Chair''||''Boat''
|-align="center"
|[[Image:pic2a.jpg]]||[[Image:pic2b.jpg]]
|}

When the rearrangment is thought as one that is concerted and proceeds with synchronicity of bond breaking and formation, these two transition states come to mind.

There is indeed a third type of transition state for this reaction, dubbed the '''''Dewar''''' transition state, that reportedly proceeds via a diradical tight-chair intermediate species that was discovered by Dupuis et al.<ref>The Cope Rearrangement Revisited, DOI:10.1021/ja00026a007</ref>, and is thus NOT concerted.

===The Chair Transition State===

The chair transition state was initally synthesised using two "resonant" '''allyl''' ('''C3H5''') fragments in gaussview, which were optimised under the 3-21G basis set, and subsequently placed in appropriate positions relative to one another to be optimised into the transition state.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=2|'''Allyl Fragment('''C3H5''')'''
|-align="center"
|colspan=2|[[Image:Allyl Frag.jpg|275px]]
|-align="center"
|'''sp3''' Centre Bond Angle (o)||109.5
|-align="center"
|'''Allyl''' C-C Bond Angle (o)||124.3
|-align="center"
|'''C-C''' Bond Length (A)||1.54
|-align="center"
|'''C=C''' Bond Length (A)||1.34
|-align="center"
|'''Allyl C-C''' Bond Length (A)||1.39
|}

The optimised allyl C-C bond angle is greater than the normal sp3 bond angle of '''109.5o''' due to resonance.

More importantly, the allyl C-C bond length is between that of a single bond and a double bond.

This optimised allyl fragment was then taken, duplicated and placed, facing opposite directions, with roughly a 2.2A distance between the terminal carbons of each fragment to mimic the shape of the chair transition state.

Thus, a frequency + optimisation was run that directed gaussian to reach a '''TS(Berny)''', with the elucidation of force constants.

The optimisation was then repeated but this time using the '''"frozen" coordinate method''', effectively fixing the two pairs of ends at a distance of 2.2A.

Clearly, this method requires that the general structure of the transition state in question be known. However, by fixing the positions in such a way, a more accurate optimisation can be done that avoids divergence.

Finally, the optimisation was performed a third time, using a normal guess '''Hessian''' method, which involves calculation of the force constant matrix.

After the three optimisations were complete, the following data were obtained:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=3|'''<big>Optimisation of Chair Transition State</big>'''
|-align="center"
|colspan=3|'''Structure'''
|-align="center"
|'''TS(Berny)'''||'''Frozen Coordinate'''||'''Hessian'''
|-align="center"
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Berny.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Frozen.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>red</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Hessian.mol</uploadedFileContents>
</jmolApplet></jmol>
|-align="center"
|colspan=3|'''Energies''' (a.u.)
|-align="center"
| -231.61932||-231.61932||-231.69167
|}

The structures returned from the normal force-constant calculation optimisation and the frozen coordinate method are indistinguishable from first glance, and their energies are virtually identical. However, the structure obtained from the Hessian is different, and so is its energy value.

'''FREQUENCY ANALYSIS'''

Vibration-wise, there was one negative, and thus "imaginary" frequency found with a wavenumber of '''-817.91cm-1'''.

When the stretch was visualised, the following was observed:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Imaginary Stretch at -817.91cm-1'''
|-
|[[Image:Imag Stretch.jpg|300px]]
|}

The appearance of this imaginary frequency resembles the motion that would ensue under the Cope Rearrangment - the action of bond breaking of the C-C σ bond in the middle of the 1,5-hexadiene with the subsequent formation of a new σ bond on across the terminals on the other side.

The fact that there is only '''one''' imaginary frequency value points to the notion that the cope rearrangment through this transition state is ''concerted'', and thus occurs in one ''synchronous'' step.

The number of imaginary frequencies produced from a frequency analysis for the midpoint of a reaction points towards the ''synchronicity'' of the reaction<ref>Cope Rearrangement Revisited, DOI:10.1021/ja00026a007</ref>.

===The Boat Transition State===

For the boat transition state, a different method was utilised, namely the '''QST2 method''', whereby reactant and product are defined, and the transition state is converged from these.

This required the previously optimised '''1,5-hexadiene''' molecule to be taken and numbered along its carbon skeleton.

The molecule was then duplicated and juxtaposed to the intial "reactant" molecule and renumbered in such a way to reflect the reacted, rearragement "product" as shown below:-

{|style="margin:1em auto 1em auto" border="1"
|-
|[[Image:Boat Method.JPG|400px]]
|}

The first instance of optimisation failed to produce an output, and thus the molecule was changed in the following way:-

The dihedral angle of both reactant and product were changed to 0 degrees, such that the two carbon groups either side of C3 and C4 are eclipsed, and the C2-C3-C4 and C3-C4-C5 bond angles were both changed to 100 degrees.

The modified structures were once again optimised under the QST2 method to yield the following structure:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''''QST2'' Optimised Boat Transition State'''
|-
|[[Image:FINAL Boat.jpg|400px]]
|}

'''DISCUSSION'''

Here, it is illustrated that the QST2 method is automated, and is thus very efficient, but is only effective if the shape provided of the reactants and products cloely mirror that of the transition state, otherwise convergence is difficult.

The '''QST3''' method was also attempted <ref>QST Calculation, http://hdl.handle.net/10042/to-2902</ref>, whereby an addition guess of the transition state geometry was implemented. If the guess of the transition state is sound, then the QST3 is by far more accurate than the QST2 method, and does not rely so much on the shape of the reactants and products.

initally, it would seem that in order for the cope rearrangement to occur, to two carbon terminals would be required to be close to one another. This is not the case with the "''anti''" conformers of the molecule, as the terminals are situated as far apart as they may possibly be, '''antiperplanar''' to one another.

Once free rotation moves the carbon groups closer together, into the gauche configuration, where the terminals are more poised to react, cope rearrangement would proceed.

The relative orientations of the vinyl groups would ultimately dictate which transition state would be traversed. For instance, if the vectors of the vinyl groups cross, such as the case for the gauche1 conformer, then as the terminals align into a bonding orientaton, the ''chair'' conformation would be adopted by the transition state.

Conversely, if the two vinyl groups lie in a pseudo-parallel fashion, where their vectors do NOT cross, such as is the case for gauche3, then the transition state would be of ''boat'' topography.

'''IRC'''

The Intrinsic Reaction Coordinate looks more cloesly at the potential surface of a reaction and works to pinpoint the highest point in a local area of the potential surface by creeping up the steepest slopes in the immediate area.

This calculation was attempted with 50 points, and did not converge within the maximum number of points.

Thus, this was carried out again whilst calculating the force constant every single point.

{|style="margin:1em auto 1em auto" border="1"
|-
|[[Image:BO_IRC.JPG]]
|}

Here, upon closer inspection of the RMS Gradient of the energy, the gradient falls to zero, thus indicating that a minimum has been reached.

'''Energy Calculation'''

The two transition states were optimised a final time under DFT B3LYP 6-31G(d) conditions.

The geometries that were outputted were very similar to the previous examples, however, the energies received for both transition states are notably more varied:-

'''Final Energy of Chair Conformation''' (a.u.) = -234.61071
'''Final Energy of Boat Conformation''' (a.u.) = -234.54309

Here, it can be conclusively said that the ''Boat'' conformation transition state is indeed lower in energy than the ''Chair'' conformation.

This corresponds to better orbital overlap of the adjacent carbons on the framework, which are exactly eclipsed and not staggered like in the chair formation.

==The Diels-Alder Cycloaddition==

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Diels-Alder Reaction of Butadiene and Ethylene'''
|-align="center"
|[[Image:Mb_da3.jpg|400px]]
|}

The Diels-Alder Reaction is an example of a pericyclic cycloaddition, that usually involves the reaction between an electron rich diene and an electron poor dienophile.

The process is described as a [4s + 2s], 6 electron process that forms 2 new σ bonds.

When viewing the molecular orbital picture of butadiene and ethylene, the molecular orbitals are classified as either '''''s'''''(''symmetric'') or '''''a'''''(''antisymmetric'') in relation to the bisecting plane as shown below.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Bisecting Plane for Diels-alder'''
|-
|[[Image:Mb_da2.jpg|200px]]
|}

===Investigation of Molecular Orbital Symmetries of Butadiene and Ethene===

Butadiene and Ethylene were optimised and their molecular orbitals visualised:-

'''''Butadiene'''''

The HOMO and LUMO are considered with reference to the bisecting plane to determine their symmetry:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=4|'''HOMO and LUMO of Butadiene'''
|-align="center"
|width=250px|'''Molecular Orbital'''||'''2D Representation'''||'''3D Representation'''||width=250px|'''s/a'''
|-align="center"
|'''HOMO'''||[[Image:BDE HOMO.png|200px]]||[[Image:3D BDE HOMO.jpg|300px]]||a
|-align="center"
|'''LUMO'''||[[Image:BDE LUMO.png|200px]]||[[Image:3D BDE LUMO.jpg|300px]]||s
|}

'''''Ethene'''''

Similarly, the HOMO and LUMO of ethylene were characterised:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=4|'''HOMO and LUMO of Ethene'''
|-align="center"
|width=250px|'''Molecular Orbital'''||'''2D Representation'''||'''3D Representation'''||width=250px|'''s/a'''
|-align="center"
|'''HOMO'''||[[Image:ET HOMO.jpg|200px]]||[[Image:3D ET HOMO.jpg|300px]]||s
|-align="center"
|'''LUMO'''||[[Image:ET LUMO.jpg|200px]]||[[Image:3D ET LUMO.jpg|300px]]||a
|}

===Optimisation of Diels-Alder Transition State===

The '''Transition State''' whereby the concerted cycloaddition of butadiene and ethylene was modelled to a guess initially, before being optimised under HF 3-21G basis set conditions to a TS(Berny).

The initial guess was modelled from the bicyclo-system of cyclohexene and subsequently removing one of the -CH2CH2- arches to arrive at an "''envelope''"-shaped 6-membered ring.

The new σ bonds were stripped before the optimisation.

A frequency analysis was first concocted via optimisation to a transition state.

There were 3 imaginary frequencies reported by the analysis, detailed below:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=4|'''Imaginary Frequency Vibrations of Transition State'''
|-align="center"
|'''Stretches'''||[[Image:CHEX VIB 1.jpg|250px]]|||[[Image:CHEX VIB 2.jpg|250px]]|||[[Image:CHEX VIB 3.jpg|250px]]
|-align="center"
|'''Frequency''' (cm-1)||-716.19||-557.32||-198.58
|}

{|style="margin=1em auto 1em auto" border="1"
|-align="center"
|colspan=3|'''Comparisons of the Optimisations of transition state'''
|-align="center"
|'''Method'''||'''Ts(Berny)'''||'''Frozen-Coordinate'''
|-align="center"
|'''Structure'''||[[Image:CHEX BERNY.jpg|300px]]||[[Image:CHEX FROZEN.jpg|300px]]
|-align="center"
|'''Energy'''(a.u.)||-231.54077||-231.67327
|-align="center"
|'''Average new C-C σ bond length''' (A)||1.472645||1.526223
|-align="center"
|'''Literature Length of a C-C σ bond''' (A)||colspan=2|1.54
|}

By freezing the terminals, the energy of the optimised structure is markedly higher, and the length of the newly-forming σ bonds are closer to what they are experimentally measured as.

'''AM1 Molecular Orbital Analysis'''

The AM1 semi-empirical method was used to visualise the highest-occupied molecular orbital of the transition state.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''''HOMO'''''||'''''LUMO'''''
|-align="center"
|[[Image:CHEX TS HOMO.jpg|300px]]||[[Image:CHEX TS LUMO.jpg|300px]]
|-align="center"
|colspan=2|'''Symmetry'''
|-align="center"
|s||s
|-align="center"
|}

With reference to the bisecting plane as shown above, the HOMO of the transition state can be seen to be symmetric when reflected.

The individual MOs of butadiene and ethene thus must both be symmetric with respect to the plane in order to react in such a fashion.

This is consistent with the idea that only molecular orbitals with the same symmetry are allowed to react.

Given this, the pertinent orbitals that are symmetric are the '''LUMO''' of the butadiene and the '''HOMO''' of the ethene.

As such, electron density is donated into the empty π* LUMO of the butadiene.

The LUMO in this case is also symmetric, which leads me to believe that the the diels-alder reaction proceeds via the s transition state.

===Diels-Alder Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride===

The Diels-Alder reaction of '''Cyclohexa-1,3-diene and Maleic Anhydride''' is also an example of a 6-electron [4+2] Diels-Alder Cycloaddition intrinsically.

However, it differs from the reaction between butadiene and ethene in that there is now a question of regioselectivity; the bigger anhydride ring can either react to adopt an '''EXO''' or an '''ENDO''' topography:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|[[Image:EXO ENDO.jpg|400px]]
|}

The vinyl goup on maleic anhydride, in comparison to ethene, is electron-deficient, thanks to the highly electron-withdrawing nature of the anhydride group.

==References==

<references />

File:CHEX VIB 3.jpg

2009-11-14T15:56:15Z

Byt07:

File:CHEX VIB 2.jpg

2009-11-14T15:56:09Z

Byt07:

File:CHEX VIB 1.jpg

2009-11-14T15:56:01Z

Byt07:

File:CHEX TS OPTFREQ.mol

2009-11-14T15:49:56Z

Byt07:

Rep:Mod:parkbom

2009-11-14T15:14:25Z

Byt07:

=Module 3 - The Computation of the Transition State=

==Introduction==

The '''Transition State''' is an energetic maximum within a given reaction coordinate and represents the point at which bond breaking and bond forming is occurring simultaneously. This is represented by dotted lines in formal notation and depicts the transient nature of the bond(s) in question in the transition state.

The analysis of potential energy surfaces are effective in finding transition states; the transition states can be described graphically as "dams" that represent energy metastability, that the reactants must "cross" in order to form the products.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Potential Energy Surface Reaction Coordinate'''<ref>http://www.ch.ic.ac.uk/motm/porphyrins/TSdiscovery.html</ref>||'''Cope Rearrangement'''<ref>http://www.ch.ic.ac.uk/local/organic/pericyclic/</ref>
|-
|[[Image:PES.gif]]||[[Image:Cope Arr.gif|350px]]
|}

From the point of view of computational chemistry, the methods of molecular mechanical structure determination that employed imaginary force fields are ineffective in the calculations concerning the transition state in larger molecules, as they do not define the dynamics of bond making and bond forming that are apparent in the transient nature of the transition state.

As such, in combination with molecular orbital based calculations, the following of the potential energy surface of a molecule can be used to locate and analyse the transition state.

This computational exercise concerns the rearrangement of ''1,5-Hexadiene'' in the '''Cope Rearrangment'''(above).

The Cope Rearrangement proceeds via a [3,3]-Sigmatropic shift involving the motion of 6 electrons and is thermally-driven.

In terms of transition states, the cope rearrangement can reportedly proceed via two checkpoints, the '''boat''' or the '''chair''':-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Chair'''||'''Boat'''
|-
|[[Image:pic2a.jpg]]||[[Image:pic2b.jpg]]
|}

The purpose of this exercise is to ascertain which reaction pathway is preferred by computation of each transition state in the elucidation of geometry, repective energies and other thermochemical data.

==''Exercise 1'' - Optimising the Reactants and Products==

In this section, the "''reactant''" '''1,5-hexadiene''' and the "''product''" '''1,5-hexadiene''' are modelled in correspondence to the cope rearrangement.

The flexible nature of 1,5-hexadiene means that there is an initial dilemma of likely conformation at rest.

Several starting conformations were thus taken and their geometries optimised to find the lowest energy conformer.

Initially, 1,5-hexadiene was taken, ensuring a roughly "'''anti'''" linkage in the centre of the molecule, so that the central C atoms '''3''' and '''4''' are approximately '''antiperiplanar''' to one another. This molecule was cleaned in the gaussview builder interface and then taken and optimised using the '''Hartree-Fock''' method with basis set '''3-21G''' through Gaussian.

'''b)''' similarly, a synclinal, or "gauche" version of 1,5-hexadiene was taken and optimised at the HF 3-21G level.

The energies of both conformers and also their symmetry were noted.

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|'''"''Anti''" 1,5-Hexadiene'''||'''"''Gauche''" 1,5-Hexadiene'''
|-align="center"
|[[Image:Bo Anti.jpg|420px]]||[[Image:Bo Gauche.jpg|400px]]
|}

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|'''Confomer'''||'''''Anti''-1,5-Hexadiene'''||'''''Gauche''-1,5-Hexadiene'''
|-align="center"
|'''Total Energy'''(a.u.)||-231.69254||-231.69266
|-align="center"
|'''Point Group'''||ci||None
|}

From a purely steric-orientated argument, the ''anti'' configuration should have a lower energy than the ''gauche'', as there is less steric hindrance from the two ends of the carbon chain 180 degrees from one another than next to each other in the gauche conformation.

As it so happens, the energy for the ''anti'' conformer is reportedly lower than that of the ''gauche'' conformer, but by a minute amount of roughly a ten-thousandth of a hartree ('''1 hartree = 4.3597482E-18 J''').

Even so, it was appreciated that the central C-C-C-C linkage of 1,5-hexadiene lends itself to quite a high degree of rotational freedom, and thus, further examples of conformational isomerism. This is reinforced by the fact of the two remarkably different conformers having remarkably similar energies, and points to the fact that 1,5-hexadiene experiences conformational ambiguity often<ref>Cope Rearrangement of 1,5-Hexadiene: Full Geometry Optimizations Using Analytic MR-CISD and MR-AQCC Gradient Methods, doi:10.1021/jp0259014</ref>.

As such, 2 other conformers were taken and optimised:-

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|colspan=4|'''<big>Different Conformers of 1,5-Hexadiene</big>'''
|-align="center"
|(''g120,a,g120'')-'''Anti1'''||(''g120,a,g-120'')-'''Anti2'''||(''g120,g-60,g120'')-'''Gauche1'''||(''g120,g60,g-120'')-'''Gauche3'''
|-align="center"
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Bo Anti1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Bo Anti2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Gauche 1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Gauche 3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-align="center"
|colspan=4|'''Energy'''(a.u.)
|-align="center"
| -231.69260213||-231.69253506||-231.69166702||-231.69266120
|-align="center"
|colspan=4|'''Point Group'''
|-align="center"
|C2||Ci||C2||C1
|}

The energies here from the HF 3-21G are similarly close, but contrary to what was expected, it was found that ''gauche3'' was actually the lowest in energy.

Beneath this, the ''anti'' conformers are found to be lower in energy than the other ''gauche'' conformer. This may be rationalised as the minimalisation of steric hindrance lowering the energy of the overall molecule, as the antiperiplanar groups maintain the least steric strain.

However, the '''''cis'''''-''gauche'' conformer is lowest in energy due to stereoelectronic arguments.

The '''π'''-bonds of the C=C bonds are lying criss-crossed over each other in this conformer. This promotes effectual π-molecular orbital overlap and hence stabilises the molecule.

Next, the anti2 conformer with ci symmetry and the gauche1 conformer was taken and further optimised under the DFT B3LYP method with a more sophisticated basis set, 6-31G(d).

The resulting optimised molecule was then compared with the earlier attempt in terms of energy.

A further frequency analysis was conducted in order to check that the optimisation has proceeded to a minimum, being the case when all frequencies were real and positive.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=5|'''Comparison of the HF 3-21G and DFT B3LYP 6-31G(d) Optimisations'''
|-align="center"
|'''Method'''||colspan=2|'''HF 3-21G'''||colspan=2|'''DFT B3LYP 6-31G(d)'''
|-align="center"
|'''Conformer'''||'''anti2'''||'''gauche1'''||'''anti2'''||'''gauche1'''
|-align="center"
|'''Total Energy'''(a.u.)||-231.69253506||-231.69166702||-231.61170616||-231.61068821
|-align="center"
|1-2 C=C Bond Length (A)||1.3162||1.3156||1.3335||1.3331
|-align="center"
|2-3 C-C Bond Length (A)||1.5091||1.5041||1.5083||1.5044
|-align="center"
|3-4 C-C Bond Length (A)||1.5527||1.5481||1.5508||1.5482
|-align="center"
|4-5 C-C Bond Length (A)||1.5091||1.5041||1.5083||1.5044
|-align="center"
|5-6 C=C Bond Length (A)||1.3163||1.3335||1.3157||1.3331
|-align="center"
|1-2-3-4 Dihedral (o)||114.66||118.49||123.93||124.24
|-align="center"
|2-3-4-5 Dihedral (o)||180.00||180.00||-64.18||-65.01
|-align="center"
|3-4-5-6 Dihedral (o)||-114.66||-118.49||123.93||124.24
|}

As can be seen here, the energies returned differ in a noticeable manner. However, comparison of energies computed under different basis sets is unbalanced.

As such, comparisons of the geometry are notably more reliable in discerning the computed differences.

The bond lengths differ by negligible amounts. However, it can be seen that the various dihedral angles have been "tightened" more to their ideal figures (closer to 120o for the anticlinal dihedrals.

As such, using a more sophisticated basis set ensures a more accurate approximation to reality.

The log file yielded the following thermochemical data:-

'''1)Sum of electronic and zero-point Energies= -234.469195'''

'''2)Sum of electronic and thermal Energies= -234.461847'''

'''3)Sum of electronic and thermal Enthalpies= -234.460903'''

'''4)Sum of electronic and thermal Free Energies= -234.500782'''

Theses values concern the 1)Potential Energy at 0K that includes the zero-point vibrational energy term 2)Potential Energy at 298.15K and 1atm and adds contributions from translational, rotational and vibrational energy 3)Energy including a correction for RT(H=E+RT) for dissociation reactions and 4)The Entropic contribution.

==''Exercise 2'' - Optimising the "Chair" and "Boat" Transition Structures==

The Cope Rearrangement has been found to proceed via one of two possible transition states with differing conformation, as mentioned in the introduction:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=2|'''<big>Cope Rearrangement Transition State Conformations</big>'''
|-align="center"
|''Chair''||''Boat''
|-align="center"
|[[Image:pic2a.jpg]]||[[Image:pic2b.jpg]]
|}

When the rearrangment is thought as one that is concerted and proceeds with synchronicity of bond breaking and formation, these two transition states come to mind.

There is indeed a third type of transition state for this reaction, dubbed the '''''Dewar''''' transition state, that reportedly proceeds via a diradical tight-chair intermediate species that was discovered by Dupuis et al.<ref>The Cope Rearrangement Revisited, DOI:10.1021/ja00026a007</ref>, and is thus NOT concerted.

===The Chair Transition State===

The chair transition state was initally synthesised using two "resonant" '''allyl''' ('''C3H5''') fragments in gaussview, which were optimised under the 3-21G basis set, and subsequently placed in appropriate positions relative to one another to be optimised into the transition state.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=2|'''Allyl Fragment('''C3H5''')'''
|-align="center"
|colspan=2|[[Image:Allyl Frag.jpg|275px]]
|-align="center"
|'''sp3''' Centre Bond Angle (o)||109.5
|-align="center"
|'''Allyl''' C-C Bond Angle (o)||124.3
|-align="center"
|'''C-C''' Bond Length (A)||1.54
|-align="center"
|'''C=C''' Bond Length (A)||1.34
|-align="center"
|'''Allyl C-C''' Bond Length (A)||1.39
|}

The optimised allyl C-C bond angle is greater than the normal sp3 bond angle of '''109.5o''' due to resonance.

More importantly, the allyl C-C bond length is between that of a single bond and a double bond.

This optimised allyl fragment was then taken, duplicated and placed, facing opposite directions, with roughly a 2.2A distance between the terminal carbons of each fragment to mimic the shape of the chair transition state.

Thus, a frequency + optimisation was run that directed gaussian to reach a '''TS(Berny)''', with the elucidation of force constants.

The optimisation was then repeated but this time using the '''"frozen" coordinate method''', effectively fixing the two pairs of ends at a distance of 2.2A.

Clearly, this method requires that the general structure of the transition state in question be known. However, by fixing the positions in such a way, a more accurate optimisation can be done that avoids divergence.

Finally, the optimisation was performed a third time, using a normal guess '''Hessian''' method, which involves calculation of the force constant matrix.

After the three optimisations were complete, the following data were obtained:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=3|'''<big>Optimisation of Chair Transition State</big>'''
|-align="center"
|colspan=3|'''Structure'''
|-align="center"
|'''TS(Berny)'''||'''Frozen Coordinate'''||'''Hessian'''
|-align="center"
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Berny.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Frozen.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>red</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Hessian.mol</uploadedFileContents>
</jmolApplet></jmol>
|-align="center"
|colspan=3|'''Energies''' (a.u.)
|-align="center"
| -231.61932||-231.61932||-231.69167
|}

The structures returned from the normal force-constant calculation optimisation and the frozen coordinate method are indistinguishable from first glance, and their energies are virtually identical. However, the structure obtained from the Hessian is different, and so is its energy value.

'''FREQUENCY ANALYSIS'''

Vibration-wise, there was one negative, and thus "imaginary" frequency found with a wavenumber of '''-817.91cm-1'''.

When the stretch was visualised, the following was observed:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Imaginary Stretch at -817.91cm-1'''
|-
|[[Image:Imag Stretch.jpg|300px]]
|}

The appearance of this imaginary frequency resembles the motion that would ensue under the Cope Rearrangment - the action of bond breaking of the C-C σ bond in the middle of the 1,5-hexadiene with the subsequent formation of a new σ bond on across the terminals on the other side.

The fact that there is only '''one''' imaginary frequency value points to the notion that the cope rearrangment through this transition state is ''concerted'', and thus occurs in one ''synchronous'' step.

The number of imaginary frequencies produced from a frequency analysis for the midpoint of a reaction points towards the ''synchronicity'' of the reaction<ref>Cope Rearrangement Revisited, DOI:10.1021/ja00026a007</ref>.

===The Boat Transition State===

For the boat transition state, a different method was utilised, namely the '''QST2 method''', whereby reactant and product are defined, and the transition state is converged from these.

This required the previously optimised '''1,5-hexadiene''' molecule to be taken and numbered along its carbon skeleton.

The molecule was then duplicated and juxtaposed to the intial "reactant" molecule and renumbered in such a way to reflect the reacted, rearragement "product" as shown below:-

{|style="margin:1em auto 1em auto" border="1"
|-
|[[Image:Boat Method.JPG|400px]]
|}

The first instance of optimisation failed to produce an output, and thus the molecule was changed in the following way:-

The dihedral angle of both reactant and product were changed to 0 degrees, such that the two carbon groups either side of C3 and C4 are eclipsed, and the C2-C3-C4 and C3-C4-C5 bond angles were both changed to 100 degrees.

The modified structures were once again optimised under the QST2 method to yield the following structure:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''''QST2'' Optimised Boat Transition State'''
|-
|[[Image:FINAL Boat.jpg|400px]]
|}

'''DISCUSSION'''

Here, it is illustrated that the QST2 method is automated, and is thus very efficient, but is only effective if the shape provided of the reactants and products cloely mirror that of the transition state, otherwise convergence is difficult.

The '''QST3''' method was also attempted <ref>QST Calculation, http://hdl.handle.net/10042/to-2902</ref>, whereby an addition guess of the transition state geometry was implemented. If the guess of the transition state is sound, then the QST3 is by far more accurate than the QST2 method, and does not rely so much on the shape of the reactants and products.

initally, it would seem that in order for the cope rearrangement to occur, to two carbon terminals would be required to be close to one another. This is not the case with the "''anti''" conformers of the molecule, as the terminals are situated as far apart as they may possibly be, '''antiperplanar''' to one another.

Once free rotation moves the carbon groups closer together, into the gauche configuration, where the terminals are more poised to react, cope rearrangement would proceed.

The relative orientations of the vinyl groups would ultimately dictate which transition state would be traversed. For instance, if the vectors of the vinyl groups cross, such as the case for the gauche1 conformer, then as the terminals align into a bonding orientaton, the ''chair'' conformation would be adopted by the transition state.

Conversely, if the two vinyl groups lie in a pseudo-parallel fashion, where their vectors do NOT cross, such as is the case for gauche3, then the transition state would be of ''boat'' topography.

'''IRC'''

The Intrinsic Reaction Coordinate looks more cloesly at the potential surface of a reaction and works to pinpoint the highest point in a local area of the potential surface by creeping up the steepest slopes in the immediate area.

This calculation was attempted with 50 points, and did not converge within the maximum number of points.

Thus, this was carried out again whilst calculating the force constant every single point.

{|style="margin:1em auto 1em auto" border="1"
|-
|[[Image:BO_IRC.JPG]]
|}

Here, upon closer inspection of the RMS Gradient of the energy, the gradient falls to zero, thus indicating that a minimum has been reached.

'''Energy Calculation'''

The two transition states were optimised a final time under DFT B3LYP 6-31G(d) conditions.

The geometries that were outputted were very similar to the previous examples, however, the energies received for both transition states are notably more varied:-

'''Final Energy of Chair Conformation''' (a.u.) = -234.61071
'''Final Energy of Boat Conformation''' (a.u.) = -234.54309

Here, it can be conclusively said that the ''Boat'' conformation transition state is indeed lower in energy than the ''Chair'' conformation.

This corresponds to better orbital overlap of the adjacent carbons on the framework, which are exactly eclipsed and not staggered like in the chair formation.

==The Diels-Alder Cycloaddition==

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Diels-Alder Reaction of Butadiene and Ethylene'''
|-align="center"
|[[Image:Mb_da3.jpg|400px]]
|}

The Diels-Alder Reaction is an example of a pericyclic cycloaddition, that usually involves the reaction between an electron rich diene and an electron poor dienophile.

The process is described as a [4s + 2s], 6 electron process that forms 2 new σ bonds.

When viewing the molecular orbital picture of butadiene and ethylene, the molecular orbitals are classified as either '''''s'''''(''symmetric'') or '''''a'''''(''antisymmetric'') in relation to the bisecting plane as shown below.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Bisecting Plane for Diels-alder'''
|-
|[[Image:Mb_da2.jpg|200px]]
|}

===Investigation of Molecular Orbital Symmetries of Butadiene and Ethene===

Butadiene and Ethylene were optimised and their molecular orbitals visualised:-

'''''Butadiene'''''

The HOMO and LUMO are considered with reference to the bisecting plane to determine their symmetry:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=4|'''HOMO and LUMO of Butadiene'''
|-align="center"
|width=250px|'''Molecular Orbital'''||'''2D Representation'''||'''3D Representation'''||width=250px|'''s/a'''
|-align="center"
|'''HOMO'''||[[Image:BDE HOMO.png|200px]]||[[Image:3D BDE HOMO.jpg|300px]]||a
|-align="center"
|'''LUMO'''||[[Image:BDE LUMO.png|200px]]||[[Image:3D BDE LUMO.jpg|300px]]||s
|}

'''''Ethene'''''

Similarly, the HOMO and LUMO of ethylene were characterised:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=4|'''HOMO and LUMO of Ethene'''
|-align="center"
|width=250px|'''Molecular Orbital'''||'''2D Representation'''||'''3D Representation'''||width=250px|'''s/a'''
|-align="center"
|'''HOMO'''||[[Image:ET HOMO.jpg|200px]]||[[Image:3D ET HOMO.jpg|300px]]||s
|-align="center"
|'''LUMO'''||[[Image:ET LUMO.jpg|200px]]||[[Image:3D ET LUMO.jpg|300px]]||a
|}

===Optimisation of Diels-Alder Transition State===

The '''Transition State''' whereby the concerted cycloaddition of butadiene and ethylene was modelled to a guess initially, before being optimised under HF 3-21G basis set conditions to a TS(Berny).

The initial guess was modelled from the bicyclo-system of cyclohexene and subsequently removing one of the -CH2CH2- arches to arrive at an "''envelope''"-shaped 6-membered ring.

The new σ bonds were stripped before the optimisation.

When this optimisation was complete, the structure obtained was slightly distorted, and so another optimisation using the frozen coordinate method.

The newly-optimised structure represented the "envelope", boat-like transition state a lot more accurately.

{|style="margin=1em auto 1em auto" border="1"
|-align="center"
|colspan=3|'''Comparisons of the Optimisations of transition state'''
|-align="center"
|'''Method'''||'''Ts(Berny)'''||'''Frozen-Coordinate'''
|-align="center"
|'''Structure'''||[[Image:CHEX BERNY.jpg|300px]]||[[Image:CHEX FROZEN.jpg|300px]]
|-align="center"
|'''Energy'''(a.u.)||-231.54077||-231.67327
|-align="center"
|'''Average new C-C σ bond length''' (A)||1.472645||1.526223
|-align="center"
|'''Literature Length of a C-C σ bond''' (A)||colspan=2|1.54
|}

By freezing the terminals, the energy of the optimised structure is markedly higher, and the length of the newly-forming σ bonds are closer to what they are experimentally measured as.

'''AM1 Molecular Orbital Analysis'''

The AM1 semi-empirical method was used to visualise the highest-occupied molecular orbital of the transition state.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''''HOMO'''''||'''''LUMO'''''
|-align="center"
|[[Image:CHEX TS HOMO.jpg|300px]]||[[Image:CHEX TS LUMO.jpg|300px]]
|-align="center"
|colspan=2|'''Symmetry'''
|-align="center"
|s||s
|-align="center"
|}

With reference to the bisecting plane as shown above, the HOMO of the transition state can be seen to be symmetric when reflected.

The individual MOs of butadiene and ethene thus must both be symmetric with respect to the plane in order to react in such a fashion.

This is consistent with the idea that only molecular orbitals with the same symmetry are allowed to react.

Given this, the pertinent orbitals that are symmetric are the '''LUMO''' of the butadiene and the '''HOMO''' of the ethene.

As such, electron density is donated into the empty π* LUMO of the butadiene.

The LUMO in this case is also symmetric, which leads me to believe that the the diels-alder reaction proceeds via the s transition state.

===Diels-Alder Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride===

The Diels-Alder reaction of '''Cyclohexa-1,3-diene and Maleic Anhydride''' is also an example of a 6-electron [4+2] Diels-Alder Cycloaddition intrinsically.

However, it differs from the reaction between butadiene and ethene in that there is now a question of regioselectivity; the bigger anhydride ring can either react to adopt an '''EXO''' or an '''ENDO''' topography:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|[[Image:EXO ENDO.jpg|400px]]
|}

The vinyl goup on maleic anhydride, in comparison to ethene, is electron-deficient, thanks to the highly electron-withdrawing nature of the anhydride group.

==References==

<references />

Rep:Mod:chaerina

2009-11-14T15:09:54Z

Byt07:

='''Module 2''' - Molecular Modelling to Determine the Structure and Stereoelectronic Properties of Simple Molecules=

=='''Exercise 1''' - Boronic Compounds==

===A Molecule of BH3 and BCl3===

====Introduction====

'''Molecular Modelling''' is a method that employs ''Quantum Mechanical'' approximations to converge into an optimised structure that is as close to reality as possible.

Programs like '''Gaussian''' use so-called "'''basis sets'''" in order to piece together the relevant units that resemble the constituent nuclei and electron distributions and converge to optimisation.

The basis sets themselves vary in terms of sophistication, which has some bearing on the quality of the resulting optimisation. Be it the case that a more sophisticated basis set returns a more accurate representation of a molecule's structure, the time taken to reach the level of optimisation required would necessitate a longer period of time. As such, a compromise between the quality of optimisation and the amount of time to reach optimisation. With smaller molecules, a larger basis set may be used, however, with larger molecules, larger basis sets would result in extended computing times.

====The optimisation of BH3====

BH3 was optimised by way of DFT method B3LYP basis set 3-21G, returning the following values for the '''bond lengths''' and '''bond angles''':-

{|style="margin: 1em auto 1em auto" border="1"
|-align="center"
|'''Bond Length (Å)'''||'''Bond Angle (o)'''
|-align="center"
|1.19||120.0
|}

The summary of the Gaussian job was given by the following window and charts:-

{|style="float:left;" border="1"
|-
|[[Image:BH3_Summary.JPG|242px]]
|}

{| style="float:right;" border="1"
|[[Image:BH3_charts.JPG|400px]]
|}

{| border="1"
|-align="center"
|'''Question'''||style="width=400px;"| '''Answer'''
|-align="center"
|'''''What is the file type?'''''||.log
|-align="center"
|'''''What is the calculation type?'''''||FOPT
|-align="center"
|'''''What is the calculation method?'''''||RB3LYP
|-align="center"
|'''''What is the basis set?'''''||3-21G
|-align="center"
|'''''What is the final energy in a.u. (atomic units)?'''''||-26.46
|-align="center"
|'''''What is the gradient?'''''||0.00000285
|-align="center"
|'''''What is the dipole moment? (debye)'''''||0.0
|-align="center"
|'''''What is the point group?'''''||D3H
|}

The graphs shown here represent the energy and the gradient of the energy throughout the process of optimisation. Convergence of the molecular structure is indicated here by the fact that the energy is at the lowest and the gradient is at its lowest at the end of optimisation, as seen in the graphs. It should be noted that the starting B-H bond length was set to 1.5Å and was optimised to the recorded value of 1.19435Å. The placement of bonds by Gaussview is dependent on a set distance between atoms. Further than this distance, gaussview does not indicate the existence of a bond.

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|colspan=5|'''<big>Stage in Optimisation</big>'''
|-align="center"
|[[Image:BH3_OPT_1.jpg|200px]]||[[Image:BH3_OPT_2.jpg|200px]]||[[Image:BH3_OPT_3.jpg|200px]]||[[Image:BH3_OPT_4.jpg|200px]]||[[Image:BH3_OPT_5.jpg|200px]]
|-align="center"
|colspan=5|'''Bond Length (''Å'')'''
|-align="center"
|1.50||1.41||1.28||1.19||1.19
|}

As can be seen from the optimisation profile, gaussview only draws in the bonds at stage 4; the existence of a bond is not acknowledged prior to that stage. It is important to respect the fact that formal bonding is seen as a direct linkage between atoms, which is purely a stylistic representation of a REAL chemical bond. In reality, and as expressed in molecular orbital terms, a chemical bond is represented by the electron cloud between the two atoms and its shape in comparison to the atomic case.

====Molecular Orbital (MO) and Natural Bond Orbital (NBO) Analysis of BH3====

Gaussian can also predict the electronic interactions of the atoms within the molecule, building up a molecular orbital picture of the entire molecule.

The MOs are visualised as lobes with varying phase, the MOs obtained for BH3 are as follows:-

{| style="float:left;" border="1"
|-align="center"
|colspan=2|'''''<big>BH3 Molecular Orbitals</big>'''''
|-align="center"
|'''MO'''||style="width:250px"|'''MO Type'''
|-align="center"
|[[Image:Boris_BH3_MO1.jpg|200px]]||'''A1`'''
|-align="center"
|[[Image:Boris_BH3_MO2.jpg|200px]]||'''A1`'''
|-align="center"
|[[Image:Boris_BH3_MO3.jpg|200px]]||'''A2``'''
|-align="center"
|[[Image:Boris_BH3_MO4.jpg|200px]]||'''HOMO''', '''E`'''
|-align="center"
|[[Image:Boris_BH3_MO5.jpg|200px]]||'''LUMO''', '''A1`'''
|}

The NBO approach analyses the charge distribution of the molecule and is in some ways a surmising method of the molecular orbital theory method.

The scale is to the effect of the following table:-

{| style="float:right;" border="1"
|-align="center"
|'''<big>NBO Analysis of BH3'''
|-align="center"
|'''Colour Representation'''
|-align="center"
|[[Image:BH3_NBO_Colors.jpg|330px]]
|-align="center"
|'''Numerical Representation'''
|-align="center"
|[[Image:BH3_NBO_Numbers.jpg|330px]]
|}

{| style="float:right;" border="2"
|-
|[[Image:NBO_Window.JPG]]
|}

The table shows a scale of charge that ranges from very negatively charged ('''red''') to very positively charged (''' green''').
As can be seen in the diagram of BH3, the Boron is highly positively charged, whereas the Hydrogens are moderately negatively charged. This is consistent with reality as boron is very lewis acidic, being electron deficient and having a high charge/mass ratio.

Although this gives a clear picture of the relative charges of the individual atoms, the diagram is highly stylised once again, and says nothing on the electronic level. A table of orbital contributions and ratios is also given by gaussview in the NBO analysis:-

{|style="float:right;" border="1"
|-align="center"
|'''<big>Bonding Orbital Contributional Analysis</big>'''
|-
|
(Occupancy) Bond orbital/ Coefficients/ Hybrids
---------------------------------------------------------------------------------
1. (1.99851) BD ( 1) B 1 - H 2
( 44.49%) 0.6670* B 1 s( 33.33%)p 2.00( 66.67%)
0.0000 0.5774 0.0000 0.0000 0.0000
0.8165 0.0000 0.0000 0.0000
( 55.51%) 0.7451* H 2 s(100.00%)
1.0000 0.0000
2. (1.99851) BD ( 1) B 1 - H 3
( 44.49%) 0.6670* B 1 s( 33.33%)p 2.00( 66.67%)
0.0000 0.5774 0.0000 0.7071 0.0000
-0.4082 0.0000 0.0000 0.0000
( 55.51%) 0.7451* H 3 s(100.00%)
1.0000 0.0000
3. (1.99851) BD ( 1) B 1 - H 4
( 44.49%) 0.6670* B 1 s( 33.33%)p 2.00( 66.67%)
0.0000 0.5774 0.0000 -0.7071 0.0000
-0.4082 0.0000 0.0000 0.0000
( 55.51%) 0.7451* H 4 s(100.00%)
1.0000 0.0000
4. (1.99904) CR ( 1) B 1 s(100.00%)
1.0000 0.0000 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000
|}

This table shows the contributions from either atom towards each "natural bonding" orbital and their respective hybridisations. It can be seen that the first three are identical in terms of ratio and hybridisation, and represent each of the 3 B-H bonds. The 4th NBO has no contribution from hydrogen and is thus the boron ''1s'' orbital. In the first 3, the orbitals of boron are one third s-character and two thirds p-character, whereas with hydrogen there is no hybridisation.

This is consistent with the idea that the boron is sp2 hybridised, on account of its bonding order and angle, thus having the reported ratios of orbital character in its hybridisation.

====Structural analysis of BCl3====

In this exercise, a structurally-similar '''Boron Chloride (BCl3)''' (''see below'') was taken and specified a point group of '''D3h''' before having its structure optimised.

{|style="margin:1em auto 1em auto;" border="2"
|-align="center"
|'''<big>BH3</big>'''||'''<big>BCl3</big>'''||'''Questions'''||colspan=2|'''Answers'''
|-align="center"
|rowspan=5|[[Image:Boris_BH3.jpg|265px]]||rowspan=5|[[Image:Boris_BCl3.jpg|265px]]||height=40px|'''''What is the calculation method?'''''||colspan=2|R3BYLP
|-align="center"
|height=40px|'''''what is the basis set?'''''||colspan=2|3-21G
|-align="center"
|height=40px|'''''How long did the calculation take?'''''||colspan=2|12 seconds
|-align="center"
|height=40px|'''''What is the B-Cl Bond Length(Å)?'''''||1.87||1.75('''LIT.''')
|-align="center"
|height=40px|'''''What is the Bond Angle(o)?'''''||width=80px|120.0||width=80px|120.0('''LIT.''')
|}

In comparison to BH3, which has a bond length of '''1.19Å''', BCl3 has a bond length of '''1.87Å''', which is notably longer. This may be rationalised by the fact that the B_Cl bonds are more '''polar''' due to the larger electronegativity difference between boron and chlorine. The electron withdrawing nature of chlorine siphons electron density from the bond, thereby making it weaker and therefore longer. Both species are totally symmetric, however, and share a bond angle of 120o.

Both species need to optimised using the same method and basis set. This is so that their optimised structures may be reliably compared under the same environment and constituent units.

A frequency analysis was also utilised:-

{|style="margin:1em auto 1em auto;" border="1"
|-
|[[Image:BCl3_Freq.JPG|500px]]
|}

In order for a minimum to be reached, the frequencies of the stretches and bend of the molecule must all be positive. From this frequency analysis, one can know that the proposed structure is a minimum.

=='''Exercise''' 2 - Cis/Trans Isomerisation in a [Mo(CO)4(PCl3)2]==

There are 2 possible stereoisomers for '''Dicarboxytetraphosphoniumchloride Molybdenum''' [Mo(CO)4(PCl3)2], as detailed below:-

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|colspan=2|'''Isomers of <big>[Mo(CO)4(PCl3)2]</big>'''
|-align="center"
|'''''Cis-'''''||'''''Trans-'''''
|-align="center"
|[[Image:CISMO.jpg|400px]]||[[Image:TRANSMO.jpg|400px]]
|}

The two stereoisomers can be distinguished by means of Vibrational Frequency Analysis, or in this case, its simulation and paying special attention to the carbonyl stretches within the complex.

The frequency data are presented below:-

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|colspan=4|'''<big>Frequency Analysis of CIS-[Mo(CO)4(PCl3)2]</big>'''||colspan=4|'''<big>Frequency Analysis of TRANS-[Mo(CO)4(PCl3)2]</big>'''
|-align="center"
|'''CO Stretch'''||'''Description of Stretch'''||'''Frequency(cm-1)'''||'''Intensity'''||'''CO Stretch'''||'''Description of Stretch'''||'''Frequency(cm-1)'''||'''Intensity'''
|-align="center"
|[[Image:CISMO_STRETCH_1.JPG|200px]]||"CIS" Stretch||1944.84||766.72||[[Image:TRANSMO_STRETCH_1.JPG|200px]]||Asymmetric Stretch||1944.84||766.72
|-align="center"
|[[Image:CISMO_STRETCH_2.JPG|200px]]||"TRANS" Stretch||1949.48||1489.46||[[Image:TRANSMO_STRETCH_2.JPG|200px]]||Symmetric Stretch||1949.48||1489.46
|-align="center"
|[[Image:CISMO_STRETCH_3.JPG|200px]]||Asymmetric Full Stretch||1958.44||650.032||[[Image:TRANSMO_STRETCH_3.JPG|200px]]||Asymmetric Full Stretch||1958.44||650.032
|-align="center"
|[[Image:CISMO_STRETCH_4.JPG|200px]]||Symmetric Full Stretch||2023.74||568.254||[[Image:TRANSMO_STRETCH_4.JPG|200px]]||Symmetric Full Stretch||2023.74||568.254
|}

'''*NB:- The arrows here correspond to the carbonyl OXYGEN's movement in relation to the carbon.'''

The following table was also returned from the frequency analysis:-

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|'''CIS-'''[Mo(CO)4(PCl3)2]||'''TRANS-'''[Mo(CO)4(PCl3)2]
|-align="center"
|[[Image:CISMO_FREQ_CHART.JPG|400px]]||[[Image:TRANSMO_FREQ_CHART.JPG|412px]]
|}

It can be noted that the the frequency chart of the '''CIS-''' isomer contains 4 peaks around the νCO region, whereas the '''TRANS-''' isomer contains but one.

This can be explained mainly due to the overall symmetry of the complex. The trans complex has greater symmetry than the cis complex, and this is shown by the fact that the νCO stretches are not degenerate, whereas the stretches in the trans complex are.

=='''Exercise''' 3 - Mini-Project - Ammonia Borane NH3BH3 and its application as a Hydrogen Storage Medium==

'''Ammonia Borane''' (pictured below), is being researched extensively as a new medium for the storage of Hydrogen, with the onset of hydrogen as a source of energy.

Intrinsically, it is isoelectronic with '''''ethane''''', and shares a similar structure. As opposed to the non-polar bond nature of ethane, ammonia borane exists thanks to the dative bond between the boron and nitrogen. The lone pair present on the nitrogen in ammonia donates itself into the empty p orbital of the lewis-acidic boron, thus forming a dative bond.

{|style="margin:1em auto 1em auto;" border="1"
|-align:"center"
|[[Image:Ammonia_Borane.jpg|400px]]||[[Image:Boris_Ethane.jpg|400px]]
|-align="center"
|'''Ammonia Borane''', '''''NH3BH3'''''||'''Ethane''', '''''C2H6'''''
|}

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|colspan=4|'''<big>Optimised Bonding Data</big>'''
|-align="center"
|'''Property'''||colspan=2|'''NH3BH3'''||'''C2H6'''
|-align="center"
|'''B-N/C-C Bond Length'''||colspan=2|1.65||1.53||
|-align="center"
|'''H-X Bond Length (where ''X=C/B/N'')||1.21(B-H)||1.02(N-H)||1.09
|}

{| style="margin:1em auto 1em auto;" border="1"
|-align="center"
|'''Question'''||style="width=400px;"|'''Ammonia Borane'''||style="width=400px;"|'''Ethane'''
|-align="center"
|'''''What is the file type?'''''||.fch||.fch
|-align="center"
|'''''What is the calculation type?'''''||FOPT||SP
|-align="center"
|'''''What is the calculation method?'''''||RMP2-FC||RMP2-FC
|-align="center"
|'''''What is the basis set?'''''||6-311G(D,P)||6-311G(D,P)
|-align="center"
|'''''What is the final energy in a.u. (atomic units)?'''''||-86.96||-79.57
|-align="center"
|'''''What is the gradient?'''''||0.0000798||0.0000000
|-align="center"
|style="background:yellow;"|'''''What is the dipole moment? (debye)'''''||style="background:yellow;"|5.56||style="background:yellow;"|0.00
|}

====MO and NBO Analysis and Comparison of Ammonia Borane and Ethane====

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|colspan=5|'''<big>Molecular Orbital Comparison</big>'''
|-align="center"
|colspan=5|'''''NH3BH3'''''
|-align="center"
|colspan=5|'''Molecular Orbital'''
|-align="center"
|1||2||3||4||5
|-align="center"
|[[Image:AB_MO_1.jpg|200px]]||[[Image:AB_MO_2.jpg|200px]]||[[Image:AB_MO_3.jpg|200px]]||[[Image:AB_MO_4.jpg|200px]]||[[Image:AB_MO_5.jpg|200px]]
|-align="center"
|colspan=5|'''''C2H6'''''
|-align="center"
|[[Image:Ethane_MO_1.jpg|200px]]||[[Image:Ethane_MO_2.jpg|200px]]||[[Image:Ethane_MO_3.jpg|200px]]||[[Image:Ethane_MO_4.jpg|200px]]||[[Image:Ethane_MO_5.jpg|200px]]
|-align="center"
|colspan=5|'''''NH3BH3'''''
|-align="center"
|colspan=5|'''Molecular Orbital'''
|-align="center"
|6||7||8||9||10
|-align="center"
|[[Image:AB_MO_6.jpg|200px]]||[[Image:AB_MO_7.jpg|200px]]||[[Image:AB_MO_8.jpg|200px]]||[[Image:AB_MO_9.jpg|200px]]||[[Image:AB_MO_10.jpg|200px]]
|-align="center"
|colspan=5|'''''C2H6'''''
|-align="center"
|[[Image:Ethane_MO_6.jpg|200px]]||[[Image:Ethane_MO_7.jpg|200px]]||[[Image:Ethane_MO_8.jpg|200px]]||[[Image:Ethane_MO_9.jpg|200px]]||[[Image:Ethane_MO_10.jpg|200px]]
|}

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|colspan=2|'''<big>NBO Picture</big>'''
|-align="center"
|'''''NH3BH3'''''||'''''C2H6'''''
|-align="center"
|[[Image:AB_NBO.jpg|400px]]||[[Image:Ethane_NBO.jpg|400px]]
|}

As can be seen, there are two striking differences between the two NBO pictures. The first is that charge distribution-wise, ethane is completely symmetric, whereas ammonia borane is rather polar. The second is that the hydrogens attached to the boron are not equivalent to the hydrogens attached to the nitrogen in ammonia borane. From this, it is apparent that ammonia borane is polar whereas homonuclear ethane is not. This has large consequences to the physical properties of each species, as will be discussed in the following section.

===Characteristics of Ammonia Borane and its application as a hydrogen storage medium===

It is interesting that while Ammonia Borane is isoelectronic, thus having the same number of electrons and structure, with ethane, their physical properties vary substantially. Ammonia Borane, for example, exists as a '''orthorhombic crystalline solid''' at room temperature, whereas ethane is a gas. Reportedly, the melting point for Ammonia Borane has a melting point of 104o, whereas that of ethane is -181.76o, differing greatly.

Analysing the values above concerning the final energies of each molecule, it can be seen that ammonia borane has the lower energy and is therefore more thermodynamically stable than its counterpart.

The reason behind this astonishing phenomenon is due to the highly polar nature of ammonia borane. Its crystalline structure is in fact the result of a specialised form of hydrogen bonding, thought by many formally as the interaction between an electronegative atom and hydrogen, which is deshielded into a protonic state. This special form of hydrogen bonding does not follow the convention of a hydrogen, atom, hydrogen, atom alternative pattern, but is rather an interaction between respective hydrogens bonded alternatively to boron and nitrogen. Due to the difference in electronegativity of the two atoms and thus the polarity of the B-N bond, the hydrogens are placed into different environments. The hydrogens on the boron are found to be rather hydridic, as visualised in the NBO colour picture as being slightly negatively charged. Conversely, the hydrogens situated on the nitrogen are acidic. As such, there is a potential δ+, δ- relationship between respective hydrogens.

This phenomenon is known as '''Dihydrogen Bonding<ref>Study of the N−H···H−B Dihydrogen Bond Including the Crystal Structure of BH3NH3 by Neutron Diffraction, DOI:10.1021/ja9825332</ref>''' and also goes on to explain the varying B-H/N-H bond lengths.
It is found that the N-H bond is rather '''''flat''''' in the expense of the B-H bond, which is '''''bent'''''.

====Conclusion====

Although ammonia borane is isoelectric with ethane, through the use of computational chemistry in the elucidation of its various properties, we have found that ammonia borane is in fact, incredibly more stable.

Ammonia is an excellent carrier of hydrogen due to its high hydrogen content (19.6 wt%, 0.145 kg L−1 H2)<ref>The thermal decomposition of ammonia borane: A potential hydrogen storage material, DOI:doi:10.1016/j.cap.2007.10.045</ref>, as well as the fact it is a stable solid in the atmosphere, making it also easy to transport.

==References==

<references />

Rep:Mod:parkbom

2009-11-14T15:09:03Z

Byt07:

=Module 3 - The Computation of the Transition State=

==Introduction==

The '''Transition State''' is an energetic maximum within a given reaction coordinate and represents the point at which bond breaking and bond forming is occurring simultaneously. This is represented by dotted lines in formal notation and depicts the transient nature of the bond(s) in question in the transition state.

The analysis of potential energy surfaces are effective in finding transition states; the transition states can be described graphically as "dams" that represent energy metastability, that the reactants must "cross" in order to form the products.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Potential Energy Surface Reaction Coordinate'''<ref>http://www.ch.ic.ac.uk/motm/porphyrins/TSdiscovery.html</ref>||'''Cope Rearrangement'''<ref>http://www.ch.ic.ac.uk/local/organic/pericyclic/</ref>
|-
|[[Image:PES.gif]]||[[Image:Cope Arr.gif|350px]]
|}

From the point of view of computational chemistry, the methods of molecular mechanical structure determination that employed imaginary force fields are ineffective in the calculations concerning the transition state in larger molecules, as they do not define the dynamics of bond making and bond forming that are apparent in the transient nature of the transition state.

As such, in combination with molecular orbital based calculations, the following of the potential energy surface of a molecule can be used to locate and analyse the transition state.

This computational exercise concerns the rearrangement of ''1,5-Hexadiene'' in the '''Cope Rearrangment'''(above).

The Cope Rearrangement proceeds via a [3,3]-Sigmatropic shift involving the motion of 6 electrons and is thermally-driven.

In terms of transition states, the cope rearrangement can reportedly proceed via two checkpoints, the '''boat''' or the '''chair''':-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Chair'''||'''Boat'''
|-
|[[Image:pic2a.jpg]]||[[Image:pic2b.jpg]]
|}

The purpose of this exercise is to ascertain which reaction pathway is preferred by computation of each transition state in the elucidation of geometry, repective energies and other thermochemical data.

==''Exercise 1'' - Optimising the Reactants and Products==

In this section, the "''reactant''" '''1,5-hexadiene''' and the "''product''" '''1,5-hexadiene''' are modelled in correspondence to the cope rearrangement.

The flexible nature of 1,5-hexadiene means that there is an initial dilemma of likely conformation at rest.

Several starting conformations were thus taken and their geometries optimised to find the lowest energy conformer.

Initially, 1,5-hexadiene was taken, ensuring a roughly "'''anti'''" linkage in the centre of the molecule, so that the central C atoms '''3''' and '''4''' are approximately '''antiperiplanar''' to one another. This molecule was cleaned in the gaussview builder interface and then taken and optimised using the '''Hartree-Fock''' method with basis set '''3-21G''' through Gaussian.

'''b)''' similarly, a synclinal, or "gauche" version of 1,5-hexadiene was taken and optimised at the HF 3-21G level.

The energies of both conformers and also their symmetry were noted.

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|'''"''Anti''" 1,5-Hexadiene'''||'''"''Gauche''" 1,5-Hexadiene'''
|-align="center"
|[[Image:Bo Anti.jpg|420px]]||[[Image:Bo Gauche.jpg|400px]]
|}

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|'''Confomer'''||'''''Anti''-1,5-Hexadiene'''||'''''Gauche''-1,5-Hexadiene'''
|-align="center"
|'''Total Energy'''(a.u.)||-231.69254||-231.69266
|-align="center"
|'''Point Group'''||ci||None
|}

From a purely steric-orientated argument, the ''anti'' configuration should have a lower energy than the ''gauche'', as there is less steric hindrance from the two ends of the carbon chain 180 degrees from one another than next to each other in the gauche conformation.

As it so happens, the energy for the ''anti'' conformer is reportedly lower than that of the ''gauche'' conformer, but by a minute amount of roughly a ten-thousandth of a hartree ('''1 hartree = 4.3597482E-18 J''').

Even so, it was appreciated that the central C-C-C-C linkage of 1,5-hexadiene lends itself to quite a high degree of rotational freedom, and thus, further examples of conformational isomerism. This is reinforced by the fact of the two remarkably different conformers having remarkably similar energies, and points to the fact that 1,5-hexadiene experiences conformational ambiguity often<ref>Cope Rearrangement of 1,5-Hexadiene: Full Geometry Optimizations Using Analytic MR-CISD and MR-AQCC Gradient Methods, doi:10.1021/jp0259014</ref>.

As such, 2 other conformers were taken and optimised:-

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|colspan=4|'''<big>Different Conformers of 1,5-Hexadiene</big>'''
|-align="center"
|(''g120,a,g120'')-'''Anti1'''||(''g120,a,g-120'')-'''Anti2'''||(''g120,g-60,g120'')-'''Gauche1'''||(''g120,g60,g-120'')-'''Gauche3'''
|-align="center"
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Bo Anti1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Bo Anti2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Gauche 1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Gauche 3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-align="center"
|colspan=4|'''Energy'''(a.u.)
|-align="center"
| -231.69260213||-231.69253506||-231.69166702||-231.69266120
|-align="center"
|colspan=4|'''Point Group'''
|-align="center"
|C2||Ci||C2||C1
|}

The energies here from the HF 3-21G are similarly close, but contrary to what was expected, it was found that ''gauche3'' was actually the lowest in energy.

Beneath this, the ''anti'' conformers are found to be lower in energy than the other ''gauche'' conformer. This may be rationalised as the minimalisation of steric hindrance lowering the energy of the overall molecule, as the antiperiplanar groups maintain the least steric strain.

However, the '''''cis'''''-''gauche'' conformer is lowest in energy due to stereoelectronic arguments.

The '''π'''-bonds of the C=C bonds are lying criss-crossed over each other in this conformer. This promotes effectual π-molecular orbital overlap and hence stabilises the molecule.

Next, the anti2 conformer with ci symmetry and the gauche1 conformer was taken and further optimised under the DFT B3LYP method with a more sophisticated basis set, 6-31G(d).

The resulting optimised molecule was then compared with the earlier attempt in terms of energy.

A further frequency analysis was conducted in order to check that the optimisation has proceeded to a minimum, being the case when all frequencies were real and positive.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=5|'''Comparison of the HF 3-21G and DFT B3LYP 6-31G(d) Optimisations'''
|-align="center"
|'''Method'''||colspan=2|'''HF 3-21G'''||colspan=2|'''DFT B3LYP 6-31G(d)'''
|-align="center"
|'''Conformer'''||'''anti2'''||'''gauche1'''||'''anti2'''||'''gauche1'''
|-align="center"
|'''Total Energy'''(a.u.)||-231.69253506||-231.69166702||-231.61170616||-231.61068821
|-align="center"
|1-2 C=C Bond Length (A)||1.3162||1.3156||1.3335||1.3331
|-align="center"
|2-3 C-C Bond Length (A)||1.5091||1.5041||1.5083||1.5044
|-align="center"
|3-4 C-C Bond Length (A)||1.5527||1.5481||1.5508||1.5482
|-align="center"
|4-5 C-C Bond Length (A)||1.5091||1.5041||1.5083||1.5044
|-align="center"
|5-6 C=C Bond Length (A)||1.3163||1.3335||1.3157||1.3331
|-align="center"
|1-2-3-4 Dihedral (o)||114.66||118.49||123.93||124.24
|-align="center"
|2-3-4-5 Dihedral (o)||180.00||180.00||-64.18||-65.01
|-align="center"
|3-4-5-6 Dihedral (o)||-114.66||-118.49||123.93||124.24
|}

As can be seen here, the energies returned differ in a noticeable manner. However, comparison of energies computed under different basis sets is unbalanced.

As such, comparisons of the geometry are notably more reliable in discerning the computed differences.

The bond lengths differ by negligible amounts. However, it can be seen that the various dihedral angles have been "tightened" more to their ideal figures (closer to 120o for the anticlinal dihedrals.

As such, using a more sophisticated basis set ensures a more accurate approximation to reality.

The log file yielded the following thermochemical data:-

'''1)Sum of electronic and zero-point Energies= -234.469195'''

'''2)Sum of electronic and thermal Energies= -234.461847'''

'''3)Sum of electronic and thermal Enthalpies= -234.460903'''

'''4)Sum of electronic and thermal Free Energies= -234.500782'''

Theses values concern the 1)Potential Energy at 0K that includes the zero-point vibrational energy term 2)Potential Energy at 298.15K and 1atm and adds contributions from translational, rotational and vibrational energy 3)Energy including a correction for RT(H=E+RT) for dissociation reactions and 4)The Entropic contribution.

==''Exercise 2'' - Optimising the "Chair" and "Boat" Transition Structures==

The Cope Rearrangement has been found to proceed via one of two possible transition states with differing conformation, as mentioned in the introduction:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=2|'''<big>Cope Rearrangement Transition State Conformations</big>'''
|-align="center"
|''Chair''||''Boat''
|-align="center"
|[[Image:pic2a.jpg]]||[[Image:pic2b.jpg]]
|}

When the rearrangment is thought as one that is concerted and proceeds with synchronicity of bond breaking and formation, these two transition states come to mind.

There is indeed a third type of transition state for this reaction, dubbed the '''''Dewar''''' transition state, that reportedly proceeds via a diradical tight-chair intermediate species that was discovered by Dupuis et al.<ref>The Cope Rearrangement Revisited, DOI:10.1021/ja00026a007</ref>, and is thus NOT concerted.

===The Chair Transition State===

The chair transition state was initally synthesised using two "resonant" '''allyl''' ('''C3H5''') fragments in gaussview, which were optimised under the 3-21G basis set, and subsequently placed in appropriate positions relative to one another to be optimised into the transition state.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=2|'''Allyl Fragment('''C3H5''')'''
|-align="center"
|colspan=2|[[Image:Allyl Frag.jpg|275px]]
|-align="center"
|'''sp3''' Centre Bond Angle (o)||109.5
|-align="center"
|'''Allyl''' C-C Bond Angle (o)||124.3
|-align="center"
|'''C-C''' Bond Length (A)||1.54
|-align="center"
|'''C=C''' Bond Length (A)||1.34
|-align="center"
|'''Allyl C-C''' Bond Length (A)||1.39
|}

The optimised allyl C-C bond angle is greater than the normal sp3 bond angle of '''109.5o''' due to resonance.

More importantly, the allyl C-C bond length is between that of a single bond and a double bond.

This optimised allyl fragment was then taken, duplicated and placed, facing opposite directions, with roughly a 2.2A distance between the terminal carbons of each fragment to mimic the shape of the chair transition state.

Thus, a frequency + optimisation was run that directed gaussian to reach a '''TS(Berny)''', with the elucidation of force constants.

The optimisation was then repeated but this time using the '''"frozen" coordinate method''', effectively fixing the two pairs of ends at a distance of 2.2A.

Clearly, this method requires that the general structure of the transition state in question be known. However, by fixing the positions in such a way, a more accurate optimisation can be done that avoids divergence.

Finally, the optimisation was performed a third time, using a normal guess '''Hessian''' method, which involves calculation of the force constant matrix.

After the three optimisations were complete, the following data were obtained:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=3|'''<big>Optimisation of Chair Transition State</big>'''
|-align="center"
|colspan=3|'''Structure'''
|-align="center"
|'''TS(Berny)'''||'''Frozen Coordinate'''||'''Hessian'''
|-align="center"
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Berny.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Frozen.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>red</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Hessian.mol</uploadedFileContents>
</jmolApplet></jmol>
|-align="center"
|colspan=3|'''Energies''' (a.u.)
|-align="center"
| -231.61932||-231.61932||-231.69167
|}

The structures returned from the normal force-constant calculation optimisation and the frozen coordinate method are indistinguishable from first glance, and their energies are virtually identical. However, the structure obtained from the Hessian is different, and so is its energy value.

'''FREQUENCY ANALYSIS'''

Vibration-wise, there was one negative, and thus "imaginary" frequency found with a wavenumber of '''-817.91cm-1'''.

When the stretch was visualised, the following was observed:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Imaginary Stretch at -817.91cm-1'''
|-
|[[Image:Imag Stretch.jpg|300px]]
|}

The appearance of this imaginary frequency resembles the motion that would ensue under the Cope Rearrangment - the action of bond breaking of the C-C σ bond in the middle of the 1,5-hexadiene with the subsequent formation of a new σ bond on across the terminals on the other side.

The fact that there is only '''one''' imaginary frequency value points to the notion that the cope rearrangment through this transition state is ''concerted'', and thus occurs in one ''synchronous'' step.

The number of imaginary frequencies produced from a frequency analysis for the midpoint of a reaction points towards the ''synchronicity'' of the reaction<ref>Cope Rearrangement Revisited, DOI:10.1021/ja00026a007</ref>.

===The Boat Transition State===

For the boat transition state, a different method was utilised, namely the '''QST2 method''', whereby reactant and product are defined, and the transition state is converged from these.

This required the previously optimised '''1,5-hexadiene''' molecule to be taken and numbered along its carbon skeleton.

The molecule was then duplicated and juxtaposed to the intial "reactant" molecule and renumbered in such a way to reflect the reacted, rearragement "product" as shown below:-

{|style="margin:1em auto 1em auto" border="1"
|-
|[[Image:Boat Method.JPG|400px]]
|}

The first instance of optimisation failed to produce an output, and thus the molecule was changed in the following way:-

The dihedral angle of both reactant and product were changed to 0 degrees, such that the two carbon groups either side of C3 and C4 are eclipsed, and the C2-C3-C4 and C3-C4-C5 bond angles were both changed to 100 degrees.

The modified structures were once again optimised under the QST2 method to yield the following structure:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''''QST2'' Optimised Boat Transition State'''
|-
|[[Image:FINAL Boat.jpg|400px]]
|}

'''DISCUSSION'''

Here, it is illustrated that the QST2 method is automated, and is thus very efficient, but is only effective if the shape provided of the reactants and products cloely mirror that of the transition state, otherwise convergence is difficult.

The '''QST3''' method was also attempted <ref>QST Calculation, http://hdl.handle.net/10042/to-2902</ref>, whereby an addition guess of the transition state geometry was implemented. If the guess of the transition state is sound, then the QST3 is by far more accurate than the QST2 method, and does not rely so much on the shape of the reactants and products.

initally, it would seem that in order for the cope rearrangement to occur, to two carbon terminals would be required to be close to one another. This is not the case with the "''anti''" conformers of the molecule, as the terminals are situated as far apart as they may possibly be, '''antiperplanar''' to one another.

Once free rotation moves the carbon groups closer together, into the gauche configuration, where the terminals are more poised to react, cope rearrangement would proceed.

The relative orientations of the vinyl groups would ultimately dictate which transition state would be traversed. For instance, if the vectors of the vinyl groups cross, such as the case for the gauche1 conformer, then as the terminals align into a bonding orientaton, the ''chair'' conformation would be adopted by the transition state.

Conversely, if the two vinyl groups lie in a pseudo-parallel fashion, where their vectors do NOT cross, such as is the case for gauche3, then the transition state would be of ''boat'' topography.

'''IRC'''

The Intrinsic Reaction Coordinate looks more cloesly at the potential surface of a reaction and works to pinpoint the highest point in a local area of the potential surface by creeping up the steepest slopes in the immediate area.

This calculation was attempted with 50 points, and did not converge within the maximum number of points.

Thus, this was carried out again whilst calculating the force constant every single point.

{|style="margin:1em auto 1em auto" border="1"
|-
|[[Image:BO_IRC.JPG]]
|}

Here, upon closer inspection of the RMS Gradient of the energy, the gradient falls to zero, thus indicating that a minimum has been reached.

'''Energy Calculation'''

The two transition states were optimised a final time under DFT B3LYP 6-31G(d) conditions.

The geometries that were outputted were very similar to the previous examples, however, the energies received for both transition states are notably more varied:-

'''Final Energy of Chair Conformation''' (a.u.) = -234.61071
'''Final Energy of Boat Conformation''' (a.u.) = -234.54309

Here, it can be conclusively said that the ''Boat'' conformation transition state is indeed lower in energy than the ''Chair'' conformation.

This corresponds to better orbital overlap of the adjacent carbons on the framework, which are exactly eclipsed and not staggered like in the chair formation.

==The Diels-Alder Cycloaddition==

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Diels-Alder Reaction of Butadiene and Ethylene'''
|-align="center"
|[[Image:Mb_da3.jpg|400px]]
|}

The Diels-Alder Reaction is an example of a pericyclic cycloaddition, that usually involves the reaction between an electron rich diene and an electron poor dienophile.

The process is described as a [4s + 2s], 6 electron process that forms 2 new σ bonds.

When viewing the molecular orbital picture of butadiene and ethylene, the molecular orbitals are classified as either '''''s'''''(''symmetric'') or '''''a'''''(''antisymmetric'') in relation to the bisecting plane as shown below.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Bisecting Plane for Diels-alder'''
|-
|[[Image:Mb_da2.jpg|200px]]
|}

===Investigation of Molecular Orbital Symmetries of Butadiene and Ethene===

Butadiene and Ethylene were optimised and their molecular orbitals visualised:-

'''''Butadiene'''''

The HOMO and LUMO are considered with reference to the bisecting plane to determine their symmetry:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=4|'''HOMO and LUMO of Butadiene'''
|-align="center"
|width=250px|'''Molecular Orbital'''||'''2D Representation'''||'''3D Representation'''||width=250px|'''s/a'''
|-align="center"
|'''HOMO'''||[[Image:BDE HOMO.png|200px]]||[[Image:3D BDE HOMO.jpg|300px]]||a
|-align="center"
|'''LUMO'''||[[Image:BDE LUMO.png|200px]]||[[Image:3D BDE LUMO.jpg|300px]]||s
|}

'''''Ethene'''''

Similarly, the HOMO and LUMO of ethylene were characterised:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=4|'''HOMO and LUMO of Ethene'''
|-align="center"
|width=250px|'''Molecular Orbital'''||'''2D Representation'''||'''3D Representation'''||width=250px|'''s/a'''
|-align="center"
|'''HOMO'''||[[Image:ET HOMO.jpg|200px]]||[[Image:3D ET HOMO.jpg|300px]]||s
|-align="center"
|'''LUMO'''||[[Image:ET LUMO.jpg|200px]]||[[Image:3D ET LUMO.jpg|300px]]||a
|}

===Optimisation of Diels-Alder Transition State===

The '''Transition State''' whereby the concerted cycloaddition of butadiene and ethylene was modelled to a guess initially, before being optimised under HF 3-21G basis set conditions to a TS(Berny).

The initial guess was modelled from the bicyclo-system of cyclohexene and subsequently removing one of the -CH2CH2- arches to arrive at an "''envelope''"-shaped 6-membered ring.

The new σ bonds were stripped before the optimisation.

When this optimisation was complete, the structure obtained was slightly distorted, and so another optimisation using the frozen coordinate method.

The newly-optimised structure represented the "envelope", boat-like transition state a lot more accurately.

{|style="margin=1em auto 1em auto" border="1"
|-align="center"
|colspan=3|'''Comparisons of the Optimisations of transition state'''
|-align="center"
|'''Method'''||'''Ts(Berny)'''||'''Frozen-Coordinate'''
|-align="center"
|'''Structure'''||[[Image:CHEX BERNY.jpg|300px]]||[[Image:CHEX FROZEN.jpg|300px]]
|-align="center"
|'''Energy'''(a.u.)||-231.54077||-231.67327
|-align="center"
|'''Average new C-C σ bond length''' (A)||1.472645||1.526223
|-align="center"
|'''Literature Length of a C-C σ bond''' (A)||colspan=2|1.54
|}

By freezing the terminals, the energy of the optimised structure is markedly higher, and the length of the newly-forming σ bonds are closer to what they are experimentally measured as.

'''AM1 Molecular Orbital Analysis'''

The AM1 semi-empirical method was used to visualise the highest-occupied molecular orbital of the transition state.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''''HOMO'''''||'''''LUMO'''''
|-align="center"
|[[Image:CHEX TS HOMO.jpg|300px]]||[[Image:CHEX TS LUMO.jpg|300px]]
|-align="center"
|colspan=2|'''Symmetry'''
|-align="center"
|s||s
|-align="center"
|}

With reference to the bisecting plane as shown above, the HOMO of the transition state can be seen to be symmetric when reflected.

The individual MOs of butadiene and ethene thus must both be symmetric with respect to the plane in order to react in such a fashion.

This is consistent with the idea that only molecular orbitals with the same symmetry are allowed to react.

Given this, the pertinent orbitals that are symmetric are the '''LUMO''' of the butadiene and the '''HOMO''' of the ethene.

As such, electron density is donated into the empty π* LUMO of the butadiene.

The LUMO in this case is also symmetric, which leads me to believe that the the diels-alder reaction proceeds via the s transition state.

===Diels-Alder Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride===

The Diels-Alder reaction of '''Cyclohexa-1,3-diene and Maleic Anhydride''' is also an example of a 6-electron [4+2] Diels-Alder Cycloaddition intrinsically.

However, it differs from the reaction between butadiene and ethene in that there is now a question of regioselectivity; the bigger anhydride ring can either react to adopt an '''EXO''' or an '''ENDO''' topography:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|[[Image:EXO ENDO.jpg|400px]]
|}

The vinyl goup on maleic anhydride, in comparison to ethene, is electron-deficient, thanks to the highly electron-withdrawing nature of the anhydride group.

<references />

Rep:Mod:parkbom

2009-11-13T16:49:41Z

Byt07:

=Module 3 - The Computation of the Transition State=

==Introduction==

The '''Transition State''' is an energetic maximum within a given reaction coordinate and represents the point at which bond breaking and bond forming is occurring simultaneously. This is represented by dotted lines in formal notation and depicts the transient nature of the bond(s) in question in the transition state.

The analysis of potential energy surfaces are effective in finding transition states; the transition states can be described graphically as "dams" that represent energy metastability, that the reactants must "cross" in order to form the products.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Potential Energy Surface Reaction Coordinate'''<ref>http://www.ch.ic.ac.uk/motm/porphyrins/TSdiscovery.html</ref>||'''Cope Rearrangement'''<ref>http://www.ch.ic.ac.uk/local/organic/pericyclic/</ref>
|-
|[[Image:PES.gif]]||[[Image:Cope Arr.gif|350px]]
|}

From the point of view of computational chemistry, the methods of molecular mechanical structure determination that employed imaginary force fields are ineffective in the calculations concerning the transition state in larger molecules, as they do not define the dynamics of bond making and bond forming that are apparent in the transient nature of the transition state.

As such, in combination with molecular orbital based calculations, the following of the potential energy surface of a molecule can be used to locate and analyse the transition state.

This computational exercise concerns the rearrangement of ''1,5-Hexadiene'' in the '''Cope Rearrangment'''(above).

The Cope Rearrangement proceeds via a [3,3]-Sigmatropic shift involving the motion of 6 electrons and is thermally-driven.

In terms of transition states, the cope rearrangement can reportedly proceed via two checkpoints, the '''boat''' or the '''chair''':-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Chair'''||'''Boat'''
|-
|[[Image:pic2a.jpg]]||[[Image:pic2b.jpg]]
|}

The purpose of this exercise is to ascertain which reaction pathway is preferred by computation of each transition state in the elucidation of geometry, repective energies and other thermochemical data.

==''Exercise 1'' - Optimising the Reactants and Products==

In this section, the "''reactant''" '''1,5-hexadiene''' and the "''product''" '''1,5-hexadiene''' are modelled in correspondence to the cope rearrangement.

The flexible nature of 1,5-hexadiene means that there is an initial dilemma of likely conformation at rest.

Several starting conformations were thus taken and their geometries optimised to find the lowest energy conformer.

Initially, 1,5-hexadiene was taken, ensuring a roughly "'''anti'''" linkage in the centre of the molecule, so that the central C atoms '''3''' and '''4''' are approximately '''antiperiplanar''' to one another. This molecule was cleaned in the gaussview builder interface and then taken and optimised using the '''Hartree-Fock''' method with basis set '''3-21G''' through Gaussian.

'''b)''' similarly, a synclinal, or "gauche" version of 1,5-hexadiene was taken and optimised at the HF 3-21G level.

The energies of both conformers and also their symmetry were noted.

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|'''"''Anti''" 1,5-Hexadiene'''||'''"''Gauche''" 1,5-Hexadiene'''
|-align="center"
|[[Image:Bo Anti.jpg|420px]]||[[Image:Bo Gauche.jpg|400px]]
|}

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|'''Confomer'''||'''''Anti''-1,5-Hexadiene'''||'''''Gauche''-1,5-Hexadiene'''
|-align="center"
|'''Total Energy'''(a.u.)||-231.69254||-231.69266
|-align="center"
|'''Point Group'''||ci||None
|}

From a purely steric-orientated argument, the ''anti'' configuration should have a lower energy than the ''gauche'', as there is less steric hindrance from the two ends of the carbon chain 180 degrees from one another than next to each other in the gauche conformation.

As it so happens, the energy for the ''anti'' conformer is reportedly lower than that of the ''gauche'' conformer, but by a minute amount of roughly a ten-thousandth of a hartree ('''1 hartree = 4.3597482E-18 J''').

Even so, it was appreciated that the central C-C-C-C linkage of 1,5-hexadiene lends itself to quite a high degree of rotational freedom, and thus, further examples of conformational isomerism. This is reinforced by the fact of the two remarkably different conformers having remarkably similar energies, and points to the fact that 1,5-hexadiene experiences conformational ambiguity often<ref>Cope Rearrangement of 1,5-Hexadiene: Full Geometry Optimizations Using Analytic MR-CISD and MR-AQCC Gradient Methods, doi:10.1021/jp0259014</ref>.

As such, 2 other conformers were taken and optimised:-

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|colspan=4|'''<big>Different Conformers of 1,5-Hexadiene</big>'''
|-align="center"
|(''g120,a,g120'')-'''Anti1'''||(''g120,a,g-120'')-'''Anti2'''||(''g120,g-60,g120'')-'''Gauche1'''||(''g120,g60,g-120'')-'''Gauche3'''
|-align="center"
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Bo Anti1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Bo Anti2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Gauche 1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Gauche 3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-align="center"
|colspan=4|'''Energy'''(a.u.)
|-align="center"
| -231.69260213||-231.69253506||-231.69166702||-231.69266120
|-align="center"
|colspan=4|'''Point Group'''
|-align="center"
|C2||Ci||C2||C1
|}

The energies here from the HF 3-21G are similarly close, but contrary to what was expected, it was found that ''gauche3'' was actually the lowest in energy.

Beneath this, the ''anti'' conformers are found to be lower in energy than the other ''gauche'' conformer. This may be rationalised as the minimalisation of steric hindrance lowering the energy of the overall molecule, as the antiperiplanar groups maintain the least steric strain.

However, the '''''cis'''''-''gauche'' conformer is lowest in energy due to stereoelectronic arguments.

The '''π'''-bonds of the C=C bonds are lying criss-crossed over each other in this conformer. This promotes effectual π-molecular orbital overlap and hence stabilises the molecule.

Next, the anti2 conformer with ci symmetry and the gauche1 conformer was taken and further optimised under the DFT B3LYP method with a more sophisticated basis set, 6-31G(d).

The resulting optimised molecule was then compared with the earlier attempt in terms of energy.

A further frequency analysis was conducted in order to check that the optimisation has proceeded to a minimum, being the case when all frequencies were real and positive.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=5|'''Comparison of the HF 3-21G and DFT B3LYP 6-31G(d) Optimisations'''
|-align="center"
|'''Method'''||colspan=2|'''HF 3-21G'''||colspan=2|'''DFT B3LYP 6-31G(d)'''
|-align="center"
|'''Conformer'''||'''anti2'''||'''gauche1'''||'''anti2'''||'''gauche1'''
|-align="center"
|'''Total Energy'''(a.u.)||-231.69253506||-231.69166702||-231.61170616||-231.61068821
|-align="center"
|1-2 C=C Bond Length (A)||1.3162||1.3156||1.3335||1.3331
|-align="center"
|2-3 C-C Bond Length (A)||1.5091||1.5041||1.5083||1.5044
|-align="center"
|3-4 C-C Bond Length (A)||1.5527||1.5481||1.5508||1.5482
|-align="center"
|4-5 C-C Bond Length (A)||1.5091||1.5041||1.5083||1.5044
|-align="center"
|5-6 C=C Bond Length (A)||1.3163||1.3335||1.3157||1.3331
|-align="center"
|1-2-3-4 Dihedral (o)||114.66||118.49||123.93||124.24
|-align="center"
|2-3-4-5 Dihedral (o)||180.00||180.00||-64.18||-65.01
|-align="center"
|3-4-5-6 Dihedral (o)||-114.66||-118.49||123.93||124.24
|}

As can be seen here, the energies returned differ in a noticeable manner. However, comparison of energies computed under different basis sets is unbalanced.

As such, comparisons of the geometry are notably more reliable in discerning the computed differences.

The bond lengths differ by negligible amounts. However, it can be seen that the various dihedral angles have been "tightened" more to their ideal figures (closer to 120o for the anticlinal dihedrals.

As such, using a more sophisticated basis set ensures a more accurate approximation to reality.

The log file yielded the following thermochemical data:-

'''1)Sum of electronic and zero-point Energies= -234.469195'''

'''2)Sum of electronic and thermal Energies= -234.461847'''

'''3)Sum of electronic and thermal Enthalpies= -234.460903'''

'''4)Sum of electronic and thermal Free Energies= -234.500782'''

Theses values concern the 1)Potential Energy at 0K that includes the zero-point vibrational energy term 2)Potential Energy at 298.15K and 1atm and adds contributions from translational, rotational and vibrational energy 3)Energy including a correction for RT(H=E+RT) for dissociation reactions and 4)The Entropic contribution.

==''Exercise 2'' - Optimising the "Chair" and "Boat" Transition Structures==

The Cope Rearrangement has been found to proceed via one of two possible transition states with differing conformation, as mentioned in the introduction:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=2|'''<big>Cope Rearrangement Transition State Conformations</big>'''
|-align="center"
|''Chair''||''Boat''
|-align="center"
|[[Image:pic2a.jpg]]||[[Image:pic2b.jpg]]
|}

When the rearrangment is thought as one that is concerted and proceeds with synchronicity of bond breaking and formation, these two transition states come to mind.

There is indeed a third type of transition state for this reaction, dubbed the '''''Dewar''''' transition state, that reportedly proceeds via a diradical tight-chair intermediate species that was discovered by Dupuis et al.<ref>The Cope Rearrangement Revisited, DOI:10.1021/ja00026a007</ref>, and is thus NOT concerted.

===The Chair Transition State===

The chair transition state was initally synthesised using two "resonant" '''allyl''' ('''C3H5''') fragments in gaussview, which were optimised under the 3-21G basis set, and subsequently placed in appropriate positions relative to one another to be optimised into the transition state.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=2|'''Allyl Fragment('''C3H5''')'''
|-align="center"
|colspan=2|[[Image:Allyl Frag.jpg|275px]]
|-align="center"
|'''sp3''' Centre Bond Angle (o)||109.5
|-align="center"
|'''Allyl''' C-C Bond Angle (o)||124.3
|-align="center"
|'''C-C''' Bond Length (A)||1.54
|-align="center"
|'''C=C''' Bond Length (A)||1.34
|-align="center"
|'''Allyl C-C''' Bond Length (A)||1.39
|}

The optimised allyl C-C bond angle is greater than the normal sp3 bond angle of '''109.5o''' due to resonance.

More importantly, the allyl C-C bond length is between that of a single bond and a double bond.

This optimised allyl fragment was then taken, duplicated and placed, facing opposite directions, with roughly a 2.2A distance between the terminal carbons of each fragment to mimic the shape of the chair transition state.

Thus, a frequency + optimisation was run that directed gaussian to reach a '''TS(Berny)''', with the elucidation of force constants.

The optimisation was then repeated but this time using the '''"frozen" coordinate method''', effectively fixing the two pairs of ends at a distance of 2.2A.

Clearly, this method requires that the general structure of the transition state in question be known. However, by fixing the positions in such a way, a more accurate optimisation can be done that avoids divergence.

Finally, the optimisation was performed a third time, using a normal guess '''Hessian''' method, which involves calculation of the force constant matrix.

After the three optimisations were complete, the following data were obtained:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=3|'''<big>Optimisation of Chair Transition State</big>'''
|-align="center"
|colspan=3|'''Structure'''
|-align="center"
|'''TS(Berny)'''||'''Frozen Coordinate'''||'''Hessian'''
|-align="center"
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Berny.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Frozen.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>red</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Hessian.mol</uploadedFileContents>
</jmolApplet></jmol>
|-align="center"
|colspan=3|'''Energies''' (a.u.)
|-align="center"
| -231.61932||-231.61932||-231.69167
|}

The structures returned from the normal force-constant calculation optimisation and the frozen coordinate method are indistinguishable from first glance, and their energies are virtually identical. However, the structure obtained from the Hessian is different, and so is its energy value.

'''FREQUENCY ANALYSIS'''

Vibration-wise, there was one negative, and thus "imaginary" frequency found with a wavenumber of '''-817.91cm-1'''.

When the stretch was visualised, the following was observed:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Imaginary Stretch at -817.91cm-1'''
|-
|[[Image:Imag Stretch.jpg|300px]]
|}

The appearance of this imaginary frequency resembles the motion that would ensue under the Cope Rearrangment - the action of bond breaking of the C-C σ bond in the middle of the 1,5-hexadiene with the subsequent formation of a new σ bond on across the terminals on the other side.

The fact that there is only '''one''' imaginary frequency value points to the notion that the cope rearrangment through this transition state is ''concerted'', and thus occurs in one ''synchronous'' step.

The number of imaginary frequencies produced from a frequency analysis for the midpoint of a reaction points towards the ''synchronicity'' of the reaction<ref>Cope Rearrangement Revisited, DOI:10.1021/ja00026a007</ref>.

===The Boat Transition State===

For the boat transition state, a different method was utilised, namely the '''QST2 method''', whereby reactant and product are defined, and the transition state is converged from these.

This required the previously optimised '''1,5-hexadiene''' molecule to be taken and numbered along its carbon skeleton.

The molecule was then duplicated and juxtaposed to the intial "reactant" molecule and renumbered in such a way to reflect the reacted, rearragement "product" as shown below:-

{|style="margin:1em auto 1em auto" border="1"
|-
|[[Image:Boat Method.JPG|400px]]
|}

The first instance of optimisation failed to produce an output, and thus the molecule was changed in the following way:-

The dihedral angle of both reactant and product were changed to 0 degrees, such that the two carbon groups either side of C3 and C4 are eclipsed, and the C2-C3-C4 and C3-C4-C5 bond angles were both changed to 100 degrees.

The modified structures were once again optimised under the QST2 method to yield the following structure:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''''QST2'' Optimised Boat Transition State'''
|-
|[[Image:FINAL Boat.jpg|400px]]
|}

'''DISCUSSION'''

Here, it is illustrated that the QST2 method is automated, and is thus very efficient, but is only effective if the shape provided of the reactants and products cloely mirror that of the transition state, otherwise convergence is difficult.

The '''QST3''' method was also attempted <ref>QST Calculation, http://hdl.handle.net/10042/to-2902</ref>, whereby an addition guess of the transition state geometry was implemented. If the guess of the transition state is sound, then the QST3 is by far more accurate than the QST2 method, and does not rely so much on the shape of the reactants and products.

initally, it would seem that in order for the cope rearrangement to occur, to two carbon terminals would be required to be close to one another. This is not the case with the "''anti''" conformers of the molecule, as the terminals are situated as far apart as they may possibly be, '''antiperplanar''' to one another.

Once free rotation moves the carbon groups closer together, into the gauche configuration, where the terminals are more poised to react, cope rearrangement would proceed.

The relative orientations of the vinyl groups would ultimately dictate which transition state would be traversed. For instance, if the vectors of the vinyl groups cross, such as the case for the gauche1 conformer, then as the terminals align into a bonding orientaton, the ''chair'' conformation would be adopted by the transition state.

Conversely, if the two vinyl groups lie in a pseudo-parallel fashion, where their vectors do NOT cross, such as is the case for gauche3, then the transition state would be of ''boat'' topography.

'''IRC'''

The Intrinsic Reaction Coordinate looks more cloesly at the potential surface of a reaction and works to pinpoint the highest point in a local area of the potential surface by creeping up the steepest slopes in the immediate area.

This calculation was attempted with 50 points, and did not converge within the maximum number of points.

Thus, this was carried out again whilst calculating the force constant every single point.

{|style="margin:1em auto 1em auto" border="1"
|-
|[[Image:BO_IRC.JPG]]
|}

Here, upon closer inspection of the RMS Gradient of the energy, the gradient falls to zero, thus indicating that a minimum has been reached.

'''Energy Calculation'''

The two transition states were optimised a final time under DFT B3LYP 6-31G(d) conditions.

The geometries that were outputted were very similar to the previous examples, however, the energies received for both transition states are notably more varied:-

'''Final Energy of Chair Conformation''' (a.u.) = -234.61071
'''Final Energy of Boat Conformation''' (a.u.) = -234.54309

Here, it can be conclusively said that the ''Boat'' conformation transition state is indeed lower in energy than the ''Chair'' conformation.

This corresponds to better orbital overlap of the adjacent carbons on the framework, which are exactly eclipsed and not staggered like in the chair formation.

==The Diels-Alder Cycloaddition==

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Diels-Alder Reaction of Butadiene and Ethylene'''
|-align="center"
|[[Image:Mb_da3.jpg|400px]]
|}

The Diels-Alder Reaction is an example of a pericyclic cycloaddition, that usually involves the reaction between an electron rich diene and an electron poor dienophile.

The process is described as a [4s + 2s], 6 electron process that forms 2 new σ bonds.

When viewing the molecular orbital picture of butadiene and ethylene, the molecular orbitals are classified as either '''''s'''''(''symmetric'') or '''''a'''''(''antisymmetric'') in relation to the bisecting plane as shown below.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Bisecting Plane for Diels-alder'''
|-
|[[Image:Mb_da2.jpg|200px]]
|}

===Investigation of Molecular Orbital Symmetries of Butadiene and Ethene===

Butadiene and Ethylene were optimised and their molecular orbitals visualised:-

'''''Butadiene'''''

The HOMO and LUMO are considered with reference to the bisecting plane to determine their symmetry:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=4|'''HOMO and LUMO of Butadiene'''
|-align="center"
|width=250px|'''Molecular Orbital'''||'''2D Representation'''||'''3D Representation'''||width=250px|'''s/a'''
|-align="center"
|'''HOMO'''||[[Image:BDE HOMO.png|200px]]||[[Image:3D BDE HOMO.jpg|300px]]||a
|-align="center"
|'''LUMO'''||[[Image:BDE LUMO.png|200px]]||[[Image:3D BDE LUMO.jpg|300px]]||s
|}

'''''Ethene'''''

Similarly, the HOMO and LUMO of ethylene were characterised:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=4|'''HOMO and LUMO of Ethene'''
|-align="center"
|width=250px|'''Molecular Orbital'''||'''2D Representation'''||'''3D Representation'''||width=250px|'''s/a'''
|-align="center"
|'''HOMO'''||[[Image:ET HOMO.jpg|200px]]||[[Image:3D ET HOMO.jpg|300px]]||s
|-align="center"
|'''LUMO'''||[[Image:ET LUMO.jpg|200px]]||[[Image:3D ET LUMO.jpg|300px]]||a
|}

===Optimisation of Diels-Alder Transition State===

The '''Transition State''' whereby the concerted cycloaddition of butadiene and ethylene was modelled to a guess initially, before being optimised under HF 3-21G basis set conditions to a TS(Berny).

The initial guess was modelled from the bicyclo-system of cyclohexene and subsequently removing one of the -CH2CH2- arches to arrive at an "''envelope''"-shaped 6-membered ring.

The new σ bonds were stripped before the optimisation.

When this optimisation was complete, the structure obtained was slightly distorted, and so another optimisation using the frozen coordinate method.

The newly-optimised structure represented the "envelope", boat-like transition state a lot more accurately.

{|style="margin=1em auto 1em auto" border="1"
|-align="center"
|colspan=3|'''Comparisons of the Optimisations of transition state'''
|-align="center"
|'''Method'''||'''Ts(Berny)'''||'''Frozen-Coordinate'''
|-align="center"
|'''Structure'''||[[Image:CHEX BERNY.jpg|300px]]||[[Image:CHEX FROZEN.jpg|300px]]
|-align="center"
|'''Energy'''(a.u.)||-231.54077||-231.67327
|-align="center"
|'''Average new C-C σ bond length''' (A)||1.472645||1.526223
|-align="center"
|'''Literature Length of a C-C σ bond''' (A)||colspan=2|1.54
|}

By freezing the terminals, the energy of the optimised structure is markedly higher, and the length of the newly-forming σ bonds are closer to what they are experimentally measured as.

'''AM1 Molecular Orbital Analysis'''

The AM1 semi-empirical method was used to visualise the highest-occupied molecular orbital of the transition state.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''''HOMO'''''||'''''LUMO'''''
|-align="center"
|[[Image:CHEX TS HOMO.jpg|300px]]||[[Image:CHEX TS LUMO.jpg|300px]]
|-align="center"
|colspan=2|'''Symmetry'''
|-align="center"
|s||s
|-align="center"
|}

With reference to the bisecting plane as shown above, the HOMO of the transition state can be seen to be symmetric when reflected.

The individual MOs of butadiene and ethene thus must both be symmetric with respect to the plane in order to react in such a fashion.

This is consistent with the idea that only molecular orbitals with the same symmetry are allowed to react.

Given this, the pertinent orbitals that are symmetric are the '''LUMO''' of the butadiene and the '''HOMO''' of the ethene.

As such, electron density is donated into the empty π* LUMO of the butadiene.

The LUMO in this case is also symmetric, which leads me to believe that the the diels-alder reaction proceeds via the s transition state.

===Diels-Alder Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride===

The Diels-Alder reaction of '''Cyclohexa-1,3-diene and Maleic Anhydride''' is also an example of a 6-electron [4+2] Diels-Alder Cycloaddition intrinsically.

However, it differs from the reaction between butadiene and ethene in that there is now a question of regioselectivity; the bigger anhydride ring can either react to adopt an '''EXO''' or an '''ENDO''' topography:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|[[Image:EXO ENDO.jpg|400px]]
|}

The vinyl goup on maleic anhydride, in comparison to ethene, is electron-deficient, thanks to the highly electron-withdrawing nature of the anhydride group.

File:BO IRC.JPG

2009-11-13T16:37:44Z

Byt07:

Rep:Mod:parkbom

2009-11-13T16:25:58Z

Byt07:

=Module 3 - The Computation of the Transition State=

==Introduction==

The '''Transition State''' is an energetic maximum within a given reaction coordinate and represents the point at which bond breaking and bond forming is occurring simultaneously. This is represented by dotted lines in formal notation and depicts the transient nature of the bond(s) in question in the transition state.

The analysis of potential energy surfaces are effective in finding transition states; the transition states can be described graphically as "dams" that represent energy metastability, that the reactants must "cross" in order to form the products.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Potential Energy Surface Reaction Coordinate'''<ref>http://www.ch.ic.ac.uk/motm/porphyrins/TSdiscovery.html</ref>||'''Cope Rearrangement'''<ref>http://www.ch.ic.ac.uk/local/organic/pericyclic/</ref>
|-
|[[Image:PES.gif]]||[[Image:Cope Arr.gif|350px]]
|}

From the point of view of computational chemistry, the methods of molecular mechanical structure determination that employed imaginary force fields are ineffective in the calculations concerning the transition state in larger molecules, as they do not define the dynamics of bond making and bond forming that are apparent in the transient nature of the transition state.

As such, in combination with molecular orbital based calculations, the following of the potential energy surface of a molecule can be used to locate and analyse the transition state.

This computational exercise concerns the rearrangement of ''1,5-Hexadiene'' in the '''Cope Rearrangment'''(above).

The Cope Rearrangement proceeds via a [3,3]-Sigmatropic shift involving the motion of 6 electrons and is thermally-driven.

In terms of transition states, the cope rearrangement can reportedly proceed via two checkpoints, the '''boat''' or the '''chair''':-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Chair'''||'''Boat'''
|-
|[[Image:pic2a.jpg]]||[[Image:pic2b.jpg]]
|}

The purpose of this exercise is to ascertain which reaction pathway is preferred by computation of each transition state in the elucidation of geometry, repective energies and other thermochemical data.

==''Exercise 1'' - Optimising the Reactants and Products==

In this section, the "''reactant''" '''1,5-hexadiene''' and the "''product''" '''1,5-hexadiene''' are modelled in correspondence to the cope rearrangement.

The flexible nature of 1,5-hexadiene means that there is an initial dilemma of likely conformation at rest.

Several starting conformations were thus taken and their geometries optimised to find the lowest energy conformer.

Initially, 1,5-hexadiene was taken, ensuring a roughly "'''anti'''" linkage in the centre of the molecule, so that the central C atoms '''3''' and '''4''' are approximately '''antiperiplanar''' to one another. This molecule was cleaned in the gaussview builder interface and then taken and optimised using the '''Hartree-Fock''' method with basis set '''3-21G''' through Gaussian.

'''b)''' similarly, a synclinal, or "gauche" version of 1,5-hexadiene was taken and optimised at the HF 3-21G level.

The energies of both conformers and also their symmetry were noted.

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|'''"''Anti''" 1,5-Hexadiene'''||'''"''Gauche''" 1,5-Hexadiene'''
|-align="center"
|[[Image:Bo Anti.jpg|420px]]||[[Image:Bo Gauche.jpg|400px]]
|}

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|'''Confomer'''||'''''Anti''-1,5-Hexadiene'''||'''''Gauche''-1,5-Hexadiene'''
|-align="center"
|'''Total Energy'''(a.u.)||-231.69254||-231.69266
|-align="center"
|'''Point Group'''||ci||None
|}

From a purely steric-orientated argument, the ''anti'' configuration should have a lower energy than the ''gauche'', as there is less steric hindrance from the two ends of the carbon chain 180 degrees from one another than next to each other in the gauche conformation.

As it so happens, the energy for the ''anti'' conformer is reportedly lower than that of the ''gauche'' conformer, but by a minute amount of roughly a ten-thousandth of a hartree ('''1 hartree = 4.3597482E-18 J''').

Even so, it was appreciated that the central C-C-C-C linkage of 1,5-hexadiene lends itself to quite a high degree of rotational freedom, and thus, further examples of conformational isomerism. This is reinforced by the fact of the two remarkably different conformers having remarkably similar energies, and points to the fact that 1,5-hexadiene experiences conformational ambiguity often<ref>Cope Rearrangement of 1,5-Hexadiene: Full Geometry Optimizations Using Analytic MR-CISD and MR-AQCC Gradient Methods, doi:10.1021/jp0259014</ref>.

As such, 2 other conformers were taken and optimised:-

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|colspan=4|'''<big>Different Conformers of 1,5-Hexadiene</big>'''
|-align="center"
|(''g120,a,g120'')-'''Anti1'''||(''g120,a,g-120'')-'''Anti2'''||(''g120,g-60,g120'')-'''Gauche1'''||(''g120,g60,g-120'')-'''Gauche3'''
|-align="center"
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Bo Anti1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Bo Anti2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Gauche 1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Gauche 3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-align="center"
|colspan=4|'''Energy'''(a.u.)
|-align="center"
| -231.69260213||-231.69253506||-231.69166702||-231.69266120
|-align="center"
|colspan=4|'''Point Group'''
|-align="center"
|C2||Ci||C2||C1
|}

The energies here from the HF 3-21G are similarly close, but contrary to what was expected, it was found that ''gauche3'' was actually the lowest in energy.

Beneath this, the ''anti'' conformers are found to be lower in energy than the other ''gauche'' conformer. This may be rationalised as the minimalisation of steric hindrance lowering the energy of the overall molecule, as the antiperiplanar groups maintain the least steric strain.

However, the '''''cis'''''-''gauche'' conformer is lowest in energy due to stereoelectronic arguments.

The '''π'''-bonds of the C=C bonds are lying criss-crossed over each other in this conformer. This promotes effectual π-molecular orbital overlap and hence stabilises the molecule.

Next, the anti2 conformer with ci symmetry and the gauche1 conformer was taken and further optimised under the DFT B3LYP method with a more sophisticated basis set, 6-31G(d).

The resulting optimised molecule was then compared with the earlier attempt in terms of energy.

A further frequency analysis was conducted in order to check that the optimisation has proceeded to a minimum, being the case when all frequencies were real and positive.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=5|'''Comparison of the HF 3-21G and DFT B3LYP 6-31G(d) Optimisations'''
|-align="center"
|'''Method'''||colspan=2|'''HF 3-21G'''||colspan=2|'''DFT B3LYP 6-31G(d)'''
|-align="center"
|'''Conformer'''||'''anti2'''||'''gauche1'''||'''anti2'''||'''gauche1'''
|-align="center"
|'''Total Energy'''(a.u.)||-231.69253506||-231.69166702||-231.61170616||-231.61068821
|-align="center"
|1-2 C=C Bond Length (A)||1.3162||1.3156||1.3335||1.3331
|-align="center"
|2-3 C-C Bond Length (A)||1.5091||1.5041||1.5083||1.5044
|-align="center"
|3-4 C-C Bond Length (A)||1.5527||1.5481||1.5508||1.5482
|-align="center"
|4-5 C-C Bond Length (A)||1.5091||1.5041||1.5083||1.5044
|-align="center"
|5-6 C=C Bond Length (A)||1.3163||1.3335||1.3157||1.3331
|-align="center"
|1-2-3-4 Dihedral (o)||114.66||118.49||123.93||124.24
|-align="center"
|2-3-4-5 Dihedral (o)||180.00||180.00||-64.18||-65.01
|-align="center"
|3-4-5-6 Dihedral (o)||-114.66||-118.49||123.93||124.24
|}

As can be seen here, the energies returned differ in a noticeable manner. However, comparison of energies computed under different basis sets is unbalanced.

As such, comparisons of the geometry are notably more reliable in discerning the computed differences.

The bond lengths differ by negligible amounts. However, it can be seen that the various dihedral angles have been "tightened" more to their ideal figures (closer to 120o for the anticlinal dihedrals.

As such, using a more sophisticated basis set ensures a more accurate approximation to reality.

The log file yielded the following thermochemical data:-

'''1)Sum of electronic and zero-point Energies= -234.469195'''

'''2)Sum of electronic and thermal Energies= -234.461847'''

'''3)Sum of electronic and thermal Enthalpies= -234.460903'''

'''4)Sum of electronic and thermal Free Energies= -234.500782'''

Theses values concern the 1)Potential Energy at 0K that includes the zero-point vibrational energy term 2)Potential Energy at 298.15K and 1atm and adds contributions from translational, rotational and vibrational energy 3)Energy including a correction for RT(H=E+RT) for dissociation reactions and 4)The Entropic contribution.

==''Exercise 2'' - Optimising the "Chair" and "Boat" Transition Structures==

The Cope Rearrangement has been found to proceed via one of two possible transition states with differing conformation, as mentioned in the introduction:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=2|'''<big>Cope Rearrangement Transition State Conformations</big>'''
|-align="center"
|''Chair''||''Boat''
|-align="center"
|[[Image:pic2a.jpg]]||[[Image:pic2b.jpg]]
|}

When the rearrangment is thought as one that is concerted and proceeds with synchronicity of bond breaking and formation, these two transition states come to mind.

There is indeed a third type of transition state for this reaction, dubbed the '''''Dewar''''' transition state, that reportedly proceeds via a diradical tight-chair intermediate species that was discovered by Dupuis et al.<ref>The Cope Rearrangement Revisited, DOI:10.1021/ja00026a007</ref>, and is thus NOT concerted.

===The Chair Transition State===

The chair transition state was initally synthesised using two "resonant" '''allyl''' ('''C3H5''') fragments in gaussview, which were optimised under the 3-21G basis set, and subsequently placed in appropriate positions relative to one another to be optimised into the transition state.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=2|'''Allyl Fragment('''C3H5''')'''
|-align="center"
|colspan=2|[[Image:Allyl Frag.jpg|275px]]
|-align="center"
|'''sp3''' Centre Bond Angle (o)||109.5
|-align="center"
|'''Allyl''' C-C Bond Angle (o)||124.3
|-align="center"
|'''C-C''' Bond Length (A)||1.54
|-align="center"
|'''C=C''' Bond Length (A)||1.34
|-align="center"
|'''Allyl C-C''' Bond Length (A)||1.39
|}

The optimised allyl C-C bond angle is greater than the normal sp3 bond angle of '''109.5o''' due to resonance.

More importantly, the allyl C-C bond length is between that of a single bond and a double bond.

This optimised allyl fragment was then taken, duplicated and placed, facing opposite directions, with roughly a 2.2A distance between the terminal carbons of each fragment to mimic the shape of the chair transition state.

Thus, a frequency + optimisation was run that directed gaussian to reach a '''TS(Berny)''', with the elucidation of force constants.

The optimisation was then repeated but this time using the '''"frozen" coordinate method''', effectively fixing the two pairs of ends at a distance of 2.2A.

Clearly, this method requires that the general structure of the transition state in question be known. However, by fixing the positions in such a way, a more accurate optimisation can be done that avoids divergence.

Finally, the optimisation was performed a third time, using a normal guess '''Hessian''' method, which involves calculation of the force constant matrix.

After the three optimisations were complete, the following data were obtained:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=3|'''<big>Optimisation of Chair Transition State</big>'''
|-align="center"
|colspan=3|'''Structure'''
|-align="center"
|'''TS(Berny)'''||'''Frozen Coordinate'''||'''Hessian'''
|-align="center"
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Berny.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Frozen.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>red</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Hessian.mol</uploadedFileContents>
</jmolApplet></jmol>
|-align="center"
|colspan=3|'''Energies''' (a.u.)
|-align="center"
| -231.61932||-231.61932||-231.69167
|}

The structures returned from the normal force-constant calculation optimisation and the frozen coordinate method are indistinguishable from first glance, and their energies are virtually identical. However, the structure obtained from the Hessian is different, and so is its energy value.

'''FREQUENCY ANALYSIS'''

Vibration-wise, there was one negative, and thus "imaginary" frequency found with a wavenumber of '''-817.91cm-1'''.

When the stretch was visualised, the following was observed:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Imaginary Stretch at -817.91cm-1'''
|-
|[[Image:Imag Stretch.jpg|300px]]
|}

The appearance of this imaginary frequency resembles the motion that would ensue under the Cope Rearrangment - the action of bond breaking of the C-C σ bond in the middle of the 1,5-hexadiene with the subsequent formation of a new σ bond on across the terminals on the other side.

The fact that there is only '''one''' imaginary frequency value points to the notion that the cope rearrangment through this transition state is ''concerted'', and thus occurs in one ''synchronous'' step.

The number of imaginary frequencies produced from a frequency analysis for the midpoint of a reaction points towards the ''synchronicity'' of the reaction<ref>Cope Rearrangement Revisited, DOI:10.1021/ja00026a007</ref>.

===The Boat Transition State===

For the boat transition state, a different method was utilised, namely the '''QST2 method''', whereby reactant and product are defined, and the transition state is converged from these.

This required the previously optimised '''1,5-hexadiene''' molecule to be taken and numbered along its carbon skeleton.

The molecule was then duplicated and juxtaposed to the intial "reactant" molecule and renumbered in such a way to reflect the reacted, rearragement "product" as shown below:-

{|style="margin:1em auto 1em auto" border="1"
|-
|[[Image:Boat Method.JPG|400px]]
|}

The first instance of optimisation failed to produce an output, and thus the molecule was changed in the following way:-

The dihedral angle of both reactant and product were changed to 0 degrees, such that the two carbon groups either side of C3 and C4 are eclipsed, and the C2-C3-C4 and C3-C4-C5 bond angles were both changed to 100 degrees.

The modified structures were once again optimised under the QST2 method to yield the following structure:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''''QST2'' Optimised Boat Transition State'''
|-
|[[Image:FINAL Boat.jpg|400px]]
|}

'''DISCUSSION'''

Here, it is illustrated that the QST2 method is automated, and is thus very efficient, but is only effective if the shape provided of the reactants and products cloely mirror that of the transition state, otherwise convergence is difficult.

The '''QST3''' method was also attempted <ref>QST Calculation, http://hdl.handle.net/10042/to-2902</ref>, whereby an addition guess of the transition state geometry was implemented. If the guess of the transition state is sound, then the QST3 is by far more accurate than the QST2 method, and does not rely so much on the shape of the reactants and products.

initally, it would seem that in order for the cope rearrangement to occur, to two carbon terminals would be required to be close to one another. This is not the case with the "''anti''" conformers of the molecule, as the terminals are situated as far apart as they may possibly be, '''antiperplanar''' to one another.

Once free rotation moves the carbon groups closer together, into the gauche configuration, where the terminals are more poised to react, cope rearrangement would proceed.

The relative orientations of the vinyl groups would ultimately dictate which transition state would be traversed. For instance, if the vectors of the vinyl groups cross, such as the case for the gauche1 conformer, then as the terminals align into a bonding orientaton, the ''chair'' conformation would be adopted by the transition state.

Conversely, if the two vinyl groups lie in a pseudo-parallel fashion, where their vectors do NOT cross, such as is the case for gauche3, then the transition state would be of ''boat'' topography.

'''IRC'''

The Intrinsic Reaction Coordinate looks more cloesly at the potential surface of a reaction and works to pinpoint the highest point in a local area of the potential surface by creeping up the steepest slopes in the immediate area.

This calculation was attempted with 50 points, and did not converge

'''Energy Calculation'''

The two transition states were optimised a final time under DFT B3LYP 6-31G(d) conditions.

The geometries that were outputted were very similar to the previous examples, however

==The Diels-Alder Cycloaddition==

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Diels-Alder Reaction of Butadiene and Ethylene'''
|-align="center"
|[[Image:Mb_da3.jpg|400px]]
|}

The Diels-Alder Reaction is an example of a pericyclic cycloaddition, that usually involves the reaction between an electron rich diene and an electron poor dienophile.

The process is described as a [4s + 2s], 6 electron process that forms 2 new σ bonds.

When viewing the molecular orbital picture of butadiene and ethylene, the molecular orbitals are classified as either '''''s'''''(''symmetric'') or '''''a'''''(''antisymmetric'') in relation to the bisecting plane as shown below.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Bisecting Plane for Diels-alder'''
|-
|[[Image:Mb_da2.jpg|200px]]
|}

===Investigation of Molecular Orbital Symmetries of Butadiene and Ethene===

Butadiene and Ethylene were optimised and their molecular orbitals visualised:-

'''''Butadiene'''''

The HOMO and LUMO are considered with reference to the bisecting plane to determine their symmetry:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=4|'''HOMO and LUMO of Butadiene'''
|-align="center"
|width=250px|'''Molecular Orbital'''||'''2D Representation'''||'''3D Representation'''||width=250px|'''s/a'''
|-align="center"
|'''HOMO'''||[[Image:BDE HOMO.png|200px]]||[[Image:3D BDE HOMO.jpg|300px]]||a
|-align="center"
|'''LUMO'''||[[Image:BDE LUMO.png|200px]]||[[Image:3D BDE LUMO.jpg|300px]]||s
|}

'''''Ethene'''''

Similarly, the HOMO and LUMO of ethylene were characterised:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=4|'''HOMO and LUMO of Ethene'''
|-align="center"
|width=250px|'''Molecular Orbital'''||'''2D Representation'''||'''3D Representation'''||width=250px|'''s/a'''
|-align="center"
|'''HOMO'''||[[Image:ET HOMO.jpg|200px]]||[[Image:3D ET HOMO.jpg|300px]]||s
|-align="center"
|'''LUMO'''||[[Image:ET LUMO.jpg|200px]]||[[Image:3D ET LUMO.jpg|300px]]||a
|}

===Optimisation of Diels-Alder Transition State===

The '''Transition State''' whereby the concerted cycloaddition of butadiene and ethylene was modelled to a guess initially, before being optimised under HF 3-21G basis set conditions to a TS(Berny).

The initial guess was modelled from the bicyclo-system of cyclohexene and subsequently removing one of the -CH2CH2- arches to arrive at an "''envelope''"-shaped 6-membered ring.

The new σ bonds were stripped before the optimisation.

When this optimisation was complete, the structure obtained was slightly distorted, and so another optimisation using the frozen coordinate method.

The newly-optimised structure represented the "envelope", boat-like transition state a lot more accurately.

{|style="margin=1em auto 1em auto" border="1"
|-align="center"
|colspan=3|'''Comparisons of the Optimisations of transition state'''
|-align="center"
|'''Method'''||'''Ts(Berny)'''||'''Frozen-Coordinate'''
|-align="center"
|'''Structure'''||[[Image:CHEX BERNY.jpg|300px]]||[[Image:CHEX FROZEN.jpg|300px]]
|-align="center"
|'''Energy'''(a.u.)||-231.54077||-231.67327
|-align="center"
|'''Average new C-C σ bond length''' (A)||1.472645||1.526223
|-align="center"
|'''Literature Length of a C-C σ bond''' (A)||colspan=2|1.54
|}

By freezing the terminals, the energy of the optimised structure is markedly higher, and the length of the newly-forming σ bonds are closer to what they are experimentally measured as.

'''AM1 Molecular Orbital Analysis'''

The AM1 semi-empirical method was used to visualise the highest-occupied molecular orbital of the transition state.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''''HOMO'''''||'''''LUMO'''''
|-align="center"
|[[Image:CHEX TS HOMO.jpg|300px]]||[[Image:CHEX TS LUMO.jpg|300px]]
|-align="center"
|colspan=2|'''Symmetry'''
|-align="center"
|s||s
|-align="center"
|}

With reference to the bisecting plane as shown above, the HOMO of the transition state can be seen to be symmetric when reflected.

The individual MOs of butadiene and ethene thus must both be symmetric with respect to the plane in order to react in such a fashion.

This is consistent with the idea that only molecular orbitals with the same symmetry are allowed to react.

Given this, the pertinent orbitals that are symmetric are the '''LUMO''' of the butadiene and the '''HOMO''' of the ethene.

As such, electron density is donated into the empty π* LUMO of the butadiene.

===Diels-Alder Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride===

The Diels-Alder reaction of '''Cyclohexa-1,3-diene and Maleic Anhydride''' is also an example of a 6-electron [4+2] Diels-Alder Cycloaddition intrinsically.

However, it differs from the reaction between butadiene and ethene in that there is now a question of regioselectivity; the bigger anhydride ring can either react to adopt an '''EXO''' or an '''ENDO''' topography:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|[[Image:EXO ENDO.jpg|400px]]
|}

The vinyl goup on maleic anhydride, in comparison to ethene, is electron-deficient, thanks to the highly electron-withdrawing nature of the anhydride group.

Rep:Mod:parkbom

2009-11-13T15:53:58Z

Byt07:

=Module 3 - The Computation of the Transition State=

==Introduction==

The '''Transition State''' is an energetic maximum within a given reaction coordinate and represents the point at which bond breaking and bond forming is occurring simultaneously. This is represented by dotted lines in formal notation and depicts the transient nature of the bond(s) in question in the transition state.

The analysis of potential energy surfaces are effective in finding transition states; the transition states can be described graphically as "dams" that represent energy metastability, that the reactants must "cross" in order to form the products.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Potential Energy Surface Reaction Coordinate'''<ref>http://www.ch.ic.ac.uk/motm/porphyrins/TSdiscovery.html</ref>||'''Cope Rearrangement'''<ref>http://www.ch.ic.ac.uk/local/organic/pericyclic/</ref>
|-
|[[Image:PES.gif]]||[[Image:Cope Arr.gif|350px]]
|}

From the point of view of computational chemistry, the methods of molecular mechanical structure determination that employed imaginary force fields are ineffective in the calculations concerning the transition state in larger molecules, as they do not define the dynamics of bond making and bond forming that are apparent in the transient nature of the transition state.

As such, in combination with molecular orbital based calculations, the following of the potential energy surface of a molecule can be used to locate and analyse the transition state.

This computational exercise concerns the rearrangement of ''1,5-Hexadiene'' in the '''Cope Rearrangment'''(above).

The Cope Rearrangement proceeds via a [3,3]-Sigmatropic shift involving the motion of 6 electrons and is thermally-driven.

In terms of transition states, the cope rearrangement can reportedly proceed via two checkpoints, the '''boat''' or the '''chair''':-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Chair'''||'''Boat'''
|-
|[[Image:pic2a.jpg]]||[[Image:pic2b.jpg]]
|}

The purpose of this exercise is to ascertain which reaction pathway is preferred by computation of each transition state in the elucidation of geometry, repective energies and other thermochemical data.

==''Exercise 1'' - Optimising the Reactants and Products==

In this section, the "''reactant''" '''1,5-hexadiene''' and the "''product''" '''1,5-hexadiene''' are modelled in correspondence to the cope rearrangement.

The flexible nature of 1,5-hexadiene means that there is an initial dilemma of likely conformation at rest.

Several starting conformations were thus taken and their geometries optimised to find the lowest energy conformer.

Initially, 1,5-hexadiene was taken, ensuring a roughly "'''anti'''" linkage in the centre of the molecule, so that the central C atoms '''3''' and '''4''' are approximately '''antiperiplanar''' to one another. This molecule was cleaned in the gaussview builder interface and then taken and optimised using the '''Hartree-Fock''' method with basis set '''3-21G''' through Gaussian.

'''b)''' similarly, a synclinal, or "gauche" version of 1,5-hexadiene was taken and optimised at the HF 3-21G level.

The energies of both conformers and also their symmetry were noted.

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|'''"''Anti''" 1,5-Hexadiene'''||'''"''Gauche''" 1,5-Hexadiene'''
|-align="center"
|[[Image:Bo Anti.jpg|420px]]||[[Image:Bo Gauche.jpg|400px]]
|}

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|'''Confomer'''||'''''Anti''-1,5-Hexadiene'''||'''''Gauche''-1,5-Hexadiene'''
|-align="center"
|'''Total Energy'''(a.u.)||-231.69254||-231.69266
|-align="center"
|'''Point Group'''||ci||None
|}

From a purely steric-orientated argument, the ''anti'' configuration should have a lower energy than the ''gauche'', as there is less steric hindrance from the two ends of the carbon chain 180 degrees from one another than next to each other in the gauche conformation.

As it so happens, the energy for the ''anti'' conformer is reportedly lower than that of the ''gauche'' conformer, but by a minute amount of roughly a ten-thousandth of a hartree ('''1 hartree = 4.3597482E-18 J''').

Even so, it was appreciated that the central C-C-C-C linkage of 1,5-hexadiene lends itself to quite a high degree of rotational freedom, and thus, further examples of conformational isomerism. This is reinforced by the fact of the two remarkably different conformers having remarkably similar energies, and points to the fact that 1,5-hexadiene experiences conformational ambiguity often<ref>Cope Rearrangement of 1,5-Hexadiene: Full Geometry Optimizations Using Analytic MR-CISD and MR-AQCC Gradient Methods, doi:10.1021/jp0259014</ref>.

As such, 2 other conformers were taken and optimised:-

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|colspan=4|'''<big>Different Conformers of 1,5-Hexadiene</big>'''
|-align="center"
|(''g120,a,g120'')-'''Anti1'''||(''g120,a,g-120'')-'''Anti2'''||(''g120,g-60,g120'')-'''Gauche1'''||(''g120,g60,g-120'')-'''Gauche3'''
|-align="center"
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Bo Anti1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Bo Anti2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Gauche 1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Gauche 3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-align="center"
|colspan=4|'''Energy'''(a.u.)
|-align="center"
| -231.69260213||-231.69253506||-231.69166702||-231.69266120
|-align="center"
|colspan=4|'''Point Group'''
|-align="center"
|C2||Ci||C2||C1
|}

The energies here from the HF 3-21G are similarly close, but contrary to what was expected, it was found that ''gauche3'' was actually the lowest in energy.

Beneath this, the ''anti'' conformers are found to be lower in energy than the other ''gauche'' conformer. This may be rationalised as the minimalisation of steric hindrance lowering the energy of the overall molecule, as the antiperiplanar groups maintain the least steric strain.

However, the '''''cis'''''-''gauche'' conformer is lowest in energy due to stereoelectronic arguments.

The '''π'''-bonds of the C=C bonds are lying criss-crossed over each other in this conformer. This promotes effectual π-molecular orbital overlap and hence stabilises the molecule.

Next, the anti2 conformer with ci symmetry and the gauche1 conformer was taken and further optimised under the DFT B3LYP method with a more sophisticated basis set, 6-31G(d).

The resulting optimised molecule was then compared with the earlier attempt in terms of energy.

A further frequency analysis was conducted in order to check that the optimisation has proceeded to a minimum, being the case when all frequencies were real and positive.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=5|'''Comparison of the HF 3-21G and DFT B3LYP 6-31G(d) Optimisations'''
|-align="center"
|'''Method'''||colspan=2|'''HF 3-21G'''||colspan=2|'''DFT B3LYP 6-31G(d)'''
|-align="center"
|'''Conformer'''||'''anti2'''||'''gauche1'''||'''anti2'''||'''gauche1'''
|-align="center"
|'''Total Energy'''(a.u.)||-231.69253506||-231.69166702||-231.61170616||-231.61068821
|-align="center"
|1-2 C=C Bond Length (A)||1.3162||1.3156||1.3335||1.3331
|-align="center"
|2-3 C-C Bond Length (A)||1.5091||1.5041||1.5083||1.5044
|-align="center"
|3-4 C-C Bond Length (A)||1.5527||1.5481||1.5508||1.5482
|-align="center"
|4-5 C-C Bond Length (A)||1.5091||1.5041||1.5083||1.5044
|-align="center"
|5-6 C=C Bond Length (A)||1.3163||1.3335||1.3157||1.3331
|-align="center"
|1-2-3-4 Dihedral (o)||114.66||118.49||123.93||124.24
|-align="center"
|2-3-4-5 Dihedral (o)||180.00||180.00||-64.18||-65.01
|-align="center"
|3-4-5-6 Dihedral (o)||-114.66||-118.49||123.93||124.24
|}

As can be seen here, the energies returned differ in a noticeable manner. However, comparison of energies computed under different basis sets is unbalanced.

As such, comparisons of the geometry are notably more reliable in discerning the computed differences.

The bond lengths differ by negligible amounts. However, it can be seen that the various dihedral angles have been "tightened" more to their ideal figures (closer to 120o for the anticlinal dihedrals.

As such, using a more sophisticated basis set ensures a more accurate approximation to reality.

The log file yielded the following thermochemical data:-

'''1)Sum of electronic and zero-point Energies= -234.469195'''

'''2)Sum of electronic and thermal Energies= -234.461847'''

'''3)Sum of electronic and thermal Enthalpies= -234.460903'''

'''4)Sum of electronic and thermal Free Energies= -234.500782'''

Theses values concern the 1)Potential Energy at 0K that includes the zero-point vibrational energy term 2)Potential Energy at 298.15K and 1atm and adds contributions from translational, rotational and vibrational energy 3)Energy including a correction for RT(H=E+RT) for dissociation reactions and 4)The Entropic contribution.

==''Exercise 2'' - Optimising the "Chair" and "Boat" Transition Structures==

The Cope Rearrangement has been found to proceed via one of two possible transition states with differing conformation, as mentioned in the introduction:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=2|'''<big>Cope Rearrangement Transition State Conformations</big>'''
|-align="center"
|''Chair''||''Boat''
|-align="center"
|[[Image:pic2a.jpg]]||[[Image:pic2b.jpg]]
|}

When the rearrangment is thought as one that is concerted and proceeds with synchronicity of bond breaking and formation, these two transition states come to mind.

There is indeed a third type of transition state for this reaction, dubbed the '''''Dewar''''' transition state, that reportedly proceeds via a diradical tight-chair intermediate species that was discovered by Dupuis et al.<ref>The Cope Rearrangement Revisited, DOI:10.1021/ja00026a007</ref>, and is thus NOT concerted.

===The Chair Transition State===

The chair transition state was initally synthesised using two "resonant" '''allyl''' ('''C3H5''') fragments in gaussview, which were optimised under the 3-21G basis set, and subsequently placed in appropriate positions relative to one another to be optimised into the transition state.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=2|'''Allyl Fragment('''C3H5''')'''
|-align="center"
|colspan=2|[[Image:Allyl Frag.jpg|275px]]
|-align="center"
|'''sp3''' Centre Bond Angle (o)||109.5
|-align="center"
|'''Allyl''' C-C Bond Angle (o)||124.3
|-align="center"
|'''C-C''' Bond Length (A)||1.54
|-align="center"
|'''C=C''' Bond Length (A)||1.34
|-align="center"
|'''Allyl C-C''' Bond Length (A)||1.39
|}

The optimised allyl C-C bond angle is greater than the normal sp3 bond angle of '''109.5o''' due to resonance.

More importantly, the allyl C-C bond length is between that of a single bond and a double bond.

This optimised allyl fragment was then taken, duplicated and placed, facing opposite directions, with roughly a 2.2A distance between the terminal carbons of each fragment to mimic the shape of the chair transition state.

Thus, a frequency + optimisation was run that directed gaussian to reach a '''TS(Berny)''', with the elucidation of force constants.

The optimisation was then repeated but this time using the '''"frozen" coordinate method''', effectively fixing the two pairs of ends at a distance of 2.2A.

Clearly, this method requires that the general structure of the transition state in question be known. However, by fixing the positions in such a way, a more accurate optimisation can be done that avoids divergence.

Finally, the optimisation was performed a third time, using a normal guess '''Hessian''' method, which involves calculation of the force constant matrix.

After the three optimisations were complete, the following data were obtained:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=3|'''<big>Optimisation of Chair Transition State</big>'''
|-align="center"
|colspan=3|'''Structure'''
|-align="center"
|'''TS(Berny)'''||'''Frozen Coordinate'''||'''Hessian'''
|-align="center"
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Berny.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Frozen.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>red</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Hessian.mol</uploadedFileContents>
</jmolApplet></jmol>
|-align="center"
|colspan=3|'''Energies''' (a.u.)
|-align="center"
| -231.61932||-231.61932||-231.69167
|}

The structures returned from the normal force-constant calculation optimisation and the frozen coordinate method are indistinguishable from first glance, and their energies are virtually identical. However, the structure obtained from the Hessian is different, and so is its energy value.

'''FREQUENCY ANALYSIS'''

Vibration-wise, there was one negative, and thus "imaginary" frequency found with a wavenumber of '''-817.91cm-1'''.

When the stretch was visualised, the following was observed:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Imaginary Stretch at -817.91cm-1'''
|-
|[[Image:Imag Stretch.jpg|300px]]
|}

The appearance of this imaginary frequency resembles the motion that would ensue under the Cope Rearrangment - the action of bond breaking of the C-C σ bond in the middle of the 1,5-hexadiene with the subsequent formation of a new σ bond on across the terminals on the other side.

The fact that there is only '''one''' imaginary frequency value points to the notion that the cope rearrangment through this transition state is ''concerted'', and thus occurs in one ''synchronous'' step.

The number of imaginary frequencies produced from a frequency analysis for the midpoint of a reaction points towards the ''synchronicity'' of the reaction<ref>Cope Rearrangement Revisited, DOI:10.1021/ja00026a007</ref>.

===The Boat Transition State===

For the boat transition state, a different method was utilised, namely the '''QST2 method''', whereby reactant and product are defined, and the transition state is converged from these.

This required the previously optimised '''1,5-hexadiene''' molecule to be taken and numbered along its carbon skeleton.

The molecule was then duplicated and juxtaposed to the intial "reactant" molecule and renumbered in such a way to reflect the reacted, rearragement "product" as shown below:-

{|style="margin:1em auto 1em auto" border="1"
|-
|[[Image:Boat Method.JPG|400px]]
|}

The first instance of optimisation failed to produce an output, and thus the molecule was changed in the following way:-

The dihedral angle of both reactant and product were changed to 0 degrees, such that the two carbon groups either side of C3 and C4 are eclipsed, and the C2-C3-C4 and C3-C4-C5 bond angles were both changed to 100 degrees.

The modified structures were once again optimised under the QST2 method to yield the following structure:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''''QST2'' Optimised Boat Transition State'''
|-
|[[Image:FINAL Boat.jpg|400px]]
|}

'''DISCUSSION'''

Here, it is illustrated that the QST2 method is automated, and is thus very efficient, but is only effective if the shape provided of the reactants and products cloely mirror that of the transition state, otherwise convergence is difficult.

The '''QST3''' method was also attempted <ref>QST Calculation, http://hdl.handle.net/10042/to-2902</ref>, whereby an addition guess of the transition state geometry was implemented. If the guess of the transition state is sound, then the QST3 is by far more accurate than the QST2 method, and does not rely so much on the shape of the reactants and products.

initally, it would seem that in order for the cope rearrangement to occur, to two carbon terminals would be required to be close to one another. This is not the case with the "''anti''" conformers of the molecule, as the terminals are situated as far apart as they may possibly be, '''antiperplanar''' to one another.

Once free rotation moves the carbon groups closer together, into the gauche configuration, where the terminals are more poised to react, cope rearrangement would proceed.

The relative orientations of the vinyl groups would ultimately dictate which transition state would be traversed. For instance, if the vectors of the vinyl groups cross, such as the case for the gauche1 conformer, then as the terminals align into a bonding orientaton, the ''chair'' conformation would be adopted by the transition state.

Conversely, if the two vinyl groups lie in a pseudo-parallel fashion, where their vectors do NOT cross, such as is the case for gauche3, then the transition state would be of ''boat'' topography.

'''IRC'''

The Intrinsic Reaction Coordinate looks more cloesly at the potential surface of a reaction and works to pinpoint the highest point in a local area of the potential surface by creeping up the steepest slopes in the immediate area.

This calculation was attempted with 50 points, and did not converge

'''Energy Calculation'''

The two transition states were optimised a final time under DFT B3LYP 6-31G(d) conditions.

The geometries that were outputted were very similar to the previous examples, however

==The Diels-Alder Cycloaddition==

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Diels-Alder Reaction of Butadiene and Ethylene'''
|-align="center"
|[[Image:Mb_da3.jpg|400px]]
|}

The Diels-Alder Reaction is an example of a pericyclic cycloaddition, that usually involves the reaction between an electron rich diene and an electron poor dienophile.

The process is described as a [4s + 2s], 6 electron process that forms 2 new σ bonds.

When viewing the molecular orbital picture of butadiene and ethylene, the molecular orbitals are classified as either '''''s'''''(''symmetric'') or '''''a'''''(''antisymmetric'') in relation to the bisecting plane as shown below.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Bisecting Plane for Diels-alder'''
|-
|[[Image:Mb_da2.jpg|200px]]
|}

===Investigation of Molecular Orbital Symmetries of Butadiene and Ethene===

Butadiene and Ethylene were optimised and their molecular orbitals visualised:-

'''''Butadiene'''''

The HOMO and LUMO are considered with reference to the bisecting plane to determine their symmetry:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=4|'''HOMO and LUMO of Butadiene'''
|-align="center"
|width=250px|'''Molecular Orbital'''||'''2D Representation'''||'''3D Representation'''||width=250px|'''s/a'''
|-align="center"
|'''HOMO'''||[[Image:BDE HOMO.png|200px]]||[[Image:3D BDE HOMO.jpg|300px]]||a
|-align="center"
|'''LUMO'''||[[Image:BDE LUMO.png|200px]]||[[Image:3D BDE LUMO.jpg|300px]]||s
|}

'''''Ethene'''''

Similarly, the HOMO and LUMO of ethylene were characterised:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=4|'''HOMO and LUMO of Ethene'''
|-align="center"
|width=250px|'''Molecular Orbital'''||'''2D Representation'''||'''3D Representation'''||width=250px|'''s/a'''
|-align="center"
|'''HOMO'''||[[Image:ET HOMO.jpg|200px]]||[[Image:3D ET HOMO.jpg|300px]]||s
|-align="center"
|'''LUMO'''||[[Image:ET LUMO.jpg|200px]]||[[Image:3D ET LUMO.jpg|300px]]||a
|}

===Optimisation of Diels-Alder Transition State===

The '''Transition State''' whereby the concerted cycloaddition of butadiene and ethylene was modelled to a guess initially, before being optimised under HF 3-21G basis set conditions to a TS(Berny).

The initial guess was modelled from the bicyclo-system of cyclohexene and subsequently removing one of the -CH2CH2- arches to arrive at an "''envelope''"-shaped 6-membered ring.

The new σ bonds were stripped before the optimisation.

When this optimisation was complete, the structure obtained was slightly distorted, and so another optimisation using the frozen coordinate method.

The newly-optimised structure represented the "envelope", boat-like transition state a lot more accurately.

{|style="margin=1em auto 1em auto" border="1"
|-align="center"
|colspan=3|'''Comparisons of the Optimisations of transition state'''
|-align="center"
|'''Method'''||'''Ts(Berny)'''||'''Frozen-Coordinate'''
|-align="center"
|'''Structure'''||[[Image:CHEX BERNY.jpg|300px]]||[[Image:CHEX FROZEN.jpg|300px]]
|-align="center"
|'''Energy'''(a.u.)||-231.54077||-231.67327
|-align="center"
|'''Average new C-C σ bond length''' (A)||1.472645||1.526223
|-align="center"
|'''Literature Length of a C-C σ bond''' (A)||colspan=2|1.54
|}

By freezing the terminals, the energy of the optimised structure is markedly higher, and the length of the newly-forming σ bonds are closer to what they are experimentally measured as.

'''AM1 Molecular Orbital Analysis'''

The AM1 semi-empirical method was used to visualise the highest-occupied molecular orbital of the transition state.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''''HOMO'''''||'''''LUMO'''''
|-align="center"
|[[Image:CHEX TS HOMO.jpg|300px]]||[[Image:CHEX TS LUMO.jpg|300px]]
|-align="center"
|colspan=2|'''Symmetry'''
|-align="center"
|s||s
|-align="center"
|}

With reference to the bisecting plane as shown above, the HOMO of the transition state can be seen to be symmetric when reflected.

The individual MOs of butadiene and ethene thus must both be symmetric with respect to the plane in order to react in such a fashion.

This is consistent with the idea that only molecular orbitals with the same symmetry are allowed to react.

Given this, the pertinent orbitals that are symmetric are the '''LUMO''' of the butadiene and the '''HOMO''' of the ethene.

As such, electron density is donated into the empty π* LUMO of the butadiene.

===Diels-Alder Cycloaddition of Cyclohexa-1,3-diene and Maleic Anhydride===

The Diels-Alder reaction of '''Cyclohexa-1,3-diene and Maleic Anhydride''' is also an example of a 6-electron [4+2] Diels-Alder Cycloaddition intrinsically.

However, it differs from the reaction between butadiene and ethene in that there is now a question of regioselectivity; the bigger anhydride ring can either react to adopt an '''EXO''' or an '''ENDO''' topography:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|[[Image:EXO ENDO.jpg]]
|}

File:EXO ENDO.jpg

2009-11-13T15:51:56Z

Byt07:

File:LUMOHO.mol

2009-11-13T15:42:14Z

Byt07:

Rep:Mod:parkbom

2009-11-13T15:40:00Z

Byt07:

=Module 3 - The Computation of the Transition State=

==Introduction==

The '''Transition State''' is an energetic maximum within a given reaction coordinate and represents the point at which bond breaking and bond forming is occurring simultaneously. This is represented by dotted lines in formal notation and depicts the transient nature of the bond(s) in question in the transition state.

The analysis of potential energy surfaces are effective in finding transition states; the transition states can be described graphically as "dams" that represent energy metastability, that the reactants must "cross" in order to form the products.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Potential Energy Surface Reaction Coordinate'''<ref>http://www.ch.ic.ac.uk/motm/porphyrins/TSdiscovery.html</ref>||'''Cope Rearrangement'''<ref>http://www.ch.ic.ac.uk/local/organic/pericyclic/</ref>
|-
|[[Image:PES.gif]]||[[Image:Cope Arr.gif|350px]]
|}

From the point of view of computational chemistry, the methods of molecular mechanical structure determination that employed imaginary force fields are ineffective in the calculations concerning the transition state in larger molecules, as they do not define the dynamics of bond making and bond forming that are apparent in the transient nature of the transition state.

As such, in combination with molecular orbital based calculations, the following of the potential energy surface of a molecule can be used to locate and analyse the transition state.

This computational exercise concerns the rearrangement of ''1,5-Hexadiene'' in the '''Cope Rearrangment'''(above).

The Cope Rearrangement proceeds via a [3,3]-Sigmatropic shift involving the motion of 6 electrons and is thermally-driven.

In terms of transition states, the cope rearrangement can reportedly proceed via two checkpoints, the '''boat''' or the '''chair''':-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Chair'''||'''Boat'''
|-
|[[Image:pic2a.jpg]]||[[Image:pic2b.jpg]]
|}

The purpose of this exercise is to ascertain which reaction pathway is preferred by computation of each transition state in the elucidation of geometry, repective energies and other thermochemical data.

==''Exercise 1'' - Optimising the Reactants and Products==

In this section, the "''reactant''" '''1,5-hexadiene''' and the "''product''" '''1,5-hexadiene''' are modelled in correspondence to the cope rearrangement.

The flexible nature of 1,5-hexadiene means that there is an initial dilemma of likely conformation at rest.

Several starting conformations were thus taken and their geometries optimised to find the lowest energy conformer.

Initially, 1,5-hexadiene was taken, ensuring a roughly "'''anti'''" linkage in the centre of the molecule, so that the central C atoms '''3''' and '''4''' are approximately '''antiperiplanar''' to one another. This molecule was cleaned in the gaussview builder interface and then taken and optimised using the '''Hartree-Fock''' method with basis set '''3-21G''' through Gaussian.

'''b)''' similarly, a synclinal, or "gauche" version of 1,5-hexadiene was taken and optimised at the HF 3-21G level.

The energies of both conformers and also their symmetry were noted.

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|'''"''Anti''" 1,5-Hexadiene'''||'''"''Gauche''" 1,5-Hexadiene'''
|-align="center"
|[[Image:Bo Anti.jpg|420px]]||[[Image:Bo Gauche.jpg|400px]]
|}

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|'''Confomer'''||'''''Anti''-1,5-Hexadiene'''||'''''Gauche''-1,5-Hexadiene'''
|-align="center"
|'''Total Energy'''(a.u.)||-231.69254||-231.69266
|-align="center"
|'''Point Group'''||ci||None
|}

From a purely steric-orientated argument, the ''anti'' configuration should have a lower energy than the ''gauche'', as there is less steric hindrance from the two ends of the carbon chain 180 degrees from one another than next to each other in the gauche conformation.

As it so happens, the energy for the ''anti'' conformer is reportedly lower than that of the ''gauche'' conformer, but by a minute amount of roughly a ten-thousandth of a hartree ('''1 hartree = 4.3597482E-18 J''').

Even so, it was appreciated that the central C-C-C-C linkage of 1,5-hexadiene lends itself to quite a high degree of rotational freedom, and thus, further examples of conformational isomerism. This is reinforced by the fact of the two remarkably different conformers having remarkably similar energies, and points to the fact that 1,5-hexadiene experiences conformational ambiguity often<ref>Cope Rearrangement of 1,5-Hexadiene: Full Geometry Optimizations Using Analytic MR-CISD and MR-AQCC Gradient Methods, doi:10.1021/jp0259014</ref>.

As such, 2 other conformers were taken and optimised:-

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|colspan=4|'''<big>Different Conformers of 1,5-Hexadiene</big>'''
|-align="center"
|(''g120,a,g120'')-'''Anti1'''||(''g120,a,g-120'')-'''Anti2'''||(''g120,g-60,g120'')-'''Gauche1'''||(''g120,g60,g-120'')-'''Gauche3'''
|-align="center"
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Bo Anti1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Bo Anti2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Gauche 1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Gauche 3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-align="center"
|colspan=4|'''Energy'''(a.u.)
|-align="center"
| -231.69260213||-231.69253506||-231.69166702||-231.69266120
|-align="center"
|colspan=4|'''Point Group'''
|-align="center"
|C2||Ci||C2||C1
|}

The energies here from the HF 3-21G are similarly close, but contrary to what was expected, it was found that ''gauche3'' was actually the lowest in energy.

Beneath this, the ''anti'' conformers are found to be lower in energy than the other ''gauche'' conformer. This may be rationalised as the minimalisation of steric hindrance lowering the energy of the overall molecule, as the antiperiplanar groups maintain the least steric strain.

However, the '''''cis'''''-''gauche'' conformer is lowest in energy due to stereoelectronic arguments.

The '''π'''-bonds of the C=C bonds are lying criss-crossed over each other in this conformer. This promotes effectual π-molecular orbital overlap and hence stabilises the molecule.

Next, the anti2 conformer with ci symmetry and the gauche1 conformer was taken and further optimised under the DFT B3LYP method with a more sophisticated basis set, 6-31G(d).

The resulting optimised molecule was then compared with the earlier attempt in terms of energy.

A further frequency analysis was conducted in order to check that the optimisation has proceeded to a minimum, being the case when all frequencies were real and positive.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=5|'''Comparison of the HF 3-21G and DFT B3LYP 6-31G(d) Optimisations'''
|-align="center"
|'''Method'''||colspan=2|'''HF 3-21G'''||colspan=2|'''DFT B3LYP 6-31G(d)'''
|-align="center"
|'''Conformer'''||'''anti2'''||'''gauche1'''||'''anti2'''||'''gauche1'''
|-align="center"
|'''Total Energy'''(a.u.)||-231.69253506||-231.69166702||-231.61170616||-231.61068821
|-align="center"
|1-2 C=C Bond Length (A)||1.3162||1.3156||1.3335||1.3331
|-align="center"
|2-3 C-C Bond Length (A)||1.5091||1.5041||1.5083||1.5044
|-align="center"
|3-4 C-C Bond Length (A)||1.5527||1.5481||1.5508||1.5482
|-align="center"
|4-5 C-C Bond Length (A)||1.5091||1.5041||1.5083||1.5044
|-align="center"
|5-6 C=C Bond Length (A)||1.3163||1.3335||1.3157||1.3331
|-align="center"
|1-2-3-4 Dihedral (o)||114.66||118.49||123.93||124.24
|-align="center"
|2-3-4-5 Dihedral (o)||180.00||180.00||-64.18||-65.01
|-align="center"
|3-4-5-6 Dihedral (o)||-114.66||-118.49||123.93||124.24
|}

As can be seen here, the energies returned differ in a noticeable manner. However, comparison of energies computed under different basis sets is unbalanced.

As such, comparisons of the geometry are notably more reliable in discerning the computed differences.

The bond lengths differ by negligible amounts. However, it can be seen that the various dihedral angles have been "tightened" more to their ideal figures (closer to 120o for the anticlinal dihedrals.

As such, using a more sophisticated basis set ensures a more accurate approximation to reality.

The log file yielded the following thermochemical data:-

'''1)Sum of electronic and zero-point Energies= -234.469195'''

'''2)Sum of electronic and thermal Energies= -234.461847'''

'''3)Sum of electronic and thermal Enthalpies= -234.460903'''

'''4)Sum of electronic and thermal Free Energies= -234.500782'''

Theses values concern the 1)Potential Energy at 0K that includes the zero-point vibrational energy term 2)Potential Energy at 298.15K and 1atm and adds contributions from translational, rotational and vibrational energy 3)Energy including a correction for RT(H=E+RT) for dissociation reactions and 4)The Entropic contribution.

==''Exercise 2'' - Optimising the "Chair" and "Boat" Transition Structures==

The Cope Rearrangement has been found to proceed via one of two possible transition states with differing conformation, as mentioned in the introduction:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=2|'''<big>Cope Rearrangement Transition State Conformations</big>'''
|-align="center"
|''Chair''||''Boat''
|-align="center"
|[[Image:pic2a.jpg]]||[[Image:pic2b.jpg]]
|}

When the rearrangment is thought as one that is concerted and proceeds with synchronicity of bond breaking and formation, these two transition states come to mind.

There is indeed a third type of transition state for this reaction, dubbed the '''''Dewar''''' transition state, that reportedly proceeds via a diradical tight-chair intermediate species that was discovered by Dupuis et al.<ref>The Cope Rearrangement Revisited, DOI:10.1021/ja00026a007</ref>, and is thus NOT concerted.

===The Chair Transition State===

The chair transition state was initally synthesised using two "resonant" '''allyl''' ('''C3H5''') fragments in gaussview, which were optimised under the 3-21G basis set, and subsequently placed in appropriate positions relative to one another to be optimised into the transition state.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=2|'''Allyl Fragment('''C3H5''')'''
|-align="center"
|colspan=2|[[Image:Allyl Frag.jpg|275px]]
|-align="center"
|'''sp3''' Centre Bond Angle (o)||109.5
|-align="center"
|'''Allyl''' C-C Bond Angle (o)||124.3
|-align="center"
|'''C-C''' Bond Length (A)||1.54
|-align="center"
|'''C=C''' Bond Length (A)||1.34
|-align="center"
|'''Allyl C-C''' Bond Length (A)||1.39
|}

The optimised allyl C-C bond angle is greater than the normal sp3 bond angle of '''109.5o''' due to resonance.

More importantly, the allyl C-C bond length is between that of a single bond and a double bond.

This optimised allyl fragment was then taken, duplicated and placed, facing opposite directions, with roughly a 2.2A distance between the terminal carbons of each fragment to mimic the shape of the chair transition state.

Thus, a frequency + optimisation was run that directed gaussian to reach a '''TS(Berny)''', with the elucidation of force constants.

The optimisation was then repeated but this time using the '''"frozen" coordinate method''', effectively fixing the two pairs of ends at a distance of 2.2A.

Clearly, this method requires that the general structure of the transition state in question be known. However, by fixing the positions in such a way, a more accurate optimisation can be done that avoids divergence.

Finally, the optimisation was performed a third time, using a normal guess '''Hessian''' method, which involves calculation of the force constant matrix.

After the three optimisations were complete, the following data were obtained:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=3|'''<big>Optimisation of Chair Transition State</big>'''
|-align="center"
|colspan=3|'''Structure'''
|-align="center"
|'''TS(Berny)'''||'''Frozen Coordinate'''||'''Hessian'''
|-align="center"
||<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Berny.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Frozen.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>red</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Hessian.mol</uploadedFileContents>
</jmolApplet></jmol>
|-align="center"
|colspan=3|'''Energies''' (a.u.)
|-align="center"
| -231.61932||-231.61932||-231.69167
|}

The structures returned from the normal force-constant calculation optimisation and the frozen coordinate method are indistinguishable from first glance, and their energies are virtually identical. However, the structure obtained from the Hessian is different, and so is its energy value.

'''FREQUENCY ANALYSIS'''

Vibration-wise, there was one negative, and thus "imaginary" frequency found with a wavenumber of '''-817.91cm-1'''.

When the stretch was visualised, the following was observed:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Imaginary Stretch at -817.91cm-1'''
|-
|[[Image:Imag Stretch.jpg|300px]]
|}

The appearance of this imaginary frequency resembles the motion that would ensue under the Cope Rearrangment - the action of bond breaking of the C-C σ bond in the middle of the 1,5-hexadiene with the subsequent formation of a new σ bond on across the terminals on the other side.

The fact that there is only '''one''' imaginary frequency value points to the notion that the cope rearrangment through this transition state is ''concerted'', and thus occurs in one ''synchronous'' step.

The number of imaginary frequencies produced from a frequency analysis for the midpoint of a reaction points towards the ''synchronicity'' of the reaction<ref>Cope Rearrangement Revisited, DOI:10.1021/ja00026a007</ref>.

===The Boat Transition State===

For the boat transition state, a different method was utilised, namely the '''QST2 method''', whereby reactant and product are defined, and the transition state is converged from these.

This required the previously optimised '''1,5-hexadiene''' molecule to be taken and numbered along its carbon skeleton.

The molecule was then duplicated and juxtaposed to the intial "reactant" molecule and renumbered in such a way to reflect the reacted, rearragement "product" as shown below:-

{|style="margin:1em auto 1em auto" border="1"
|-
|[[Image:Boat Method.JPG|400px]]
|}

The first instance of optimisation failed to produce an output, and thus the molecule was changed in the following way:-

The dihedral angle of both reactant and product were changed to 0 degrees, such that the two carbon groups either side of C3 and C4 are eclipsed, and the C2-C3-C4 and C3-C4-C5 bond angles were both changed to 100 degrees.

The modified structures were once again optimised under the QST2 method to yield the following structure:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''''QST2'' Optimised Boat Transition State'''
|-
|[[Image:FINAL Boat.jpg|400px]]
|}

'''DISCUSSION'''

Here, it is illustrated that the QST2 method is automated, and is thus very efficient, but is only effective if the shape provided of the reactants and products cloely mirror that of the transition state, otherwise convergence is difficult.

The '''QST3''' method was also attempted <ref>QST Calculation, http://hdl.handle.net/10042/to-2902</ref>, whereby an addition guess of the transition state geometry was implemented. If the guess of the transition state is sound, then the QST3 is by far more accurate than the QST2 method, and does not rely so much on the shape of the reactants and products.

initally, it would seem that in order for the cope rearrangement to occur, to two carbon terminals would be required to be close to one another. This is not the case with the "''anti''" conformers of the molecule, as the terminals are situated as far apart as they may possibly be, '''antiperplanar''' to one another.

Once free rotation moves the carbon groups closer together, into the gauche configuration, where the terminals are more poised to react, cope rearrangement would proceed.

The relative orientations of the vinyl groups would ultimately dictate which transition state would be traversed. For instance, if the vectors of the vinyl groups cross, such as the case for the gauche1 conformer, then as the terminals align into a bonding orientaton, the ''chair'' conformation would be adopted by the transition state.

Conversely, if the two vinyl groups lie in a pseudo-parallel fashion, where their vectors do NOT cross, such as is the case for gauche3, then the transition state would be of ''boat'' topography.

'''IRC'''

The Intrinsic Reaction Coordinate looks more cloesly at the potential surface of a reaction and works to pinpoint the highest point in a local area of the potential surface by creeping up the steepest slopes in the immediate area.

This calculation was attempted with 50 points, and did not converge

'''Energy Calculation'''

The two transition states were optimised a final time under DFT B3LYP 6-31G(d) conditions.

The geometries that were outputted were very similar to the previous examples, however

==The Diels-Alder Cycloaddition==

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Diels-Alder Reaction of Butadiene and Ethylene'''
|-align="center"
|[[Image:Mb_da3.jpg|400px]]
|}

The Diels-Alder Reaction is an example of a pericyclic cycloaddition, that usually involves the reaction between an electron rich diene and an electron poor dienophile.

The process is described as a [4s + 2s], 6 electron process that forms 2 new σ bonds.

When viewing the molecular orbital picture of butadiene and ethylene, the molecular orbitals are classified as either '''''s'''''(''symmetric'') or '''''a'''''(''antisymmetric'') in relation to the bisecting plane as shown below.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Bisecting Plane for Diels-alder'''
|-
|[[Image:Mb_da2.jpg|200px]]
|}

===Investigation of Molecular Orbital Symmetries of Butadiene and Ethene===

Butadiene and Ethylene were optimised and their molecular orbitals visualised:-

'''''Butadiene'''''

The HOMO and LUMO are considered with reference to the bisecting plane to determine their symmetry:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=4|'''HOMO and LUMO of Butadiene'''
|-align="center"
|width=250px|'''Molecular Orbital'''||'''2D Representation'''||'''3D Representation'''||width=250px|'''s/a'''
|-align="center"
|'''HOMO'''||[[Image:BDE HOMO.png|200px]]||[[Image:3D BDE HOMO.jpg|300px]]||a
|-align="center"
|'''LUMO'''||[[Image:BDE LUMO.png|200px]]||[[Image:3D BDE LUMO.jpg|300px]]||s
|}

'''''Ethene'''''

Similarly, the HOMO and LUMO of ethylene were characterised:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=4|'''HOMO and LUMO of Ethene'''
|-align="center"
|width=250px|'''Molecular Orbital'''||'''2D Representation'''||'''3D Representation'''||width=250px|'''s/a'''
|-align="center"
|'''HOMO'''||[[Image:ET HOMO.jpg|200px]]||[[Image:3D ET HOMO.jpg|300px]]||s
|-align="center"
|'''LUMO'''||[[Image:ET LUMO.jpg|200px]]||[[Image:3D ET LUMO.jpg|300px]]||a
|}

===Optimisation of Diels-Alder Transition State===

The '''Transition State''' whereby the concerted cycloaddition of butadiene and ethylene was modelled to a guess initially, before being optimised under HF 3-21G basis set conditions to a TS(Berny).

The initial guess was modelled from the bicyclo-system of cyclohexene and subsequently removing one of the -CH2CH2- arches to arrive at an "''envelope''"-shaped 6-membered ring.

The new σ bonds were stripped before the optimisation.

When this optimisation was complete, the structure obtained was slightly distorted, and so another optimisation using the frozen coordinate method.

The newly-optimised structure represented the "envelope", boat-like transition state a lot more accurately.

{|style="margin=1em auto 1em auto" border="1"
|-align="center"
|colspan=3|'''Comparisons of the Optimisations of transition state'''
|-align="center"
|'''Method'''||'''Ts(Berny)'''||'''Frozen-Coordinate'''
|-align="center"
|'''Structure'''||[[Image:CHEX BERNY.jpg|300px]]||[[Image:CHEX FROZEN.jpg|300px]]
|-align="center"
|'''Energy'''(a.u.)||-231.54077||-231.67327
|-align="center"
|'''Average new C-C σ bond length''' (A)||1.472645||1.526223
|-align="center"
|'''Literature Length of a C-C σ bond''' (A)||colspan=2|1.54
|}

By freezing the terminals, the energy of the optimised structure is markedly higher, and the length of the newly-forming σ bonds are closer to what they are experimentally measured as.

'''AM1 Molecular Orbital Analysis'''

The AM1 semi-empirical method was used to visualise the highest-occupied molecular orbital of the transition state.

{|style="margin:1em autp 1em auto" border="1"
|-align="center"
|'''''HOMO'''''||'''''LUMO'''''
|-align="center"
|[[Image:CHEX TS HOMO.jpg|300px]]||[[Image:CHEX TS LUMO.jpg|300px]]
|-align="center"
|colspan=2|'''Symmetry'''
|-align="center"
|s||s
|}

With reference to the bisecting plane as shown above, the HOMO of the transition state can be seen to be symmetric when reflected.

The individual MOs of butadiene and ethene thus must both be symmetric with respect to the plane in order to react in such a fashion.

This is consistent with the idea that only molecular orbitals with the same symmetry are allowed to react.

Given this, the pertinent orbitals that are symmetric are the '''LUMO''' of the butadiene and the '''HOMO''' of the ethene.

As such, electron density is donated into the empty π* LUMO of the butadiene.

File:CHEX TS LUMO.jpg

2009-11-13T15:38:37Z

Byt07:

Rep:Mod:parkbom

2009-11-13T15:25:25Z

Byt07:

=Module 3 - The Computation of the Transition State=

==Introduction==

The '''Transition State''' is an energetic maximum within a given reaction coordinate and represents the point at which bond breaking and bond forming is occurring simultaneously. This is represented by dotted lines in formal notation and depicts the transient nature of the bond(s) in question in the transition state.

The analysis of potential energy surfaces are effective in finding transition states; the transition states can be described graphically as "dams" that represent energy metastability, that the reactants must "cross" in order to form the products.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Potential Energy Surface Reaction Coordinate'''<ref>http://www.ch.ic.ac.uk/motm/porphyrins/TSdiscovery.html</ref>||'''Cope Rearrangement'''<ref>http://www.ch.ic.ac.uk/local/organic/pericyclic/</ref>
|-
|[[Image:PES.gif]]||[[Image:Cope Arr.gif|350px]]
|}

From the point of view of computational chemistry, the methods of molecular mechanical structure determination that employed imaginary force fields are ineffective in the calculations concerning the transition state in larger molecules, as they do not define the dynamics of bond making and bond forming that are apparent in the transient nature of the transition state.

As such, in combination with molecular orbital based calculations, the following of the potential energy surface of a molecule can be used to locate and analyse the transition state.

This computational exercise concerns the rearrangement of ''1,5-Hexadiene'' in the '''Cope Rearrangment'''(above).

The Cope Rearrangement proceeds via a [3,3]-Sigmatropic shift involving the motion of 6 electrons and is thermally-driven.

In terms of transition states, the cope rearrangement can reportedly proceed via two checkpoints, the '''boat''' or the '''chair''':-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Chair'''||'''Boat'''
|-
|[[Image:pic2a.jpg]]||[[Image:pic2b.jpg]]
|}

The purpose of this exercise is to ascertain which reaction pathway is preferred by computation of each transition state in the elucidation of geometry, repective energies and other thermochemical data.

==''Exercise 1'' - Optimising the Reactants and Products==

In this section, the "''reactant''" '''1,5-hexadiene''' and the "''product''" '''1,5-hexadiene''' are modelled in correspondence to the cope rearrangement.

The flexible nature of 1,5-hexadiene means that there is an initial dilemma of likely conformation at rest.

Several starting conformations were thus taken and their geometries optimised to find the lowest energy conformer.

Initially, 1,5-hexadiene was taken, ensuring a roughly "'''anti'''" linkage in the centre of the molecule, so that the central C atoms '''3''' and '''4''' are approximately '''antiperiplanar''' to one another. This molecule was cleaned in the gaussview builder interface and then taken and optimised using the '''Hartree-Fock''' method with basis set '''3-21G''' through Gaussian.

'''b)''' similarly, a synclinal, or "gauche" version of 1,5-hexadiene was taken and optimised at the HF 3-21G level.

The energies of both conformers and also their symmetry were noted.

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|'''"''Anti''" 1,5-Hexadiene'''||'''"''Gauche''" 1,5-Hexadiene'''
|-align="center"
|[[Image:Bo Anti.jpg|420px]]||[[Image:Bo Gauche.jpg|400px]]
|}

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|'''Confomer'''||'''''Anti''-1,5-Hexadiene'''||'''''Gauche''-1,5-Hexadiene'''
|-align="center"
|'''Total Energy'''(a.u.)||-231.69254||-231.69266
|-align="center"
|'''Point Group'''||ci||None
|}

From a purely steric-orientated argument, the ''anti'' configuration should have a lower energy than the ''gauche'', as there is less steric hindrance from the two ends of the carbon chain 180 degrees from one another than next to each other in the gauche conformation.

As it so happens, the energy for the ''anti'' conformer is reportedly lower than that of the ''gauche'' conformer, but by a minute amount of roughly a ten-thousandth of a hartree ('''1 hartree = 4.3597482E-18 J''').

Even so, it was appreciated that the central C-C-C-C linkage of 1,5-hexadiene lends itself to quite a high degree of rotational freedom, and thus, further examples of conformational isomerism. This is reinforced by the fact of the two remarkably different conformers having remarkably similar energies, and points to the fact that 1,5-hexadiene experiences conformational ambiguity often<ref>Cope Rearrangement of 1,5-Hexadiene: Full Geometry Optimizations Using Analytic MR-CISD and MR-AQCC Gradient Methods, doi:10.1021/jp0259014</ref>.

As such, 2 other conformers were taken and optimised:-

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|colspan=4|'''<big>Different Conformers of 1,5-Hexadiene</big>'''
|-align="center"
|(''g120,a,g120'')-'''Anti1'''||(''g120,a,g-120'')-'''Anti2'''||(''g120,g-60,g120'')-'''Gauche1'''||(''g120,g60,g-120'')-'''Gauche3'''
|-align="center"
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Bo Anti1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Bo Anti2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Gauche 1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Gauche 3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-align="center"
|colspan=4|'''Energy'''(a.u.)
|-align="center"
| -231.69260213||-231.69253506||-231.69166702||-231.69266120
|-align="center"
|colspan=4|'''Point Group'''
|-align="center"
|C2||Ci||C2||C1
|}

The energies here from the HF 3-21G are similarly close, but contrary to what was expected, it was found that ''gauche3'' was actually the lowest in energy.

Beneath this, the ''anti'' conformers are found to be lower in energy than the other ''gauche'' conformer. This may be rationalised as the minimalisation of steric hindrance lowering the energy of the overall molecule, as the antiperiplanar groups maintain the least steric strain.

However, the '''''cis'''''-''gauche'' conformer is lowest in energy due to stereoelectronic arguments.

The '''π'''-bonds of the C=C bonds are lying criss-crossed over each other in this conformer. This promotes effectual π-molecular orbital overlap and hence stabilises the molecule.

Next, the anti2 conformer with ci symmetry and the gauche1 conformer was taken and further optimised under the DFT B3LYP method with a more sophisticated basis set, 6-31G(d).

The resulting optimised molecule was then compared with the earlier attempt in terms of energy.

A further frequency analysis was conducted in order to check that the optimisation has proceeded to a minimum, being the case when all frequencies were real and positive.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=5|'''Comparison of the HF 3-21G and DFT B3LYP 6-31G(d) Optimisations'''
|-align="center"
|'''Method'''||colspan=2|'''HF 3-21G'''||colspan=2|'''DFT B3LYP 6-31G(d)'''
|-align="center"
|'''Conformer'''||'''anti2'''||'''gauche1'''||'''anti2'''||'''gauche1'''
|-align="center"
|'''Total Energy'''(a.u.)||-231.69253506||-231.69166702||-231.61170616||-231.61068821
|-align="center"
|1-2 C=C Bond Length (A)||1.3162||1.3156||1.3335||1.3331
|-align="center"
|2-3 C-C Bond Length (A)||1.5091||1.5041||1.5083||1.5044
|-align="center"
|3-4 C-C Bond Length (A)||1.5527||1.5481||1.5508||1.5482
|-align="center"
|4-5 C-C Bond Length (A)||1.5091||1.5041||1.5083||1.5044
|-align="center"
|5-6 C=C Bond Length (A)||1.3163||1.3335||1.3157||1.3331
|-align="center"
|1-2-3-4 Dihedral (o)||114.66||118.49||123.93||124.24
|-align="center"
|2-3-4-5 Dihedral (o)||180.00||180.00||-64.18||-65.01
|-align="center"
|3-4-5-6 Dihedral (o)||-114.66||-118.49||123.93||124.24
|}

As can be seen here, the energies returned differ in a noticeable manner. However, comparison of energies computed under different basis sets is unbalanced.

As such, comparisons of the geometry are notably more reliable in discerning the computed differences.

The bond lengths differ by negligible amounts. However, it can be seen that the various dihedral angles have been "tightened" more to their ideal figures (closer to 120o for the anticlinal dihedrals.

As such, using a more sophisticated basis set ensures a more accurate approximation to reality.

The log file yielded the following thermochemical data:-

'''1)Sum of electronic and zero-point Energies= -234.469195'''

'''2)Sum of electronic and thermal Energies= -234.461847'''

'''3)Sum of electronic and thermal Enthalpies= -234.460903'''

'''4)Sum of electronic and thermal Free Energies= -234.500782'''

Theses values concern the 1)Potential Energy at 0K that includes the zero-point vibrational energy term 2)Potential Energy at 298.15K and 1atm and adds contributions from translational, rotational and vibrational energy 3)Energy including a correction for RT(H=E+RT) for dissociation reactions and 4)The Entropic contribution.

==''Exercise 2'' - Optimising the "Chair" and "Boat" Transition Structures==

The Cope Rearrangement has been found to proceed via one of two possible transition states with differing conformation, as mentioned in the introduction:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=2|'''<big>Cope Rearrangement Transition State Conformations</big>'''
|-align="center"
|''Chair''||''Boat''
|-align="center"
|[[Image:pic2a.jpg]]||[[Image:pic2b.jpg]]
|}

When the rearrangment is thought as one that is concerted and proceeds with synchronicity of bond breaking and formation, these two transition states come to mind.

There is indeed a third type of transition state for this reaction, dubbed the '''''Dewar''''' transition state, that reportedly proceeds via a diradical tight-chair intermediate species that was discovered by Dupuis et al.<ref>The Cope Rearrangement Revisited, DOI:10.1021/ja00026a007</ref>, and is thus NOT concerted.

===The Chair Transition State===

The chair transition state was initally synthesised using two "resonant" '''allyl''' ('''C3H5''') fragments in gaussview, which were optimised under the 3-21G basis set, and subsequently placed in appropriate positions relative to one another to be optimised into the transition state.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=2|'''Allyl Fragment('''C3H5''')'''
|-align="center"
|colspan=2|[[Image:Allyl Frag.jpg|275px]]
|-align="center"
|'''sp3''' Centre Bond Angle (o)||109.5
|-align="center"
|'''Allyl''' C-C Bond Angle (o)||124.3
|-align="center"
|'''C-C''' Bond Length (A)||1.54
|-align="center"
|'''C=C''' Bond Length (A)||1.34
|-align="center"
|'''Allyl C-C''' Bond Length (A)||1.39
|}

The optimised allyl C-C bond angle is greater than the normal sp3 bond angle of '''109.5o''' due to resonance.

More importantly, the allyl C-C bond length is between that of a single bond and a double bond.

This optimised allyl fragment was then taken, duplicated and placed, facing opposite directions, with roughly a 2.2A distance between the terminal carbons of each fragment to mimic the shape of the chair transition state.

Thus, a frequency + optimisation was run that directed gaussian to reach a '''TS(Berny)''', with the elucidation of force constants.

The optimisation was then repeated but this time using the '''"frozen" coordinate method''', effectively fixing the two pairs of ends at a distance of 2.2A.

Clearly, this method requires that the general structure of the transition state in question be known. However, by fixing the positions in such a way, a more accurate optimisation can be done that avoids divergence.

Finally, the optimisation was performed a third time, using a normal guess '''Hessian''' method, which involves calculation of the force constant matrix.

After the three optimisations were complete, the following data were obtained:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=3|'''<big>Optimisation of Chair Transition State</big>'''
|-align="center"
|colspan=3|'''Structure'''
|-align="center"
|'''TS(Berny)'''||'''Frozen Coordinate'''||'''Hessian'''
|-align="center"
||<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Berny.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Frozen.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>red</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Hessian.mol</uploadedFileContents>
</jmolApplet></jmol>
|-align="center"
|colspan=3|'''Energies''' (a.u.)
|-align="center"
| -231.61932||-231.61932||-231.69167
|}

The structures returned from the normal force-constant calculation optimisation and the frozen coordinate method are indistinguishable from first glance, and their energies are virtually identical. However, the structure obtained from the Hessian is different, and so is its energy value.

'''FREQUENCY ANALYSIS'''

Vibration-wise, there was one negative, and thus "imaginary" frequency found with a wavenumber of '''-817.91cm-1'''.

When the stretch was visualised, the following was observed:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Imaginary Stretch at -817.91cm-1'''
|-
|[[Image:Imag Stretch.jpg|300px]]
|}

The appearance of this imaginary frequency resembles the motion that would ensue under the Cope Rearrangment - the action of bond breaking of the C-C σ bond in the middle of the 1,5-hexadiene with the subsequent formation of a new σ bond on across the terminals on the other side.

The fact that there is only '''one''' imaginary frequency value points to the notion that the cope rearrangment through this transition state is ''concerted'', and thus occurs in one ''synchronous'' step.

The number of imaginary frequencies produced from a frequency analysis for the midpoint of a reaction points towards the ''synchronicity'' of the reaction<ref>Cope Rearrangement Revisited, DOI:10.1021/ja00026a007</ref>.

===The Boat Transition State===

For the boat transition state, a different method was utilised, namely the '''QST2 method''', whereby reactant and product are defined, and the transition state is converged from these.

This required the previously optimised '''1,5-hexadiene''' molecule to be taken and numbered along its carbon skeleton.

The molecule was then duplicated and juxtaposed to the intial "reactant" molecule and renumbered in such a way to reflect the reacted, rearragement "product" as shown below:-

{|style="margin:1em auto 1em auto" border="1"
|-
|[[Image:Boat Method.JPG|400px]]
|}

The first instance of optimisation failed to produce an output, and thus the molecule was changed in the following way:-

The dihedral angle of both reactant and product were changed to 0 degrees, such that the two carbon groups either side of C3 and C4 are eclipsed, and the C2-C3-C4 and C3-C4-C5 bond angles were both changed to 100 degrees.

The modified structures were once again optimised under the QST2 method to yield the following structure:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''''QST2'' Optimised Boat Transition State'''
|-
|[[Image:FINAL Boat.jpg|400px]]
|}

'''DISCUSSION'''

Here, it is illustrated that the QST2 method is automated, and is thus very efficient, but is only effective if the shape provided of the reactants and products cloely mirror that of the transition state, otherwise convergence is difficult.

The '''QST3''' method was also attempted <ref>QST Calculation, http://hdl.handle.net/10042/to-2902</ref>, whereby an addition guess of the transition state geometry was implemented. If the guess of the transition state is sound, then the QST3 is by far more accurate than the QST2 method, and does not rely so much on the shape of the reactants and products.

initally, it would seem that in order for the cope rearrangement to occur, to two carbon terminals would be required to be close to one another. This is not the case with the "''anti''" conformers of the molecule, as the terminals are situated as far apart as they may possibly be, '''antiperplanar''' to one another.

Once free rotation moves the carbon groups closer together, into the gauche configuration, where the terminals are more poised to react, cope rearrangement would proceed.

The relative orientations of the vinyl groups would ultimately dictate which transition state would be traversed. For instance, if the vectors of the vinyl groups cross, such as the case for the gauche1 conformer, then as the terminals align into a bonding orientaton, the ''chair'' conformation would be adopted by the transition state.

Conversely, if the two vinyl groups lie in a pseudo-parallel fashion, where their vectors do NOT cross, such as is the case for gauche3, then the transition state would be of ''boat'' topography.

'''IRC'''

The Intrinsic Reaction Coordinate looks more cloesly at the potential surface of a reaction and works to pinpoint the highest point in a local area of the potential surface by creeping up the steepest slopes in the immediate area.

This calculation was attempted with 50 points, and did not converge

'''Energy Calculation'''

The two transition states were optimised a final time under DFT B3LYP 6-31G(d) conditions.

The geometries that were outputted were very similar to the previous examples, however

==The Diels-Alder Cycloaddition==

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Diels-Alder Reaction of Butadiene and Ethylene'''
|-align="center"
|[[Image:Mb_da3.jpg|400px]]
|}

The Diels-Alder Reaction is an example of a pericyclic cycloaddition, that usually involves the reaction between an electron rich diene and an electron poor dienophile.

The process is described as a [4s + 2s], 6 electron process that forms 2 new σ bonds.

When viewing the molecular orbital picture of butadiene and ethylene, the molecular orbitals are classified as either '''''s'''''(''symmetric'') or '''''a'''''(''antisymmetric'') in relation to the bisecting plane as shown below.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Bisecting Plane for Diels-alder'''
|-
|[[Image:Mb_da2.jpg|200px]]
|}

===Investigation of Molecular Orbital Symmetries of Butadiene and Ethene===

Butadiene and Ethylene were optimised and their molecular orbitals visualised:-

'''''Butadiene'''''

The HOMO and LUMO are considered with reference to the bisecting plane to determine their symmetry:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=4|'''HOMO and LUMO of Butadiene'''
|-align="center"
|width=250px|'''Molecular Orbital'''||'''2D Representation'''||'''3D Representation'''||width=250px|'''s/a'''
|-align="center"
|'''HOMO'''||[[Image:BDE HOMO.png|200px]]||[[Image:3D BDE HOMO.jpg|300px]]||a
|-align="center"
|'''LUMO'''||[[Image:BDE LUMO.png|200px]]||[[Image:3D BDE LUMO.jpg|300px]]||s
|}

'''''Ethene'''''

Similarly, the HOMO and LUMO of ethylene were characterised:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=4|'''HOMO and LUMO of Ethene'''
|-align="center"
|width=250px|'''Molecular Orbital'''||'''2D Representation'''||'''3D Representation'''||width=250px|'''s/a'''
|-align="center"
|'''HOMO'''||[[Image:ET HOMO.jpg|200px]]||[[Image:3D ET HOMO.jpg|300px]]||s
|-align="center"
|'''LUMO'''||[[Image:ET LUMO.jpg|200px]]||[[Image:3D ET LUMO.jpg|300px]]||a
|}

===Optimisation of Diels-Alder Transition State===

The '''Transition State''' whereby the concerted cycloaddition of butadiene and ethylene was modelled to a guess initially, before being optimised under HF 3-21G basis set conditions to a TS(Berny).

The initial guess was modelled from the bicyclo-system of cyclohexene and subsequently removing one of the -CH2CH2- arches to arrive at an "''envelope''"-shaped 6-membered ring.

The new σ bonds were stripped before the optimisation.

When this optimisation was complete, the structure obtained was slightly distorted, and so another optimisation using the frozen coordinate method.

The newly-optimised structure represented the "envelope", boat-like transition state a lot more accurately.

{|style="margin=1em auto 1em auto" border="1"
|-align="center"
|colspan=3|'''Comparisons of the Optimisations of transition state'''
|-align="center"
|'''Method'''||'''Ts(Berny)'''||'''Frozen-Coordinate'''
|-align="center"
|'''Structure'''||[[Image:CHEX BERNY.jpg|300px]]||[[Image:CHEX FROZEN.jpg|300px]]
|-align="center"
|'''Energy'''(a.u.)||-231.54077||-231.67327
|-align="center"
|'''Average new C-C σ bond length''' (A)||1.472645||1.526223
|-align="center"
|'''Literature Length of a C-C σ bond''' (A)||colspan=2|1.54
|}

By freezing the terminals, the energy of the optimised structure is markedly higher, and the length of the newly-forming σ bonds are closer to what they are experimentally measured as.

'''AM1 Molecular Orbital Analysis'''

The AM1 semi-empirical method was used to visualise the highest-occupied molecular orbital of the transition state.

{|style="margin:1em autp 1em auto" border="1"
|-align="center"
|'''''HOMO'' of Diels-Alder transition state'''
|-align="center"
|[[Image:CHEX TS HOMO.jpg]]
|-align="center"
|'''Symmetry'''
|-align="center"
|s
|}

With reference to the bisecting plane as shown above, the HOMO of the transition state can be seen to be symmetric when reflected.

The individual MOs of butadiene and ethene thus must both be symmetric with respect to the plane in order to react in such a fashion.

This is consistent with the idea that only molecular orbitals with the same symmetry are allowed to react.

Given this, the pertinent orbitals that are symmetric are the '''LUMO''' of the butadiene and the '''HOMO''' of the ethene.

This is also consistent with reality, as the electron

File:CHEX TS HOMO.jpg

2009-11-13T15:13:17Z

Byt07:

Rep:Mod:parkbom

2009-11-13T14:38:36Z

Byt07:

=Module 3 - The Computation of the Transition State=

==Introduction==

The '''Transition State''' is an energetic maximum within a given reaction coordinate and represents the point at which bond breaking and bond forming is occurring simultaneously. This is represented by dotted lines in formal notation and depicts the transient nature of the bond(s) in question in the transition state.

The analysis of potential energy surfaces are effective in finding transition states; the transition states can be described graphically as "dams" that represent energy metastability, that the reactants must "cross" in order to form the products.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Potential Energy Surface Reaction Coordinate'''<ref>http://www.ch.ic.ac.uk/motm/porphyrins/TSdiscovery.html</ref>||'''Cope Rearrangement'''<ref>http://www.ch.ic.ac.uk/local/organic/pericyclic/</ref>
|-
|[[Image:PES.gif]]||[[Image:Cope Arr.gif|350px]]
|}

From the point of view of computational chemistry, the methods of molecular mechanical structure determination that employed imaginary force fields are ineffective in the calculations concerning the transition state in larger molecules, as they do not define the dynamics of bond making and bond forming that are apparent in the transient nature of the transition state.

As such, in combination with molecular orbital based calculations, the following of the potential energy surface of a molecule can be used to locate and analyse the transition state.

This computational exercise concerns the rearrangement of ''1,5-Hexadiene'' in the '''Cope Rearrangment'''(above).

The Cope Rearrangement proceeds via a [3,3]-Sigmatropic shift involving the motion of 6 electrons and is thermally-driven.

In terms of transition states, the cope rearrangement can reportedly proceed via two checkpoints, the '''boat''' or the '''chair''':-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Chair'''||'''Boat'''
|-
|[[Image:pic2a.jpg]]||[[Image:pic2b.jpg]]
|}

The purpose of this exercise is to ascertain which reaction pathway is preferred by computation of each transition state in the elucidation of geometry, repective energies and other thermochemical data.

==''Exercise 1'' - Optimising the Reactants and Products==

In this section, the "''reactant''" '''1,5-hexadiene''' and the "''product''" '''1,5-hexadiene''' are modelled in correspondence to the cope rearrangement.

The flexible nature of 1,5-hexadiene means that there is an initial dilemma of likely conformation at rest.

Several starting conformations were thus taken and their geometries optimised to find the lowest energy conformer.

Initially, 1,5-hexadiene was taken, ensuring a roughly "'''anti'''" linkage in the centre of the molecule, so that the central C atoms '''3''' and '''4''' are approximately '''antiperiplanar''' to one another. This molecule was cleaned in the gaussview builder interface and then taken and optimised using the '''Hartree-Fock''' method with basis set '''3-21G''' through Gaussian.

'''b)''' similarly, a synclinal, or "gauche" version of 1,5-hexadiene was taken and optimised at the HF 3-21G level.

The energies of both conformers and also their symmetry were noted.

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|'''"''Anti''" 1,5-Hexadiene'''||'''"''Gauche''" 1,5-Hexadiene'''
|-align="center"
|[[Image:Bo Anti.jpg|420px]]||[[Image:Bo Gauche.jpg|400px]]
|}

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|'''Confomer'''||'''''Anti''-1,5-Hexadiene'''||'''''Gauche''-1,5-Hexadiene'''
|-align="center"
|'''Total Energy'''(a.u.)||-231.69254||-231.69266
|-align="center"
|'''Point Group'''||ci||None
|}

From a purely steric-orientated argument, the ''anti'' configuration should have a lower energy than the ''gauche'', as there is less steric hindrance from the two ends of the carbon chain 180 degrees from one another than next to each other in the gauche conformation.

As it so happens, the energy for the ''anti'' conformer is reportedly lower than that of the ''gauche'' conformer, but by a minute amount of roughly a ten-thousandth of a hartree ('''1 hartree = 4.3597482E-18 J''').

Even so, it was appreciated that the central C-C-C-C linkage of 1,5-hexadiene lends itself to quite a high degree of rotational freedom, and thus, further examples of conformational isomerism. This is reinforced by the fact of the two remarkably different conformers having remarkably similar energies, and points to the fact that 1,5-hexadiene experiences conformational ambiguity often<ref>Cope Rearrangement of 1,5-Hexadiene: Full Geometry Optimizations Using Analytic MR-CISD and MR-AQCC Gradient Methods, doi:10.1021/jp0259014</ref>.

As such, 2 other conformers were taken and optimised:-

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|colspan=4|'''<big>Different Conformers of 1,5-Hexadiene</big>'''
|-align="center"
|(''g120,a,g120'')-'''Anti1'''||(''g120,a,g-120'')-'''Anti2'''||(''g120,g-60,g120'')-'''Gauche1'''||(''g120,g60,g-120'')-'''Gauche3'''
|-align="center"
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Bo Anti1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Bo Anti2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Gauche 1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Gauche 3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-align="center"
|colspan=4|'''Energy'''(a.u.)
|-align="center"
| -231.69260213||-231.69253506||-231.69166702||-231.69266120
|-align="center"
|colspan=4|'''Point Group'''
|-align="center"
|C2||Ci||C2||C1
|}

The energies here from the HF 3-21G are similarly close, but contrary to what was expected, it was found that ''gauche3'' was actually the lowest in energy.

Beneath this, the ''anti'' conformers are found to be lower in energy than the other ''gauche'' conformer. This may be rationalised as the minimalisation of steric hindrance lowering the energy of the overall molecule, as the antiperiplanar groups maintain the least steric strain.

However, the '''''cis'''''-''gauche'' conformer is lowest in energy due to stereoelectronic arguments.

The '''π'''-bonds of the C=C bonds are lying criss-crossed over each other in this conformer. This promotes effectual π-molecular orbital overlap and hence stabilises the molecule.

Next, the anti2 conformer with ci symmetry and the gauche1 conformer was taken and further optimised under the DFT B3LYP method with a more sophisticated basis set, 6-31G(d).

The resulting optimised molecule was then compared with the earlier attempt in terms of energy.

A further frequency analysis was conducted in order to check that the optimisation has proceeded to a minimum, being the case when all frequencies were real and positive.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=5|'''Comparison of the HF 3-21G and DFT B3LYP 6-31G(d) Optimisations'''
|-align="center"
|'''Method'''||colspan=2|'''HF 3-21G'''||colspan=2|'''DFT B3LYP 6-31G(d)'''
|-align="center"
|'''Conformer'''||'''anti2'''||'''gauche1'''||'''anti2'''||'''gauche1'''
|-align="center"
|'''Total Energy'''(a.u.)||-231.69253506||-231.69166702||-231.61170616||-231.61068821
|-align="center"
|1-2 C=C Bond Length (A)||1.3162||1.3156||1.3335||1.3331
|-align="center"
|2-3 C-C Bond Length (A)||1.5091||1.5041||1.5083||1.5044
|-align="center"
|3-4 C-C Bond Length (A)||1.5527||1.5481||1.5508||1.5482
|-align="center"
|4-5 C-C Bond Length (A)||1.5091||1.5041||1.5083||1.5044
|-align="center"
|5-6 C=C Bond Length (A)||1.3163||1.3335||1.3157||1.3331
|-align="center"
|1-2-3-4 Dihedral (o)||114.66||118.49||123.93||124.24
|-align="center"
|2-3-4-5 Dihedral (o)||180.00||180.00||-64.18||-65.01
|-align="center"
|3-4-5-6 Dihedral (o)||-114.66||-118.49||123.93||124.24
|}

As can be seen here, the energies returned differ in a noticeable manner. However, comparison of energies computed under different basis sets is unbalanced.

As such, comparisons of the geometry are notably more reliable in discerning the computed differences.

The bond lengths differ by negligible amounts. However, it can be seen that the various dihedral angles have been "tightened" more to their ideal figures (closer to 120o for the anticlinal dihedrals.

As such, using a more sophisticated basis set ensures a more accurate approximation to reality.

The log file yielded the following thermochemical data:-

'''1)Sum of electronic and zero-point Energies= -234.469195'''

'''2)Sum of electronic and thermal Energies= -234.461847'''

'''3)Sum of electronic and thermal Enthalpies= -234.460903'''

'''4)Sum of electronic and thermal Free Energies= -234.500782'''

Theses values concern the 1)Potential Energy at 0K that includes the zero-point vibrational energy term 2)Potential Energy at 298.15K and 1atm and adds contributions from translational, rotational and vibrational energy 3)Energy including a correction for RT(H=E+RT) for dissociation reactions and 4)The Entropic contribution.

==''Exercise 2'' - Optimising the "Chair" and "Boat" Transition Structures==

The Cope Rearrangement has been found to proceed via one of two possible transition states with differing conformation, as mentioned in the introduction:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=2|'''<big>Cope Rearrangement Transition State Conformations</big>'''
|-align="center"
|''Chair''||''Boat''
|-align="center"
|[[Image:pic2a.jpg]]||[[Image:pic2b.jpg]]
|}

When the rearrangment is thought as one that is concerted and proceeds with synchronicity of bond breaking and formation, these two transition states come to mind.

There is indeed a third type of transition state for this reaction, dubbed the '''''Dewar''''' transition state, that reportedly proceeds via a diradical tight-chair intermediate species that was discovered by Dupuis et al.<ref>The Cope Rearrangement Revisited, DOI:10.1021/ja00026a007</ref>, and is thus NOT concerted.

===The Chair Transition State===

The chair transition state was initally synthesised using two "resonant" '''allyl''' ('''C3H5''') fragments in gaussview, which were optimised under the 3-21G basis set, and subsequently placed in appropriate positions relative to one another to be optimised into the transition state.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=2|'''Allyl Fragment('''C3H5''')'''
|-align="center"
|colspan=2|[[Image:Allyl Frag.jpg|275px]]
|-align="center"
|'''sp3''' Centre Bond Angle (o)||109.5
|-align="center"
|'''Allyl''' C-C Bond Angle (o)||124.3
|-align="center"
|'''C-C''' Bond Length (A)||1.54
|-align="center"
|'''C=C''' Bond Length (A)||1.34
|-align="center"
|'''Allyl C-C''' Bond Length (A)||1.39
|}

The optimised allyl C-C bond angle is greater than the normal sp3 bond angle of '''109.5o''' due to resonance.

More importantly, the allyl C-C bond length is between that of a single bond and a double bond.

This optimised allyl fragment was then taken, duplicated and placed, facing opposite directions, with roughly a 2.2A distance between the terminal carbons of each fragment to mimic the shape of the chair transition state.

Thus, a frequency + optimisation was run that directed gaussian to reach a '''TS(Berny)''', with the elucidation of force constants.

The optimisation was then repeated but this time using the '''"frozen" coordinate method''', effectively fixing the two pairs of ends at a distance of 2.2A.

Clearly, this method requires that the general structure of the transition state in question be known. However, by fixing the positions in such a way, a more accurate optimisation can be done that avoids divergence.

Finally, the optimisation was performed a third time, using a normal guess '''Hessian''' method, which involves calculation of the force constant matrix.

After the three optimisations were complete, the following data were obtained:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=3|'''<big>Optimisation of Chair Transition State</big>'''
|-align="center"
|colspan=3|'''Structure'''
|-align="center"
|'''TS(Berny)'''||'''Frozen Coordinate'''||'''Hessian'''
|-align="center"
||<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Berny.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Frozen.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>red</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Hessian.mol</uploadedFileContents>
</jmolApplet></jmol>
|-align="center"
|colspan=3|'''Energies''' (a.u.)
|-align="center"
| -231.61932||-231.61932||-231.69167
|}

The structures returned from the normal force-constant calculation optimisation and the frozen coordinate method are indistinguishable from first glance, and their energies are virtually identical. However, the structure obtained from the Hessian is different, and so is its energy value.

'''FREQUENCY ANALYSIS'''

Vibration-wise, there was one negative, and thus "imaginary" frequency found with a wavenumber of '''-817.91cm-1'''.

When the stretch was visualised, the following was observed:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Imaginary Stretch at -817.91cm-1'''
|-
|[[Image:Imag Stretch.jpg|300px]]
|}

The appearance of this imaginary frequency resembles the motion that would ensue under the Cope Rearrangment - the action of bond breaking of the C-C σ bond in the middle of the 1,5-hexadiene with the subsequent formation of a new σ bond on across the terminals on the other side.

The fact that there is only '''one''' imaginary frequency value points to the notion that the cope rearrangment through this transition state is ''concerted'', and thus occurs in one ''synchronous'' step.

The number of imaginary frequencies produced from a frequency analysis for the midpoint of a reaction points towards the ''synchronicity'' of the reaction<ref>Cope Rearrangement Revisited, DOI:10.1021/ja00026a007</ref>.

===The Boat Transition State===

For the boat transition state, a different method was utilised, namely the '''QST2 method''', whereby reactant and product are defined, and the transition state is converged from these.

This required the previously optimised '''1,5-hexadiene''' molecule to be taken and numbered along its carbon skeleton.

The molecule was then duplicated and juxtaposed to the intial "reactant" molecule and renumbered in such a way to reflect the reacted, rearragement "product" as shown below:-

{|style="margin:1em auto 1em auto" border="1"
|-
|[[Image:Boat Method.JPG|400px]]
|}

The first instance of optimisation failed to produce an output, and thus the molecule was changed in the following way:-

The dihedral angle of both reactant and product were changed to 0 degrees, such that the two carbon groups either side of C3 and C4 are eclipsed, and the C2-C3-C4 and C3-C4-C5 bond angles were both changed to 100 degrees.

The modified structures were once again optimised under the QST2 method to yield the following structure:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''''QST2'' Optimised Boat Transition State'''
|-
|[[Image:FINAL Boat.jpg|400px]]
|}

'''DISCUSSION'''

Here, it is illustrated that the QST2 method is automated, and is thus very efficient, but is only effective if the shape provided of the reactants and products cloely mirror that of the transition state, otherwise convergence is difficult.

The '''QST3''' method was also attempted <ref>QST Calculation, http://hdl.handle.net/10042/to-2902</ref>, whereby an addition guess of the transition state geometry was implemented. If the guess of the transition state is sound, then the QST3 is by far more accurate than the QST2 method, and does not rely so much on the shape of the reactants and products.

initally, it would seem that in order for the cope rearrangement to occur, to two carbon terminals would be required to be close to one another. This is not the case with the "''anti''" conformers of the molecule, as the terminals are situated as far apart as they may possibly be, '''antiperplanar''' to one another.

Once free rotation moves the carbon groups closer together, into the gauche configuration, where the terminals are more poised to react, cope rearrangement would proceed.

The relative orientations of the vinyl groups would ultimately dictate which transition state would be traversed. For instance, if the vectors of the vinyl groups cross, such as the case for the gauche1 conformer, then as the terminals align into a bonding orientaton, the ''chair'' conformation would be adopted by the transition state.

Conversely, if the two vinyl groups lie in a pseudo-parallel fashion, where their vectors do NOT cross, such as is the case for gauche3, then the transition state would be of ''boat'' topography.

'''IRC'''

The Intrinsic Reaction Coordinate looks more cloesly at the potential surface of a reaction and works to pinpoint the highest point in a local area of the potential surface by creeping up the steepest slopes in the immediate area.

This calculation was attempted with 50 points, and did not converge

'''Energy Calculation'''

The two transition states were optimised a final time under DFT B3LYP 6-31G(d) conditions.

The geometries that were outputted were very similar to the previous examples, however

==The Diels-Alder Cycloaddition==

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Diels-Alder Reaction of Butadiene and Ethylene'''
|-align="center"
|[[Image:Mb_da3.jpg|400px]]
|}

The Diels-Alder Reaction is an example of a pericyclic cycloaddition, that usually involves the reaction between an electron rich diene and an electron poor dienophile.

The process is described as a [4s + 2s], 6 electron process that forms 2 new σ bonds.

When viewing the molecular orbital picture of butadiene and ethylene, the molecular orbitals are classified as either '''''s'''''(''symmetric'') or '''''a'''''(''antisymmetric'') in relation to the bisecting plane as shown below.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Bisecting Plane for Diels-alder'''
|-
|[[Image:Mb_da2.jpg|200px]]
|}

===Investigation of Molecular Orbital Symmetries of Butadiene and Ethene===

Butadiene and Ethylene were optimised and their molecular orbitals visualised:-

'''''Butadiene'''''

The HOMO and LUMO are considered with reference to the bisecting plane to determine their symmetry:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=4|'''HOMO and LUMO of Butadiene'''
|-align="center"
|width=250px|'''Molecular Orbital'''||'''2D Representation'''||'''3D Representation'''||width=250px|'''s/a'''
|-align="center"
|'''HOMO'''||[[Image:BDE HOMO.png|200px]]||[[Image:3D BDE HOMO.jpg|300px]]||a
|-align="center"
|'''LUMO'''||[[Image:BDE LUMO.png|200px]]||[[Image:3D BDE LUMO.jpg|300px]]||s
|}

'''''Ethene'''''

Similarly, the HOMO and LUMO of ethylene were characterised:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=4|'''HOMO and LUMO of Ethene'''
|-align="center"
|width=250px|'''Molecular Orbital'''||'''2D Representation'''||'''3D Representation'''||width=250px|'''s/a'''
|-align="center"
|'''HOMO'''||[[Image:ET HOMO.jpg|200px]]||[[Image:3D ET HOMO.jpg|300px]]||s
|-align="center"
|'''LUMO'''||[[Image:ET LUMO.jpg|200px]]||[[Image:3D ET LUMO.jpg|300px]]||a
|}

===Optimisation of Diels-Alder Transition State===

The '''Transition State''' whereby the concerted cycloaddition of butadiene and ethylene was modelled to a guess initially, before being optimised under HF 3-21G basis set conditions to a TS(Berny).

The initial guess was modelled from the bicyclo-system of cyclohexene and subsequently removing one of the -CH2CH2- arches to arrive at an "''envelope''"-shaped 6-membered ring.

The new σ bonds were stripped before the optimisation.

When this optimisation was complete, the structure obtained was slightly distorted, and so another optimisation using the frozen coordinate method.

When this was run, the same twisted variant of the starting geometry was received, but with the relative positions of the terminal methyl groups on the butadiene to the ethyl fragment reversed.

{|style="margin=1em auto 1em auto" border="1"
|-align="center"
|colspan=3|'''Comparisons of the Optimisations of transition state'''
|-align="center"
|'''Method'''||'''Ts(Berny)'''||'''Frozen-Coordinate'''
|-align="center"
|'''Structure'''||[[Image:CHEX BERNY.jpg|300px]]||[[Image:CHEX FROZEN.jpg|300px]]
|-align="center"
|'''Energy'''(a.u.)||-231.54077||-231.53787
|-align="center"
|'''Average new C-C σ bond length''' (A)||1.472645||1.526225
|-align="center"
|'''Literature Length of a C-C σ bond''' (A)||colspan=2|1.54
|}

By freezing the terminals, the energy of the optimised structure is slightly lower, and the length of the newly-forming σ bonds are closer to what they are experimentally measured as.

The twisting of the original guess geometry is to be expected as product, cyclohexadiene, wants to be in the chair, its most stable conformation.

As such, the geometry of the sp2 constrained carbon frameworks must slowly twist into the alternating configuration of the chair, and thus, one carbon is pushed up and the otehr pushed down.

Rep:Mod:parkbom

2009-11-13T14:32:00Z

Byt07:

=Module 3 - The Computation of the Transition State=

==Introduction==

The '''Transition State''' is an energetic maximum within a given reaction coordinate and represents the point at which bond breaking and bond forming is occurring simultaneously. This is represented by dotted lines in formal notation and depicts the transient nature of the bond(s) in question in the transition state.

The analysis of potential energy surfaces are effective in finding transition states; the transition states can be described graphically as "dams" that represent energy metastability, that the reactants must "cross" in order to form the products.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Potential Energy Surface Reaction Coordinate'''<ref>http://www.ch.ic.ac.uk/motm/porphyrins/TSdiscovery.html</ref>||'''Cope Rearrangement'''<ref>http://www.ch.ic.ac.uk/local/organic/pericyclic/</ref>
|-
|[[Image:PES.gif]]||[[Image:Cope Arr.gif|350px]]
|}

From the point of view of computational chemistry, the methods of molecular mechanical structure determination that employed imaginary force fields are ineffective in the calculations concerning the transition state in larger molecules, as they do not define the dynamics of bond making and bond forming that are apparent in the transient nature of the transition state.

As such, in combination with molecular orbital based calculations, the following of the potential energy surface of a molecule can be used to locate and analyse the transition state.

This computational exercise concerns the rearrangement of ''1,5-Hexadiene'' in the '''Cope Rearrangment'''(above).

The Cope Rearrangement proceeds via a [3,3]-Sigmatropic shift involving the motion of 6 electrons and is thermally-driven.

In terms of transition states, the cope rearrangement can reportedly proceed via two checkpoints, the '''boat''' or the '''chair''':-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Chair'''||'''Boat'''
|-
|[[Image:pic2a.jpg]]||[[Image:pic2b.jpg]]
|}

The purpose of this exercise is to ascertain which reaction pathway is preferred by computation of each transition state in the elucidation of geometry, repective energies and other thermochemical data.

==''Exercise 1'' - Optimising the Reactants and Products==

In this section, the "''reactant''" '''1,5-hexadiene''' and the "''product''" '''1,5-hexadiene''' are modelled in correspondence to the cope rearrangement.

The flexible nature of 1,5-hexadiene means that there is an initial dilemma of likely conformation at rest.

Several starting conformations were thus taken and their geometries optimised to find the lowest energy conformer.

Initially, 1,5-hexadiene was taken, ensuring a roughly "'''anti'''" linkage in the centre of the molecule, so that the central C atoms '''3''' and '''4''' are approximately '''antiperiplanar''' to one another. This molecule was cleaned in the gaussview builder interface and then taken and optimised using the '''Hartree-Fock''' method with basis set '''3-21G''' through Gaussian.

'''b)''' similarly, a synclinal, or "gauche" version of 1,5-hexadiene was taken and optimised at the HF 3-21G level.

The energies of both conformers and also their symmetry were noted.

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|'''"''Anti''" 1,5-Hexadiene'''||'''"''Gauche''" 1,5-Hexadiene'''
|-align="center"
|[[Image:Bo Anti.jpg|420px]]||[[Image:Bo Gauche.jpg|400px]]
|}

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|'''Confomer'''||'''''Anti''-1,5-Hexadiene'''||'''''Gauche''-1,5-Hexadiene'''
|-align="center"
|'''Total Energy'''(a.u.)||-231.69254||-231.69266
|-align="center"
|'''Point Group'''||ci||None
|}

From a purely steric-orientated argument, the ''anti'' configuration should have a lower energy than the ''gauche'', as there is less steric hindrance from the two ends of the carbon chain 180 degrees from one another than next to each other in the gauche conformation.

As it so happens, the energy for the ''anti'' conformer is reportedly lower than that of the ''gauche'' conformer, but by a minute amount of roughly a ten-thousandth of a hartree ('''1 hartree = 4.3597482E-18 J''').

Even so, it was appreciated that the central C-C-C-C linkage of 1,5-hexadiene lends itself to quite a high degree of rotational freedom, and thus, further examples of conformational isomerism. This is reinforced by the fact of the two remarkably different conformers having remarkably similar energies, and points to the fact that 1,5-hexadiene experiences conformational ambiguity often<ref>Cope Rearrangement of 1,5-Hexadiene: Full Geometry Optimizations Using Analytic MR-CISD and MR-AQCC Gradient Methods, doi:10.1021/jp0259014</ref>.

As such, 2 other conformers were taken and optimised:-

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|colspan=4|'''<big>Different Conformers of 1,5-Hexadiene</big>'''
|-align="center"
|(''g120,a,g120'')-'''Anti1'''||(''g120,a,g-120'')-'''Anti2'''||(''g120,g-60,g120'')-'''Gauche1'''||(''g120,g60,g-120'')-'''Gauche3'''
|-align="center"
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Bo Anti1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Bo Anti2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Gauche 1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Gauche 3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-align="center"
|colspan=4|'''Energy'''(a.u.)
|-align="center"
| -231.69260213||-231.69253506||-231.69166702||-231.69266120
|-align="center"
|colspan=4|'''Point Group'''
|-align="center"
|C2||Ci||C2||C1
|}

The energies here from the HF 3-21G are similarly close, but contrary to what was expected, it was found that ''gauche3'' was actually the lowest in energy.

Beneath this, the ''anti'' conformers are found to be lower in energy than the other ''gauche'' conformer. This may be rationalised as the minimalisation of steric hindrance lowering the energy of the overall molecule, as the antiperiplanar groups maintain the least steric strain.

However, the '''''cis'''''-''gauche'' conformer is lowest in energy due to stereoelectronic arguments.

The '''π'''-bonds of the C=C bonds are lying criss-crossed over each other in this conformer. This promotes effectual π-molecular orbital overlap and hence stabilises the molecule.

Next, the anti2 conformer with ci symmetry and the gauche1 conformer was taken and further optimised under the DFT B3LYP method with a more sophisticated basis set, 6-31G(d).

The resulting optimised molecule was then compared with the earlier attempt in terms of energy.

A further frequency analysis was conducted in order to check that the optimisation has proceeded to a minimum, being the case when all frequencies were real and positive.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=5|'''Comparison of the HF 3-21G and DFT B3LYP 6-31G(d) Optimisations'''
|-align="center"
|'''Method'''||colspan=2|'''HF 3-21G'''||colspan=2|'''DFT B3LYP 6-31G(d)'''
|-align="center"
|'''Conformer'''||'''anti2'''||'''gauche1'''||'''anti2'''||'''gauche1'''
|-align="center"
|'''Total Energy'''(a.u.)||-231.69253506||-231.69166702||-231.61170616||-231.61068821
|-align="center"
|1-2 C=C Bond Length (A)||1.3162||1.3156||1.3335||1.3331
|-align="center"
|2-3 C-C Bond Length (A)||1.5091||1.5041||1.5083||1.5044
|-align="center"
|3-4 C-C Bond Length (A)||1.5527||1.5481||1.5508||1.5482
|-align="center"
|4-5 C-C Bond Length (A)||1.5091||1.5041||1.5083||1.5044
|-align="center"
|5-6 C=C Bond Length (A)||1.3163||1.3335||1.3157||1.3331
|-align="center"
|1-2-3-4 Dihedral (o)||114.66||118.49||123.93||124.24
|-align="center"
|2-3-4-5 Dihedral (o)||180.00||180.00||-64.18||-65.01
|-align="center"
|3-4-5-6 Dihedral (o)||-114.66||-118.49||123.93||124.24
|}

As can be seen here, the energies returned differ in a noticeable manner. However, comparison of energies computed under different basis sets is unbalanced.

As such, comparisons of the geometry are notably more reliable in discerning the computed differences.

The bond lengths differ by negligible amounts. However, it can be seen that the various dihedral angles have been "tightened" more to their ideal figures (closer to 120o for the anticlinal dihedrals.

As such, using a more sophisticated basis set ensures a more accurate approximation to reality.

The log file yielded the following thermochemical data:-

'''1)Sum of electronic and zero-point Energies= -234.469195'''

'''2)Sum of electronic and thermal Energies= -234.461847'''

'''3)Sum of electronic and thermal Enthalpies= -234.460903'''

'''4)Sum of electronic and thermal Free Energies= -234.500782'''

Theses values concern the 1)Potential Energy at 0K that includes the zero-point vibrational energy term 2)Potential Energy at 298.15K and 1atm and adds contributions from translational, rotational and vibrational energy 3)Energy including a correction for RT(H=E+RT) for dissociation reactions and 4)The Entropic contribution.

==''Exercise 2'' - Optimising the "Chair" and "Boat" Transition Structures==

The Cope Rearrangement has been found to proceed via one of two possible transition states with differing conformation, as mentioned in the introduction:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=2|'''<big>Cope Rearrangement Transition State Conformations</big>'''
|-align="center"
|''Chair''||''Boat''
|-align="center"
|[[Image:pic2a.jpg]]||[[Image:pic2b.jpg]]
|}

When the rearrangment is thought as one that is concerted and proceeds with synchronicity of bond breaking and formation, these two transition states come to mind.

There is indeed a third type of transition state for this reaction, dubbed the '''''Dewar''''' transition state, that reportedly proceeds via a diradical tight-chair intermediate species that was discovered by Dupuis et al.<ref>The Cope Rearrangement Revisited, DOI:10.1021/ja00026a007</ref>, and is thus NOT concerted.

===The Chair Transition State===

The chair transition state was initally synthesised using two "resonant" '''allyl''' ('''C3H5''') fragments in gaussview, which were optimised under the 3-21G basis set, and subsequently placed in appropriate positions relative to one another to be optimised into the transition state.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=2|'''Allyl Fragment('''C3H5''')'''
|-align="center"
|colspan=2|[[Image:Allyl Frag.jpg|275px]]
|-align="center"
|'''sp3''' Centre Bond Angle (o)||109.5
|-align="center"
|'''Allyl''' C-C Bond Angle (o)||124.3
|-align="center"
|'''C-C''' Bond Length (A)||1.54
|-align="center"
|'''C=C''' Bond Length (A)||1.34
|-align="center"
|'''Allyl C-C''' Bond Length (A)||1.39
|}

The optimised allyl C-C bond angle is greater than the normal sp3 bond angle of '''109.5o''' due to resonance.

More importantly, the allyl C-C bond length is between that of a single bond and a double bond.

This optimised allyl fragment was then taken, duplicated and placed, facing opposite directions, with roughly a 2.2A distance between the terminal carbons of each fragment to mimic the shape of the chair transition state.

Thus, a frequency + optimisation was run that directed gaussian to reach a '''TS(Berny)''', with the elucidation of force constants.

The optimisation was then repeated but this time using the '''"frozen" coordinate method''', effectively fixing the two pairs of ends at a distance of 2.2A.

Clearly, this method requires that the general structure of the transition state in question be known. However, by fixing the positions in such a way, a more accurate optimisation can be done that avoids divergence.

Finally, the optimisation was performed a third time, using a normal guess '''Hessian''' method, which involves calculation of the force constant matrix.

After the three optimisations were complete, the following data were obtained:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=3|'''<big>Optimisation of Chair Transition State</big>'''
|-align="center"
|colspan=3|'''Structure'''
|-align="center"
|'''TS(Berny)'''||'''Frozen Coordinate'''||'''Hessian'''
|-align="center"
||<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Berny.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Frozen.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>red</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Hessian.mol</uploadedFileContents>
</jmolApplet></jmol>
|-align="center"
|colspan=3|'''Energies''' (a.u.)
|-align="center"
| -231.61932||-231.61932||-231.69167
|}

The structures returned from the normal force-constant calculation optimisation and the frozen coordinate method are indistinguishable from first glance, and their energies are virtually identical. However, the structure obtained from the Hessian is different, and so is its energy value.

'''FREQUENCY ANALYSIS'''

Vibration-wise, there was one negative, and thus "imaginary" frequency found with a wavenumber of '''-817.91cm-1'''.

When the stretch was visualised, the following was observed:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Imaginary Stretch at -817.91cm-1'''
|-
|[[Image:Imag Stretch.jpg|300px]]
|}

The appearance of this imaginary frequency resembles the motion that would ensue under the Cope Rearrangment - the action of bond breaking of the C-C σ bond in the middle of the 1,5-hexadiene with the subsequent formation of a new σ bond on across the terminals on the other side.

The fact that there is only '''one''' imaginary frequency value points to the notion that the cope rearrangment through this transition state is ''concerted'', and thus occurs in one ''synchronous'' step.

The number of imaginary frequencies produced from a frequency analysis for the midpoint of a reaction points towards the ''synchronicity'' of the reaction<ref>Cope Rearrangement Revisited, DOI:10.1021/ja00026a007</ref>.

===The Boat Transition State===

For the boat transition state, a different method was utilised, namely the '''QST2 method''', whereby reactant and product are defined, and the transition state is converged from these.

This required the previously optimised '''1,5-hexadiene''' molecule to be taken and numbered along its carbon skeleton.

The molecule was then duplicated and juxtaposed to the intial "reactant" molecule and renumbered in such a way to reflect the reacted, rearragement "product" as shown below:-

{|style="margin:1em auto 1em auto" border="1"
|-
|[[Image:Boat Method.JPG|400px]]
|}

The first instance of optimisation failed to produce an output, and thus the molecule was changed in the following way:-

The dihedral angle of both reactant and product were changed to 0 degrees, such that the two carbon groups either side of C3 and C4 are eclipsed, and the C2-C3-C4 and C3-C4-C5 bond angles were both changed to 100 degrees.

The modified structures were once again optimised under the QST2 method to yield the following structure:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''''QST2'' Optimised Boat Transition State'''
|-
|[[Image:FINAL Boat.jpg|400px]]
|}

'''DISCUSSION'''

Here, it is illustrated that the QST2 method is automated, and is thus very efficient, but is only effective if the shape provided of the reactants and products cloely mirror that of the transition state, otherwise convergence is difficult.

The '''QST3''' method was also attempted <ref>QST Calculation, http://hdl.handle.net/10042/to-2902</ref>, whereby an addition guess of the transition state geometry was implemented. If the guess of the transition state is sound, then the QST3 is by far more accurate than the QST2 method, and does not rely so much on the shape of the reactants and products.

initally, it would seem that in order for the cope rearrangement to occur, to two carbon terminals would be required to be close to one another. This is not the case with the "''anti''" conformers of the molecule, as the terminals are situated as far apart as they may possibly be, '''antiperplanar''' to one another.

Once free rotation moves the carbon groups closer together, into the gauche configuration, where the terminals are more poised to react, cope rearrangement would proceed.

The relative orientations of the vinyl groups would ultimately dictate which transition state would be traversed. For instance, if the vectors of the vinyl groups cross, such as the case for the gauche1 conformer, then as the terminals align into a bonding orientaton, the ''chair'' conformation would be adopted by the transition state.

Conversely, if the two vinyl groups lie in a pseudo-parallel fashion, where their vectors do NOT cross, such as is the case for gauche3, then the transition state would be of ''boat'' topography.

'''IRC'''

The Intrinsic Reaction Coordinate looks more cloesly at the potential surface of a reaction and works to pinpoint the highest point in a local area of the potential surface by creeping up the steepest slopes in the immediate area.

This calculation was attempted with 50 points, and did not converge

'''Energy Calculation'''

The two transition states were optimised a final time under DFT B3LYP 6-31G(d) conditions.

The geometries that were outputted were very similar to the previous examples, however

==The Diels-Alder Cycloaddition==

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Diels-Alder Reaction of Butadiene and Ethylene'''
|-align="center"
|[[Image:Mb_da3.jpg|400px]]
|}

The Diels-Alder Reaction is an example of a pericyclic cycloaddition, that usually involves the reaction between an electron rich diene and an electron poor dienophile.

The process is described as a [4s + 2s], 6 electron process that forms 2 new σ bonds.

When viewing the molecular orbital picture of butadiene and ethylene, the molecular orbitals are classified as either '''''s'''''(''symmetric'') or '''''a'''''(''antisymmetric'') in relation to the bisecting plane as shown below.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Bisecting Plane for Diels-alder'''
|-
|[[Image:Mb_da2.jpg|200px]]
|}

===Investigation of Molecular Orbital Symmetries of Butadiene and Ethene===

Butadiene and Ethylene were optimised and their molecular orbitals visualised:-

'''''Butadiene'''''

The HOMO and LUMO are considered with reference to the bisecting plane to determine their symmetry:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=4|'''HOMO and LUMO of Butadiene'''
|-align="center"
|width=250px|'''Molecular Orbital'''||'''2D Representation'''||'''3D Representation'''||width=250px|'''s/a'''
|-align="center"
|'''HOMO'''||[[Image:BDE HOMO.png|200px]]||[[Image:3D BDE HOMO.jpg|300px]]||a
|-align="center"
|'''LUMO'''||[[Image:BDE LUMO.png|200px]]||[[Image:3D BDE LUMO.jpg|300px]]||s
|}

'''''Ethene'''''

Similarly, the HOMO and LUMO of ethylene were characterised:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=4|'''HOMO and LUMO of Ethene'''
|-align="center"
|width=250px|'''Molecular Orbital'''||'''2D Representation'''||'''3D Representation'''||width=250px|'''s/a'''
|-align="center"
|'''HOMO'''||[[Image:ET HOMO.jpg|200px]]||[[Image:3D ET HOMO.jpg|300px]]||s
|-align="center"
|'''LUMO'''||[[Image:ET LUMO.jpg|200px]]||[[Image:3D ET LUMO.jpg|300px]]||a
|}

===Optimisation of Diels-Alder Transition State===

The '''Transition State''' whereby the concerted cycloaddition of butadiene and ethylene was modelled to a guess initially, before being optimised under HF 3-21G basis set conditions to a TS(Berny).

The initial guess was modelled from the bicyclo-system of cyclohexene and subsequently removing one of the -CH2CH2- arches to arrive at an "''envelope''"-shaped 6-membered ring.

The new σ bonds were stripped before the optimisation.

When this optimisation was complete, the structure obtained was slightly distorted, and so another optimisation using the frozen coordinate method.

When this was run, the same twisted variant of the starting geometry was received, but with the relative positions of the terminal methyl groups on the butadiene to the ethyl fragment reversed.

{|style="margin=1em auto 1em auto" border="1"
|-align="center"
|colspan=3|'''Comparisons of the Optimisations of transition state'''
|-align="center"
|'''Method'''||'''Ts(Berny)'''||'''Frozen-Coordinate'''
|-align="center"
|'''Structure'''||[[Image:CHEX BERNY.jpg|300px]]||[[Image:CHEX FROZEN.jpg]]
|-align="center"
|'''Energy'''(a.u.)||-231.54077||-231.53787
|-align="center"
|'''Average new C-C σ bond length''' (A)||1.472645||1.526225
|-align="center"
|'''Literature Length of a C-C σ bond''' (A)||colspan=2|1.54
|}

File:CHEX FROZEN.jpg

2009-11-13T14:24:01Z

Byt07:

File:CHEX BERNY.jpg

2009-11-13T14:23:55Z

Byt07:

Rep:Mod:parkbom

2009-11-13T13:53:39Z

Byt07:

=Module 3 - The Computation of the Transition State=

==Introduction==

The '''Transition State''' is an energetic maximum within a given reaction coordinate and represents the point at which bond breaking and bond forming is occurring simultaneously. This is represented by dotted lines in formal notation and depicts the transient nature of the bond(s) in question in the transition state.

The analysis of potential energy surfaces are effective in finding transition states; the transition states can be described graphically as "dams" that represent energy metastability, that the reactants must "cross" in order to form the products.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Potential Energy Surface Reaction Coordinate'''<ref>http://www.ch.ic.ac.uk/motm/porphyrins/TSdiscovery.html</ref>||'''Cope Rearrangement'''<ref>http://www.ch.ic.ac.uk/local/organic/pericyclic/</ref>
|-
|[[Image:PES.gif]]||[[Image:Cope Arr.gif|350px]]
|}

From the point of view of computational chemistry, the methods of molecular mechanical structure determination that employed imaginary force fields are ineffective in the calculations concerning the transition state in larger molecules, as they do not define the dynamics of bond making and bond forming that are apparent in the transient nature of the transition state.

As such, in combination with molecular orbital based calculations, the following of the potential energy surface of a molecule can be used to locate and analyse the transition state.

This computational exercise concerns the rearrangement of ''1,5-Hexadiene'' in the '''Cope Rearrangment'''(above).

The Cope Rearrangement proceeds via a [3,3]-Sigmatropic shift involving the motion of 6 electrons and is thermally-driven.

In terms of transition states, the cope rearrangement can reportedly proceed via two checkpoints, the '''boat''' or the '''chair''':-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Chair'''||'''Boat'''
|-
|[[Image:pic2a.jpg]]||[[Image:pic2b.jpg]]
|}

The purpose of this exercise is to ascertain which reaction pathway is preferred by computation of each transition state in the elucidation of geometry, repective energies and other thermochemical data.

==''Exercise 1'' - Optimising the Reactants and Products==

In this section, the "''reactant''" '''1,5-hexadiene''' and the "''product''" '''1,5-hexadiene''' are modelled in correspondence to the cope rearrangement.

The flexible nature of 1,5-hexadiene means that there is an initial dilemma of likely conformation at rest.

Several starting conformations were thus taken and their geometries optimised to find the lowest energy conformer.

Initially, 1,5-hexadiene was taken, ensuring a roughly "'''anti'''" linkage in the centre of the molecule, so that the central C atoms '''3''' and '''4''' are approximately '''antiperiplanar''' to one another. This molecule was cleaned in the gaussview builder interface and then taken and optimised using the '''Hartree-Fock''' method with basis set '''3-21G''' through Gaussian.

'''b)''' similarly, a synclinal, or "gauche" version of 1,5-hexadiene was taken and optimised at the HF 3-21G level.

The energies of both conformers and also their symmetry were noted.

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|'''"''Anti''" 1,5-Hexadiene'''||'''"''Gauche''" 1,5-Hexadiene'''
|-align="center"
|[[Image:Bo Anti.jpg|420px]]||[[Image:Bo Gauche.jpg|400px]]
|}

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|'''Confomer'''||'''''Anti''-1,5-Hexadiene'''||'''''Gauche''-1,5-Hexadiene'''
|-align="center"
|'''Total Energy'''(a.u.)||-231.69254||-231.69266
|-align="center"
|'''Point Group'''||ci||None
|}

From a purely steric-orientated argument, the ''anti'' configuration should have a lower energy than the ''gauche'', as there is less steric hindrance from the two ends of the carbon chain 180 degrees from one another than next to each other in the gauche conformation.

As it so happens, the energy for the ''anti'' conformer is reportedly lower than that of the ''gauche'' conformer, but by a minute amount of roughly a ten-thousandth of a hartree ('''1 hartree = 4.3597482E-18 J''').

Even so, it was appreciated that the central C-C-C-C linkage of 1,5-hexadiene lends itself to quite a high degree of rotational freedom, and thus, further examples of conformational isomerism. This is reinforced by the fact of the two remarkably different conformers having remarkably similar energies, and points to the fact that 1,5-hexadiene experiences conformational ambiguity often<ref>Cope Rearrangement of 1,5-Hexadiene: Full Geometry Optimizations Using Analytic MR-CISD and MR-AQCC Gradient Methods, doi:10.1021/jp0259014</ref>.

As such, 2 other conformers were taken and optimised:-

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|colspan=4|'''<big>Different Conformers of 1,5-Hexadiene</big>'''
|-align="center"
|(''g120,a,g120'')-'''Anti1'''||(''g120,a,g-120'')-'''Anti2'''||(''g120,g-60,g120'')-'''Gauche1'''||(''g120,g60,g-120'')-'''Gauche3'''
|-align="center"
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Bo Anti1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Bo Anti2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Gauche 1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Gauche 3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-align="center"
|colspan=4|'''Energy'''(a.u.)
|-align="center"
| -231.69260213||-231.69253506||-231.69166702||-231.69266120
|-align="center"
|colspan=4|'''Point Group'''
|-align="center"
|C2||Ci||C2||C1
|}

The energies here from the HF 3-21G are similarly close, but contrary to what was expected, it was found that ''gauche3'' was actually the lowest in energy.

Beneath this, the ''anti'' conformers are found to be lower in energy than the other ''gauche'' conformer. This may be rationalised as the minimalisation of steric hindrance lowering the energy of the overall molecule, as the antiperiplanar groups maintain the least steric strain.

However, the '''''cis'''''-''gauche'' conformer is lowest in energy due to stereoelectronic arguments.

The '''π'''-bonds of the C=C bonds are lying criss-crossed over each other in this conformer. This promotes effectual π-molecular orbital overlap and hence stabilises the molecule.

Next, the anti2 conformer with ci symmetry and the gauche1 conformer was taken and further optimised under the DFT B3LYP method with a more sophisticated basis set, 6-31G(d).

The resulting optimised molecule was then compared with the earlier attempt in terms of energy.

A further frequency analysis was conducted in order to check that the optimisation has proceeded to a minimum, being the case when all frequencies were real and positive.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=5|'''Comparison of the HF 3-21G and DFT B3LYP 6-31G(d) Optimisations'''
|-align="center"
|'''Method'''||colspan=2|'''HF 3-21G'''||colspan=2|'''DFT B3LYP 6-31G(d)'''
|-align="center"
|'''Conformer'''||'''anti2'''||'''gauche1'''||'''anti2'''||'''gauche1'''
|-align="center"
|'''Total Energy'''(a.u.)||-231.69253506||-231.69166702||-231.61170616||-231.61068821
|-align="center"
|1-2 C=C Bond Length (A)||1.3162||1.3156||1.3335||1.3331
|-align="center"
|2-3 C-C Bond Length (A)||1.5091||1.5041||1.5083||1.5044
|-align="center"
|3-4 C-C Bond Length (A)||1.5527||1.5481||1.5508||1.5482
|-align="center"
|4-5 C-C Bond Length (A)||1.5091||1.5041||1.5083||1.5044
|-align="center"
|5-6 C=C Bond Length (A)||1.3163||1.3335||1.3157||1.3331
|-align="center"
|1-2-3-4 Dihedral (o)||114.66||118.49||123.93||124.24
|-align="center"
|2-3-4-5 Dihedral (o)||180.00||180.00||-64.18||-65.01
|-align="center"
|3-4-5-6 Dihedral (o)||-114.66||-118.49||123.93||124.24
|}

As can be seen here, the energies returned differ in a noticeable manner. However, comparison of energies computed under different basis sets is unbalanced.

As such, comparisons of the geometry are notably more reliable in discerning the computed differences.

The bond lengths differ by negligible amounts. However, it can be seen that the various dihedral angles have been "tightened" more to their ideal figures (closer to 120o for the anticlinal dihedrals.

As such, using a more sophisticated basis set ensures a more accurate approximation to reality.

The log file yielded the following thermochemical data:-

'''1)Sum of electronic and zero-point Energies= -234.469195'''

'''2)Sum of electronic and thermal Energies= -234.461847'''

'''3)Sum of electronic and thermal Enthalpies= -234.460903'''

'''4)Sum of electronic and thermal Free Energies= -234.500782'''

Theses values concern the 1)Potential Energy at 0K that includes the zero-point vibrational energy term 2)Potential Energy at 298.15K and 1atm and adds contributions from translational, rotational and vibrational energy 3)Energy including a correction for RT(H=E+RT) for dissociation reactions and 4)The Entropic contribution.

==''Exercise 2'' - Optimising the "Chair" and "Boat" Transition Structures==

The Cope Rearrangement has been found to proceed via one of two possible transition states with differing conformation, as mentioned in the introduction:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=2|'''<big>Cope Rearrangement Transition State Conformations</big>'''
|-align="center"
|''Chair''||''Boat''
|-align="center"
|[[Image:pic2a.jpg]]||[[Image:pic2b.jpg]]
|}

When the rearrangment is thought as one that is concerted and proceeds with synchronicity of bond breaking and formation, these two transition states come to mind.

There is indeed a third type of transition state for this reaction, dubbed the '''''Dewar''''' transition state, that reportedly proceeds via a diradical tight-chair intermediate species that was discovered by Dupuis et al.<ref>The Cope Rearrangement Revisited, DOI:10.1021/ja00026a007</ref>, and is thus NOT concerted.

===The Chair Transition State===

The chair transition state was initally synthesised using two "resonant" '''allyl''' ('''C3H5''') fragments in gaussview, which were optimised under the 3-21G basis set, and subsequently placed in appropriate positions relative to one another to be optimised into the transition state.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=2|'''Allyl Fragment('''C3H5''')'''
|-align="center"
|colspan=2|[[Image:Allyl Frag.jpg|275px]]
|-align="center"
|'''sp3''' Centre Bond Angle (o)||109.5
|-align="center"
|'''Allyl''' C-C Bond Angle (o)||124.3
|-align="center"
|'''C-C''' Bond Length (A)||1.54
|-align="center"
|'''C=C''' Bond Length (A)||1.34
|-align="center"
|'''Allyl C-C''' Bond Length (A)||1.39
|}

The optimised allyl C-C bond angle is greater than the normal sp3 bond angle of '''109.5o''' due to resonance.

More importantly, the allyl C-C bond length is between that of a single bond and a double bond.

This optimised allyl fragment was then taken, duplicated and placed, facing opposite directions, with roughly a 2.2A distance between the terminal carbons of each fragment to mimic the shape of the chair transition state.

Thus, a frequency + optimisation was run that directed gaussian to reach a '''TS(Berny)''', with the elucidation of force constants.

The optimisation was then repeated but this time using the '''"frozen" coordinate method''', effectively fixing the two pairs of ends at a distance of 2.2A.

Clearly, this method requires that the general structure of the transition state in question be known. However, by fixing the positions in such a way, a more accurate optimisation can be done that avoids divergence.

Finally, the optimisation was performed a third time, using a normal guess '''Hessian''' method, which involves calculation of the force constant matrix.

After the three optimisations were complete, the following data were obtained:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=3|'''<big>Optimisation of Chair Transition State</big>'''
|-align="center"
|colspan=3|'''Structure'''
|-align="center"
|'''TS(Berny)'''||'''Frozen Coordinate'''||'''Hessian'''
|-align="center"
||<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Berny.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Frozen.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>red</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Hessian.mol</uploadedFileContents>
</jmolApplet></jmol>
|-align="center"
|colspan=3|'''Energies''' (a.u.)
|-align="center"
| -231.61932||-231.61932||-231.69167
|}

The structures returned from the normal force-constant calculation optimisation and the frozen coordinate method are indistinguishable from first glance, and their energies are virtually identical. However, the structure obtained from the Hessian is different, and so is its energy value.

'''FREQUENCY ANALYSIS'''

Vibration-wise, there was one negative, and thus "imaginary" frequency found with a wavenumber of '''-817.91cm-1'''.

When the stretch was visualised, the following was observed:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Imaginary Stretch at -817.91cm-1'''
|-
|[[Image:Imag Stretch.jpg|300px]]
|}

The appearance of this imaginary frequency resembles the motion that would ensue under the Cope Rearrangment - the action of bond breaking of the C-C σ bond in the middle of the 1,5-hexadiene with the subsequent formation of a new σ bond on across the terminals on the other side.

The fact that there is only '''one''' imaginary frequency value points to the notion that the cope rearrangment through this transition state is ''concerted'', and thus occurs in one ''synchronous'' step.

The number of imaginary frequencies produced from a frequency analysis for the midpoint of a reaction points towards the ''synchronicity'' of the reaction<ref>Cope Rearrangement Revisited, DOI:10.1021/ja00026a007</ref>.

===The Boat Transition State===

For the boat transition state, a different method was utilised, namely the '''QST2 method''', whereby reactant and product are defined, and the transition state is converged from these.

This required the previously optimised '''1,5-hexadiene''' molecule to be taken and numbered along its carbon skeleton.

The molecule was then duplicated and juxtaposed to the intial "reactant" molecule and renumbered in such a way to reflect the reacted, rearragement "product" as shown below:-

{|style="margin:1em auto 1em auto" border="1"
|-
|[[Image:Boat Method.JPG|400px]]
|}

The first instance of optimisation failed to produce an output, and thus the molecule was changed in the following way:-

The dihedral angle of both reactant and product were changed to 0 degrees, such that the two carbon groups either side of C3 and C4 are eclipsed, and the C2-C3-C4 and C3-C4-C5 bond angles were both changed to 100 degrees.

The modified structures were once again optimised under the QST2 method to yield the following structure:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''''QST2'' Optimised Boat Transition State'''
|-
|[[Image:FINAL Boat.jpg|400px]]
|}

'''DISCUSSION'''

Here, it is illustrated that the QST2 method is automated, and is thus very efficient, but is only effective if the shape provided of the reactants and products cloely mirror that of the transition state, otherwise convergence is difficult.

The '''QST3''' method was also attempted <ref>QST Calculation, http://hdl.handle.net/10042/to-2902</ref>, whereby an addition guess of the transition state geometry was implemented. If the guess of the transition state is sound, then the QST3 is by far more accurate than the QST2 method, and does not rely so much on the shape of the reactants and products.

initally, it would seem that in order for the cope rearrangement to occur, to two carbon terminals would be required to be close to one another. This is not the case with the "''anti''" conformers of the molecule, as the terminals are situated as far apart as they may possibly be, '''antiperplanar''' to one another.

Once free rotation moves the carbon groups closer together, into the gauche configuration, where the terminals are more poised to react, cope rearrangement would proceed.

The relative orientations of the vinyl groups would ultimately dictate which transition state would be traversed. For instance, if the vectors of the vinyl groups cross, such as the case for the gauche1 conformer, then as the terminals align into a bonding orientaton, the ''chair'' conformation would be adopted by the transition state.

Conversely, if the two vinyl groups lie in a pseudo-parallel fashion, where their vectors do NOT cross, such as is the case for gauche3, then the transition state would be of ''boat'' topography.

'''IRC'''

The Intrinsic Reaction Coordinate looks more cloesly at the potential surface of a reaction and works to pinpoint the highest point in a local area of the potential surface by creeping up the steepest slopes in the immediate area.

This calculation was attempted with 50 points, and did not converge

'''Energy Calculation'''

The two transition states were optimised a final time under DFT B3LYP 6-31G(d) conditions.

The geometries that were outputted were very similar to the previous examples, however

==The Diels-Alder Cycloaddition==

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Diels-Alder Reaction of Butadiene and Ethylene'''
|-align="center"
|[[Image:Mb_da3.jpg|400px]]
|}

The Diels-Alder Reaction is an example of a pericyclic cycloaddition, that usually involves the reaction between an electron rich diene and an electron poor dienophile.

The process is described as a [4s + 2s], 6 electron process that forms 2 new σ bonds.

When viewing the molecular orbital picture of butadiene and ethylene, the molecular orbitals are classified as either '''''s'''''(''symmetric'') or '''''a'''''(''antisymmetric'') in relation to the bisecting plane as shown below.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Bisecting Plane for Diels-alder'''
|-
|[[Image:Mb_da2.jpg|200px]]
|}

===Investigation of Molecular Orbital Symmetries of Butadiene and Ethene===

Butadiene and Ethylene were optimised and their molecular orbitals visualised:-

'''''Butadiene'''''

The HOMO and LUMO are considered with reference to the bisecting plane to determine their symmetry:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=4|'''HOMO and LUMO of Butadiene'''
|-align="center"
|width=250px|'''Molecular Orbital'''||'''2D Representation'''||'''3D Representation'''||width=250px|'''s/a'''
|-align="center"
|'''HOMO'''||[[Image:BDE HOMO.png|200px]]||[[Image:3D BDE HOMO.jpg|300px]]||a
|-align="center"
|'''LUMO'''||[[Image:BDE LUMO.png|200px]]||[[Image:3D BDE LUMO.jpg|300px]]||s
|}

'''''Ethene'''''

Similarly, the HOMO and LUMO of ethylene were characterised:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=4|'''HOMO and LUMO of Ethene'''
|-align="center"
|width=250px|'''Molecular Orbital'''||'''2D Representation'''||'''3D Representation'''||width=250px|'''s/a'''
|-align="center"
|'''HOMO'''||[[Image:ET HOMO.jpg|200px]]||[[Image:3D ET HOMO.jpg|300px]]||s
|-align="center"
|'''LUMO'''||[[Image:ET LUMO.jpg|200px]]||[[Image:3D ET LUMO.jpg|300px]]||a
|}

===Optimisation of Diels-Alder Transition State===

Rep:Mod:parkbom

2009-11-13T13:31:48Z

Byt07:

=Module 3 - The Computation of the Transition State=

==Introduction==

The '''Transition State''' is an energetic maximum within a given reaction coordinate and represents the point at which bond breaking and bond forming is occurring simultaneously. This is represented by dotted lines in formal notation and depicts the transient nature of the bond(s) in question in the transition state.

The analysis of potential energy surfaces are effective in finding transition states; the transition states can be described graphically as "dams" that represent energy metastability, that the reactants must "cross" in order to form the products.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Potential Energy Surface Reaction Coordinate'''<ref>http://www.ch.ic.ac.uk/motm/porphyrins/TSdiscovery.html</ref>||'''Cope Rearrangement'''<ref>http://www.ch.ic.ac.uk/local/organic/pericyclic/</ref>
|-
|[[Image:PES.gif]]||[[Image:Cope Arr.gif|350px]]
|}

From the point of view of computational chemistry, the methods of molecular mechanical structure determination that employed imaginary force fields are ineffective in the calculations concerning the transition state in larger molecules, as they do not define the dynamics of bond making and bond forming that are apparent in the transient nature of the transition state.

As such, in combination with molecular orbital based calculations, the following of the potential energy surface of a molecule can be used to locate and analyse the transition state.

This computational exercise concerns the rearrangement of ''1,5-Hexadiene'' in the '''Cope Rearrangment'''(above).

The Cope Rearrangement proceeds via a [3,3]-Sigmatropic shift involving the motion of 6 electrons and is thermally-driven.

In terms of transition states, the cope rearrangement can reportedly proceed via two checkpoints, the '''boat''' or the '''chair''':-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Chair'''||'''Boat'''
|-
|[[Image:pic2a.jpg]]||[[Image:pic2b.jpg]]
|}

The purpose of this exercise is to ascertain which reaction pathway is preferred by computation of each transition state in the elucidation of geometry, repective energies and other thermochemical data.

==''Exercise 1'' - Optimising the Reactants and Products==

In this section, the "''reactant''" '''1,5-hexadiene''' and the "''product''" '''1,5-hexadiene''' are modelled in correspondence to the cope rearrangement.

The flexible nature of 1,5-hexadiene means that there is an initial dilemma of likely conformation at rest.

Several starting conformations were thus taken and their geometries optimised to find the lowest energy conformer.

Initially, 1,5-hexadiene was taken, ensuring a roughly "'''anti'''" linkage in the centre of the molecule, so that the central C atoms '''3''' and '''4''' are approximately '''antiperiplanar''' to one another. This molecule was cleaned in the gaussview builder interface and then taken and optimised using the '''Hartree-Fock''' method with basis set '''3-21G''' through Gaussian.

'''b)''' similarly, a synclinal, or "gauche" version of 1,5-hexadiene was taken and optimised at the HF 3-21G level.

The energies of both conformers and also their symmetry were noted.

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|'''"''Anti''" 1,5-Hexadiene'''||'''"''Gauche''" 1,5-Hexadiene'''
|-align="center"
|[[Image:Bo Anti.jpg|420px]]||[[Image:Bo Gauche.jpg|400px]]
|}

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|'''Confomer'''||'''''Anti''-1,5-Hexadiene'''||'''''Gauche''-1,5-Hexadiene'''
|-align="center"
|'''Total Energy'''(a.u.)||-231.69254||-231.69266
|-align="center"
|'''Point Group'''||ci||None
|}

From a purely steric-orientated argument, the ''anti'' configuration should have a lower energy than the ''gauche'', as there is less steric hindrance from the two ends of the carbon chain 180 degrees from one another than next to each other in the gauche conformation.

As it so happens, the energy for the ''anti'' conformer is reportedly lower than that of the ''gauche'' conformer, but by a minute amount of roughly a ten-thousandth of a hartree ('''1 hartree = 4.3597482E-18 J''').

Even so, it was appreciated that the central C-C-C-C linkage of 1,5-hexadiene lends itself to quite a high degree of rotational freedom, and thus, further examples of conformational isomerism. This is reinforced by the fact of the two remarkably different conformers having remarkably similar energies, and points to the fact that 1,5-hexadiene experiences conformational ambiguity often<ref>Cope Rearrangement of 1,5-Hexadiene: Full Geometry Optimizations Using Analytic MR-CISD and MR-AQCC Gradient Methods, doi:10.1021/jp0259014</ref>.

As such, 2 other conformers were taken and optimised:-

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|colspan=4|'''<big>Different Conformers of 1,5-Hexadiene</big>'''
|-align="center"
|(''g120,a,g120'')-'''Anti1'''||(''g120,a,g-120'')-'''Anti2'''||(''g120,g-60,g120'')-'''Gauche1'''||(''g120,g60,g-120'')-'''Gauche3'''
|-align="center"
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Bo Anti1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Bo Anti2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Gauche 1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Gauche 3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-align="center"
|colspan=4|'''Energy'''(a.u.)
|-align="center"
| -231.69260213||-231.69253506||-231.69166702||-231.69266120
|-align="center"
|colspan=4|'''Point Group'''
|-align="center"
|C2||Ci||C2||C1
|}

The energies here from the HF 3-21G are similarly close, but contrary to what was expected, it was found that ''gauche3'' was actually the lowest in energy.

Beneath this, the ''anti'' conformers are found to be lower in energy than the other ''gauche'' conformer. This may be rationalised as the minimalisation of steric hindrance lowering the energy of the overall molecule, as the antiperiplanar groups maintain the least steric strain.

However, the '''''cis'''''-''gauche'' conformer is lowest in energy due to stereoelectronic arguments.

The '''π'''-bonds of the C=C bonds are lying criss-crossed over each other in this conformer. This promotes effectual π-molecular orbital overlap and hence stabilises the molecule.

Next, the anti2 conformer with ci symmetry and the gauche1 conformer was taken and further optimised under the DFT B3LYP method with a more sophisticated basis set, 6-31G(d).

The resulting optimised molecule was then compared with the earlier attempt in terms of energy.

A further frequency analysis was conducted in order to check that the optimisation has proceeded to a minimum, being the case when all frequencies were real and positive.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=5|'''Comparison of the HF 3-21G and DFT B3LYP 6-31G(d) Optimisations'''
|-align="center"
|'''Method'''||colspan=2|'''HF 3-21G'''||colspan=2|'''DFT B3LYP 6-31G(d)'''
|-align="center"
|'''Conformer'''||'''anti2'''||'''gauche1'''||'''anti2'''||'''gauche1'''
|-align="center"
|'''Total Energy'''(a.u.)||-231.69253506||-231.69166702||-231.61170616||-231.61068821
|-align="center"
|1-2 C=C Bond Length (A)||1.3162||1.3156||1.3335||1.3331
|-align="center"
|2-3 C-C Bond Length (A)||1.5091||1.5041||1.5083||1.5044
|-align="center"
|3-4 C-C Bond Length (A)||1.5527||1.5481||1.5508||1.5482
|-align="center"
|4-5 C-C Bond Length (A)||1.5091||1.5041||1.5083||1.5044
|-align="center"
|5-6 C=C Bond Length (A)||1.3163||1.3335||1.3157||1.3331
|-align="center"
|1-2-3-4 Dihedral (o)||114.66||118.49||123.93||124.24
|-align="center"
|2-3-4-5 Dihedral (o)||180.00||180.00||-64.18||-65.01
|-align="center"
|3-4-5-6 Dihedral (o)||-114.66||-118.49||123.93||124.24
|}

As can be seen here, the energies returned differ in a noticeable manner. However, comparison of energies computed under different basis sets is unbalanced.

As such, comparisons of the geometry are notably more reliable in discerning the computed differences.

The bond lengths differ by negligible amounts. However, it can be seen that the various dihedral angles have been "tightened" more to their ideal figures (closer to 120o for the anticlinal dihedrals.

As such, using a more sophisticated basis set ensures a more accurate approximation to reality.

The log file yielded the following thermochemical data:-

'''1)Sum of electronic and zero-point Energies= -234.469195'''

'''2)Sum of electronic and thermal Energies= -234.461847'''

'''3)Sum of electronic and thermal Enthalpies= -234.460903'''

'''4)Sum of electronic and thermal Free Energies= -234.500782'''

Theses values concern the 1)Potential Energy at 0K that includes the zero-point vibrational energy term 2)Potential Energy at 298.15K and 1atm and adds contributions from translational, rotational and vibrational energy 3)Energy including a correction for RT(H=E+RT) for dissociation reactions and 4)The Entropic contribution.

==''Exercise 2'' - Optimising the "Chair" and "Boat" Transition Structures==

The Cope Rearrangement has been found to proceed via one of two possible transition states with differing conformation, as mentioned in the introduction:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=2|'''<big>Cope Rearrangement Transition State Conformations</big>'''
|-align="center"
|''Chair''||''Boat''
|-align="center"
|[[Image:pic2a.jpg]]||[[Image:pic2b.jpg]]
|}

When the rearrangment is thought as one that is concerted and proceeds with synchronicity of bond breaking and formation, these two transition states come to mind.

There is indeed a third type of transition state for this reaction, dubbed the '''''Dewar''''' transition state, that reportedly proceeds via a diradical tight-chair intermediate species that was discovered by Dupuis et al.<ref>The Cope Rearrangement Revisited, DOI:10.1021/ja00026a007</ref>, and is thus NOT concerted.

===The Chair Transition State===

The chair transition state was initally synthesised using two "resonant" '''allyl''' ('''C3H5''') fragments in gaussview, which were optimised under the 3-21G basis set, and subsequently placed in appropriate positions relative to one another to be optimised into the transition state.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=2|'''Allyl Fragment('''C3H5''')'''
|-align="center"
|colspan=2|[[Image:Allyl Frag.jpg|275px]]
|-align="center"
|'''sp3''' Centre Bond Angle (o)||109.5
|-align="center"
|'''Allyl''' C-C Bond Angle (o)||124.3
|-align="center"
|'''C-C''' Bond Length (A)||1.54
|-align="center"
|'''C=C''' Bond Length (A)||1.34
|-align="center"
|'''Allyl C-C''' Bond Length (A)||1.39
|}

The optimised allyl C-C bond angle is greater than the normal sp3 bond angle of '''109.5o''' due to resonance.

More importantly, the allyl C-C bond length is between that of a single bond and a double bond.

This optimised allyl fragment was then taken, duplicated and placed, facing opposite directions, with roughly a 2.2A distance between the terminal carbons of each fragment to mimic the shape of the chair transition state.

Thus, a frequency + optimisation was run that directed gaussian to reach a '''TS(Berny)''', with the elucidation of force constants.

The optimisation was then repeated but this time using the '''"frozen" coordinate method''', effectively fixing the two pairs of ends at a distance of 2.2A.

Clearly, this method requires that the general structure of the transition state in question be known. However, by fixing the positions in such a way, a more accurate optimisation can be done that avoids divergence.

Finally, the optimisation was performed a third time, using a normal guess '''Hessian''' method, which involves calculation of the force constant matrix.

After the three optimisations were complete, the following data were obtained:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=3|'''<big>Optimisation of Chair Transition State</big>'''
|-align="center"
|colspan=3|'''Structure'''
|-align="center"
|'''TS(Berny)'''||'''Frozen Coordinate'''||'''Hessian'''
|-align="center"
||<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Berny.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Frozen.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>red</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Hessian.mol</uploadedFileContents>
</jmolApplet></jmol>
|-align="center"
|colspan=3|'''Energies''' (a.u.)
|-align="center"
| -231.61932||-231.61932||-231.69167
|}

The structures returned from the normal force-constant calculation optimisation and the frozen coordinate method are indistinguishable from first glance, and their energies are virtually identical. However, the structure obtained from the Hessian is different, and so is its energy value.

'''FREQUENCY ANALYSIS'''

Vibration-wise, there was one negative, and thus "imaginary" frequency found with a wavenumber of '''-817.91cm-1'''.

When the stretch was visualised, the following was observed:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Imaginary Stretch at -817.91cm-1'''
|-
|[[Image:Imag Stretch.jpg|300px]]
|}

The appearance of this imaginary frequency resembles the motion that would ensue under the Cope Rearrangment - the action of bond breaking of the C-C σ bond in the middle of the 1,5-hexadiene with the subsequent formation of a new σ bond on across the terminals on the other side.

The fact that there is only '''one''' imaginary frequency value points to the notion that the cope rearrangment through this transition state is ''concerted'', and thus occurs in one ''synchronous'' step.

The number of imaginary frequencies produced from a frequency analysis for the midpoint of a reaction points towards the ''synchronicity'' of the reaction<ref>Cope Rearrangement Revisited, DOI:10.1021/ja00026a007</ref>.

===The Boat Transition State===

For the boat transition state, a different method was utilised, namely the '''QST2 method''', whereby reactant and product are defined, and the transition state is converged from these.

This required the previously optimised '''1,5-hexadiene''' molecule to be taken and numbered along its carbon skeleton.

The molecule was then duplicated and juxtaposed to the intial "reactant" molecule and renumbered in such a way to reflect the reacted, rearragement "product" as shown below:-

{|style="margin:1em auto 1em auto" border="1"
|-
|[[Image:Boat Method.JPG|400px]]
|}

The first instance of optimisation failed to produce an output, and thus the molecule was changed in the following way:-

The dihedral angle of both reactant and product were changed to 0 degrees, such that the two carbon groups either side of C3 and C4 are eclipsed, and the C2-C3-C4 and C3-C4-C5 bond angles were both changed to 100 degrees.

The modified structures were once again optimised under the QST2 method to yield the following structure:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''''QST2'' Optimised Boat Transition State'''
|-
|[[Image:FINAL Boat.jpg|400px]]
|}

'''DISCUSSION'''

Here, it is illustrated that the QST2 method is automated, and is thus very efficient, but is only effective if the shape provided of the reactants and products cloely mirror that of the transition state, otherwise convergence is difficult.

The '''QST3''' method was also attempted <ref>QST Calculation, http://hdl.handle.net/10042/to-2902</ref>, whereby an addition guess of the transition state geometry was implemented. If the guess of the transition state is sound, then the QST3 is by far more accurate than the QST2 method, and does not rely so much on the shape of the reactants and products.

initally, it would seem that in order for the cope rearrangement to occur, to two carbon terminals would be required to be close to one another. This is not the case with the "''anti''" conformers of the molecule, as the terminals are situated as far apart as they may possibly be, '''antiperplanar''' to one another.

Once free rotation moves the carbon groups closer together, into the gauche configuration, where the terminals are more poised to react, cope rearrangement would proceed.

The relative orientations of the vinyl groups would ultimately dictate which transition state would be traversed. For instance, if the vectors of the vinyl groups cross, such as the case for the gauche1 conformer, then as the terminals align into a bonding orientaton, the ''chair'' conformation would be adopted by the transition state.

Conversely, if the two vinyl groups lie in a pseudo-parallel fashion, where their vectors do NOT cross, such as is the case for gauche3, then the transition state would be of ''boat'' topography.

'''IRC'''

The Intrinsic Reaction Coordinate looks more cloesly at the potential surface of a reaction and works to pinpoint the highest point in a local area of the potential surface by creeping up the steepest slopes in the immediate area.

This calculation was attempted with 50 points, and did not converge

'''Energy Calculation'''

The two transition states were optimised a final time under DFT B3LYP 6-31G(d) conditions.

The geometries that were outputted were very similar to the previous examples, however

==The Diels-Alder Cycloaddition==

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Diels-Alder Reaction of Butadiene and Ethylene'''
|-align="center"
|[[Image:Mb_da3.jpg|400px]]
|}

The Diels-Alder Reaction is an example of a pericyclic cycloaddition, that usually involves the reaction between an electron rich diene and an electron poor dienophile.

The process is described as a [4s + 2s], 6 electron process that forms 2 new σ bonds.

When viewing the molecular orbital picture of butadiene and ethylene, the molecular orbitals are classified as either '''''s'''''(''symmetric'') or '''''a'''''(''antisymmetric'') in relation to the bisecting plane as shown below.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Bisecting Plane for Diels-alder'''
|-
|[[Image:Mb_da2.jpg|200px]]
|}

'''Investigation of Molecular Orbital Symmetries of Butadiene and Ethene'''

Butadiene and Ethylene were optimised and their molecular orbitals visualised:-

'''''Butadiene'''''

The HOMO and LUMO are considered with reference to the bisecting plane to determine their symmetry:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=4|'''HOMO and LUMO of Butadiene'''
|-align="center"
|width=250px|'''Molecular Orbital'''||'''2D Representation'''||'''3D Representation'''||width=250px|'''s/a'''
|-align="center"
|'''HOMO'''||[[Image:BDE HOMO.png|200px]]||[[Image:3D BDE HOMO.jpg|300px]]||a
|-align="center"
|'''LUMO'''||[[Image:BDE LUMO.png|200px]]||[[Image:3D BDE LUMO.jpg|300px]]||s
|}

'''''Ethene'''''

Similarly, the HOMO and LUMO of ethylene were characterised:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=4|'''HOMO and LUMO of Ethene'''
|-align="center"
|width=250px|'''Molecular Orbital'''||'''2D Representation'''||'''3D Representation'''||width=250px|'''s/a'''
|-align="center"
|'''HOMO'''||[[Image:ET HOMO.png|200px]]||[[Image:3D ET HOMO.jpg|300px]]||s
|-align="center"
|'''LUMO'''||[[Image:ET LUMO.png|200px]]||[[Image:3D ET LUMO.jpg|300px]]||a
|}

File:3D ET LUMO.jpg

2009-11-13T13:30:16Z

Byt07:

File:3D ET HOMO.jpg

2009-11-13T13:30:06Z

Byt07:

File:ET LUMO.jpg

2009-11-13T13:29:49Z

Byt07:

File:ET HOMO.jpg

2009-11-13T13:29:43Z

Byt07:

Rep:Mod:parkbom

2009-11-13T13:15:30Z

Byt07:

=Module 3 - The Computation of the Transition State=

==Introduction==

The '''Transition State''' is an energetic maximum within a given reaction coordinate and represents the point at which bond breaking and bond forming is occurring simultaneously. This is represented by dotted lines in formal notation and depicts the transient nature of the bond(s) in question in the transition state.

The analysis of potential energy surfaces are effective in finding transition states; the transition states can be described graphically as "dams" that represent energy metastability, that the reactants must "cross" in order to form the products.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Potential Energy Surface Reaction Coordinate'''<ref>http://www.ch.ic.ac.uk/motm/porphyrins/TSdiscovery.html</ref>||'''Cope Rearrangement'''<ref>http://www.ch.ic.ac.uk/local/organic/pericyclic/</ref>
|-
|[[Image:PES.gif]]||[[Image:Cope Arr.gif|350px]]
|}

From the point of view of computational chemistry, the methods of molecular mechanical structure determination that employed imaginary force fields are ineffective in the calculations concerning the transition state in larger molecules, as they do not define the dynamics of bond making and bond forming that are apparent in the transient nature of the transition state.

As such, in combination with molecular orbital based calculations, the following of the potential energy surface of a molecule can be used to locate and analyse the transition state.

This computational exercise concerns the rearrangement of ''1,5-Hexadiene'' in the '''Cope Rearrangment'''(above).

The Cope Rearrangement proceeds via a [3,3]-Sigmatropic shift involving the motion of 6 electrons and is thermally-driven.

In terms of transition states, the cope rearrangement can reportedly proceed via two checkpoints, the '''boat''' or the '''chair''':-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Chair'''||'''Boat'''
|-
|[[Image:pic2a.jpg]]||[[Image:pic2b.jpg]]
|}

The purpose of this exercise is to ascertain which reaction pathway is preferred by computation of each transition state in the elucidation of geometry, repective energies and other thermochemical data.

==''Exercise 1'' - Optimising the Reactants and Products==

In this section, the "''reactant''" '''1,5-hexadiene''' and the "''product''" '''1,5-hexadiene''' are modelled in correspondence to the cope rearrangement.

The flexible nature of 1,5-hexadiene means that there is an initial dilemma of likely conformation at rest.

Several starting conformations were thus taken and their geometries optimised to find the lowest energy conformer.

Initially, 1,5-hexadiene was taken, ensuring a roughly "'''anti'''" linkage in the centre of the molecule, so that the central C atoms '''3''' and '''4''' are approximately '''antiperiplanar''' to one another. This molecule was cleaned in the gaussview builder interface and then taken and optimised using the '''Hartree-Fock''' method with basis set '''3-21G''' through Gaussian.

'''b)''' similarly, a synclinal, or "gauche" version of 1,5-hexadiene was taken and optimised at the HF 3-21G level.

The energies of both conformers and also their symmetry were noted.

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|'''"''Anti''" 1,5-Hexadiene'''||'''"''Gauche''" 1,5-Hexadiene'''
|-align="center"
|[[Image:Bo Anti.jpg|420px]]||[[Image:Bo Gauche.jpg|400px]]
|}

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|'''Confomer'''||'''''Anti''-1,5-Hexadiene'''||'''''Gauche''-1,5-Hexadiene'''
|-align="center"
|'''Total Energy'''(a.u.)||-231.69254||-231.69266
|-align="center"
|'''Point Group'''||ci||None
|}

From a purely steric-orientated argument, the ''anti'' configuration should have a lower energy than the ''gauche'', as there is less steric hindrance from the two ends of the carbon chain 180 degrees from one another than next to each other in the gauche conformation.

As it so happens, the energy for the ''anti'' conformer is reportedly lower than that of the ''gauche'' conformer, but by a minute amount of roughly a ten-thousandth of a hartree ('''1 hartree = 4.3597482E-18 J''').

Even so, it was appreciated that the central C-C-C-C linkage of 1,5-hexadiene lends itself to quite a high degree of rotational freedom, and thus, further examples of conformational isomerism. This is reinforced by the fact of the two remarkably different conformers having remarkably similar energies, and points to the fact that 1,5-hexadiene experiences conformational ambiguity often<ref>Cope Rearrangement of 1,5-Hexadiene: Full Geometry Optimizations Using Analytic MR-CISD and MR-AQCC Gradient Methods, doi:10.1021/jp0259014</ref>.

As such, 2 other conformers were taken and optimised:-

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|colspan=4|'''<big>Different Conformers of 1,5-Hexadiene</big>'''
|-align="center"
|(''g120,a,g120'')-'''Anti1'''||(''g120,a,g-120'')-'''Anti2'''||(''g120,g-60,g120'')-'''Gauche1'''||(''g120,g60,g-120'')-'''Gauche3'''
|-align="center"
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Bo Anti1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Bo Anti2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Gauche 1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Gauche 3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-align="center"
|colspan=4|'''Energy'''(a.u.)
|-align="center"
| -231.69260213||-231.69253506||-231.69166702||-231.69266120
|-align="center"
|colspan=4|'''Point Group'''
|-align="center"
|C2||Ci||C2||C1
|}

The energies here from the HF 3-21G are similarly close, but contrary to what was expected, it was found that ''gauche3'' was actually the lowest in energy.

Beneath this, the ''anti'' conformers are found to be lower in energy than the other ''gauche'' conformer. This may be rationalised as the minimalisation of steric hindrance lowering the energy of the overall molecule, as the antiperiplanar groups maintain the least steric strain.

However, the '''''cis'''''-''gauche'' conformer is lowest in energy due to stereoelectronic arguments.

The '''π'''-bonds of the C=C bonds are lying criss-crossed over each other in this conformer. This promotes effectual π-molecular orbital overlap and hence stabilises the molecule.

Next, the anti2 conformer with ci symmetry and the gauche1 conformer was taken and further optimised under the DFT B3LYP method with a more sophisticated basis set, 6-31G(d).

The resulting optimised molecule was then compared with the earlier attempt in terms of energy.

A further frequency analysis was conducted in order to check that the optimisation has proceeded to a minimum, being the case when all frequencies were real and positive.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=5|'''Comparison of the HF 3-21G and DFT B3LYP 6-31G(d) Optimisations'''
|-align="center"
|'''Method'''||colspan=2|'''HF 3-21G'''||colspan=2|'''DFT B3LYP 6-31G(d)'''
|-align="center"
|'''Conformer'''||'''anti2'''||'''gauche1'''||'''anti2'''||'''gauche1'''
|-align="center"
|'''Total Energy'''(a.u.)||-231.69253506||-231.69166702||-231.61170616||-231.61068821
|-align="center"
|1-2 C=C Bond Length (A)||1.3162||1.3156||1.3335||1.3331
|-align="center"
|2-3 C-C Bond Length (A)||1.5091||1.5041||1.5083||1.5044
|-align="center"
|3-4 C-C Bond Length (A)||1.5527||1.5481||1.5508||1.5482
|-align="center"
|4-5 C-C Bond Length (A)||1.5091||1.5041||1.5083||1.5044
|-align="center"
|5-6 C=C Bond Length (A)||1.3163||1.3335||1.3157||1.3331
|-align="center"
|1-2-3-4 Dihedral (o)||114.66||118.49||123.93||124.24
|-align="center"
|2-3-4-5 Dihedral (o)||180.00||180.00||-64.18||-65.01
|-align="center"
|3-4-5-6 Dihedral (o)||-114.66||-118.49||123.93||124.24
|}

As can be seen here, the energies returned differ in a noticeable manner. However, comparison of energies computed under different basis sets is unbalanced.

As such, comparisons of the geometry are notably more reliable in discerning the computed differences.

The bond lengths differ by negligible amounts. However, it can be seen that the various dihedral angles have been "tightened" more to their ideal figures (closer to 120o for the anticlinal dihedrals.

As such, using a more sophisticated basis set ensures a more accurate approximation to reality.

The log file yielded the following thermochemical data:-

'''1)Sum of electronic and zero-point Energies= -234.469195'''

'''2)Sum of electronic and thermal Energies= -234.461847'''

'''3)Sum of electronic and thermal Enthalpies= -234.460903'''

'''4)Sum of electronic and thermal Free Energies= -234.500782'''

Theses values concern the 1)Potential Energy at 0K that includes the zero-point vibrational energy term 2)Potential Energy at 298.15K and 1atm and adds contributions from translational, rotational and vibrational energy 3)Energy including a correction for RT(H=E+RT) for dissociation reactions and 4)The Entropic contribution.

==''Exercise 2'' - Optimising the "Chair" and "Boat" Transition Structures==

The Cope Rearrangement has been found to proceed via one of two possible transition states with differing conformation, as mentioned in the introduction:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=2|'''<big>Cope Rearrangement Transition State Conformations</big>'''
|-align="center"
|''Chair''||''Boat''
|-align="center"
|[[Image:pic2a.jpg]]||[[Image:pic2b.jpg]]
|}

When the rearrangment is thought as one that is concerted and proceeds with synchronicity of bond breaking and formation, these two transition states come to mind.

There is indeed a third type of transition state for this reaction, dubbed the '''''Dewar''''' transition state, that reportedly proceeds via a diradical tight-chair intermediate species that was discovered by Dupuis et al.<ref>The Cope Rearrangement Revisited, DOI:10.1021/ja00026a007</ref>, and is thus NOT concerted.

===The Chair Transition State===

The chair transition state was initally synthesised using two "resonant" '''allyl''' ('''C3H5''') fragments in gaussview, which were optimised under the 3-21G basis set, and subsequently placed in appropriate positions relative to one another to be optimised into the transition state.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=2|'''Allyl Fragment('''C3H5''')'''
|-align="center"
|colspan=2|[[Image:Allyl Frag.jpg|275px]]
|-align="center"
|'''sp3''' Centre Bond Angle (o)||109.5
|-align="center"
|'''Allyl''' C-C Bond Angle (o)||124.3
|-align="center"
|'''C-C''' Bond Length (A)||1.54
|-align="center"
|'''C=C''' Bond Length (A)||1.34
|-align="center"
|'''Allyl C-C''' Bond Length (A)||1.39
|}

The optimised allyl C-C bond angle is greater than the normal sp3 bond angle of '''109.5o''' due to resonance.

More importantly, the allyl C-C bond length is between that of a single bond and a double bond.

This optimised allyl fragment was then taken, duplicated and placed, facing opposite directions, with roughly a 2.2A distance between the terminal carbons of each fragment to mimic the shape of the chair transition state.

Thus, a frequency + optimisation was run that directed gaussian to reach a '''TS(Berny)''', with the elucidation of force constants.

The optimisation was then repeated but this time using the '''"frozen" coordinate method''', effectively fixing the two pairs of ends at a distance of 2.2A.

Clearly, this method requires that the general structure of the transition state in question be known. However, by fixing the positions in such a way, a more accurate optimisation can be done that avoids divergence.

Finally, the optimisation was performed a third time, using a normal guess '''Hessian''' method, which involves calculation of the force constant matrix.

After the three optimisations were complete, the following data were obtained:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=3|'''<big>Optimisation of Chair Transition State</big>'''
|-align="center"
|colspan=3|'''Structure'''
|-align="center"
|'''TS(Berny)'''||'''Frozen Coordinate'''||'''Hessian'''
|-align="center"
||<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Berny.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Frozen.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>red</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Hessian.mol</uploadedFileContents>
</jmolApplet></jmol>
|-align="center"
|colspan=3|'''Energies''' (a.u.)
|-align="center"
| -231.61932||-231.61932||-231.69167
|}

The structures returned from the normal force-constant calculation optimisation and the frozen coordinate method are indistinguishable from first glance, and their energies are virtually identical. However, the structure obtained from the Hessian is different, and so is its energy value.

'''FREQUENCY ANALYSIS'''

Vibration-wise, there was one negative, and thus "imaginary" frequency found with a wavenumber of '''-817.91cm-1'''.

When the stretch was visualised, the following was observed:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Imaginary Stretch at -817.91cm-1'''
|-
|[[Image:Imag Stretch.jpg|300px]]
|}

The appearance of this imaginary frequency resembles the motion that would ensue under the Cope Rearrangment - the action of bond breaking of the C-C σ bond in the middle of the 1,5-hexadiene with the subsequent formation of a new σ bond on across the terminals on the other side.

The fact that there is only '''one''' imaginary frequency value points to the notion that the cope rearrangment through this transition state is ''concerted'', and thus occurs in one ''synchronous'' step.

The number of imaginary frequencies produced from a frequency analysis for the midpoint of a reaction points towards the ''synchronicity'' of the reaction<ref>Cope Rearrangement Revisited, DOI:10.1021/ja00026a007</ref>.

===The Boat Transition State===

For the boat transition state, a different method was utilised, namely the '''QST2 method''', whereby reactant and product are defined, and the transition state is converged from these.

This required the previously optimised '''1,5-hexadiene''' molecule to be taken and numbered along its carbon skeleton.

The molecule was then duplicated and juxtaposed to the intial "reactant" molecule and renumbered in such a way to reflect the reacted, rearragement "product" as shown below:-

{|style="margin:1em auto 1em auto" border="1"
|-
|[[Image:Boat Method.JPG|400px]]
|}

The first instance of optimisation failed to produce an output, and thus the molecule was changed in the following way:-

The dihedral angle of both reactant and product were changed to 0 degrees, such that the two carbon groups either side of C3 and C4 are eclipsed, and the C2-C3-C4 and C3-C4-C5 bond angles were both changed to 100 degrees.

The modified structures were once again optimised under the QST2 method to yield the following structure:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''''QST2'' Optimised Boat Transition State'''
|-
|[[Image:FINAL Boat.jpg|400px]]
|}

'''DISCUSSION'''

Here, it is illustrated that the QST2 method is automated, and is thus very efficient, but is only effective if the shape provided of the reactants and products cloely mirror that of the transition state, otherwise convergence is difficult.

The '''QST3''' method was also attempted <ref>QST Calculation, http://hdl.handle.net/10042/to-2902</ref>, whereby an addition guess of the transition state geometry was implemented. If the guess of the transition state is sound, then the QST3 is by far more accurate than the QST2 method, and does not rely so much on the shape of the reactants and products.

initally, it would seem that in order for the cope rearrangement to occur, to two carbon terminals would be required to be close to one another. This is not the case with the "''anti''" conformers of the molecule, as the terminals are situated as far apart as they may possibly be, '''antiperplanar''' to one another.

Once free rotation moves the carbon groups closer together, into the gauche configuration, where the terminals are more poised to react, cope rearrangement would proceed.

The relative orientations of the vinyl groups would ultimately dictate which transition state would be traversed. For instance, if the vectors of the vinyl groups cross, such as the case for the gauche1 conformer, then as the terminals align into a bonding orientaton, the ''chair'' conformation would be adopted by the transition state.

Conversely, if the two vinyl groups lie in a pseudo-parallel fashion, where their vectors do NOT cross, such as is the case for gauche3, then the transition state would be of ''boat'' topography.

'''IRC'''

The Intrinsic Reaction Coordinate looks more cloesly at the potential surface of a reaction and works to pinpoint the highest point in a local area of the potential surface by creeping up the steepest slopes in the immediate area.

This calculation was attempted with 50 points, and did not converge

'''Energy Calculation'''

The two transition states were optimised a final time under DFT B3LYP 6-31G(d) conditions.

The geometries that were outputted were very similar to the previous examples, however

==The Diels-Alder Cycloaddition==

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Diels-Alder Reaction of Butadiene and Ethylene'''
|-align="center"
|[[Image:Mb_da3.jpg|400px]]
|}

The Diels-Alder Reaction is an example of a pericyclic cycloaddition, that usually involves the reaction between an electron rich diene and an electron poor dienophile.

The process is described as a [4s + 2s], 6 electron process that forms 2 new σ bonds.

When viewing the molecular orbital picture of butadiene and ethylene, the molecular orbitals are classified as either '''''s'''''(''symmetric'') or '''''a'''''(''antisymmetric'') in relation to the bisecting plane as shown below.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Bisecting Plane for Diels-alder'''
|-
|[[Image:Mb_da2.jpg|200px]]
|}

'''Investigation of Molecular Orbital Symmetries of Butadiene and Ethene'''

Butadiene and Ethylene were optimised and their molecular orbitals visualised:-

'''''Butadiene'''''

The HOMO and LUMO are considered with reference to the bisecting plane to determine their symmetry:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=4|'''HOMO and LUMO of Butadiene'''
|-align="center"
|width=250px|'''Molecular Orbital'''||'''2D Representation'''||'''3D Representation'''||width=250px|'''s/a'''
|-align="center"
|'''HOMO'''||[[Image:BDE HOMO.png|200px]]||[[Image:3D BDE HOMO.jpg|300px]]||a
|-align="center"
|'''LUMO'''||[[Image:BDE LUMO.png|200px]]||[[Image:3D BDE LUMO.jpg|300px]]||s
|}

Rep:Mod:parkbom

2009-11-13T13:13:06Z

Byt07:

=Module 3 - The Computation of the Transition State=

==Introduction==

The '''Transition State''' is an energetic maximum within a given reaction coordinate and represents the point at which bond breaking and bond forming is occurring simultaneously. This is represented by dotted lines in formal notation and depicts the transient nature of the bond(s) in question in the transition state.

The analysis of potential energy surfaces are effective in finding transition states; the transition states can be described graphically as "dams" that represent energy metastability, that the reactants must "cross" in order to form the products.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Potential Energy Surface Reaction Coordinate'''<ref>http://www.ch.ic.ac.uk/motm/porphyrins/TSdiscovery.html</ref>||'''Cope Rearrangement'''<ref>http://www.ch.ic.ac.uk/local/organic/pericyclic/</ref>
|-
|[[Image:PES.gif]]||[[Image:Cope Arr.gif|350px]]
|}

From the point of view of computational chemistry, the methods of molecular mechanical structure determination that employed imaginary force fields are ineffective in the calculations concerning the transition state in larger molecules, as they do not define the dynamics of bond making and bond forming that are apparent in the transient nature of the transition state.

As such, in combination with molecular orbital based calculations, the following of the potential energy surface of a molecule can be used to locate and analyse the transition state.

This computational exercise concerns the rearrangement of ''1,5-Hexadiene'' in the '''Cope Rearrangment'''(above).

The Cope Rearrangement proceeds via a [3,3]-Sigmatropic shift involving the motion of 6 electrons and is thermally-driven.

In terms of transition states, the cope rearrangement can reportedly proceed via two checkpoints, the '''boat''' or the '''chair''':-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Chair'''||'''Boat'''
|-
|[[Image:pic2a.jpg]]||[[Image:pic2b.jpg]]
|}

The purpose of this exercise is to ascertain which reaction pathway is preferred by computation of each transition state in the elucidation of geometry, repective energies and other thermochemical data.

==''Exercise 1'' - Optimising the Reactants and Products==

In this section, the "''reactant''" '''1,5-hexadiene''' and the "''product''" '''1,5-hexadiene''' are modelled in correspondence to the cope rearrangement.

The flexible nature of 1,5-hexadiene means that there is an initial dilemma of likely conformation at rest.

Several starting conformations were thus taken and their geometries optimised to find the lowest energy conformer.

Initially, 1,5-hexadiene was taken, ensuring a roughly "'''anti'''" linkage in the centre of the molecule, so that the central C atoms '''3''' and '''4''' are approximately '''antiperiplanar''' to one another. This molecule was cleaned in the gaussview builder interface and then taken and optimised using the '''Hartree-Fock''' method with basis set '''3-21G''' through Gaussian.

'''b)''' similarly, a synclinal, or "gauche" version of 1,5-hexadiene was taken and optimised at the HF 3-21G level.

The energies of both conformers and also their symmetry were noted.

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|'''"''Anti''" 1,5-Hexadiene'''||'''"''Gauche''" 1,5-Hexadiene'''
|-align="center"
|[[Image:Bo Anti.jpg|420px]]||[[Image:Bo Gauche.jpg|400px]]
|}

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|'''Confomer'''||'''''Anti''-1,5-Hexadiene'''||'''''Gauche''-1,5-Hexadiene'''
|-align="center"
|'''Total Energy'''(a.u.)||-231.69254||-231.69266
|-align="center"
|'''Point Group'''||ci||None
|}

From a purely steric-orientated argument, the ''anti'' configuration should have a lower energy than the ''gauche'', as there is less steric hindrance from the two ends of the carbon chain 180 degrees from one another than next to each other in the gauche conformation.

As it so happens, the energy for the ''anti'' conformer is reportedly lower than that of the ''gauche'' conformer, but by a minute amount of roughly a ten-thousandth of a hartree ('''1 hartree = 4.3597482E-18 J''').

Even so, it was appreciated that the central C-C-C-C linkage of 1,5-hexadiene lends itself to quite a high degree of rotational freedom, and thus, further examples of conformational isomerism. This is reinforced by the fact of the two remarkably different conformers having remarkably similar energies, and points to the fact that 1,5-hexadiene experiences conformational ambiguity often<ref>Cope Rearrangement of 1,5-Hexadiene: Full Geometry Optimizations Using Analytic MR-CISD and MR-AQCC Gradient Methods, doi:10.1021/jp0259014</ref>.

As such, 2 other conformers were taken and optimised:-

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|colspan=4|'''<big>Different Conformers of 1,5-Hexadiene</big>'''
|-align="center"
|(''g120,a,g120'')-'''Anti1'''||(''g120,a,g-120'')-'''Anti2'''||(''g120,g-60,g120'')-'''Gauche1'''||(''g120,g60,g-120'')-'''Gauche3'''
|-align="center"
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Bo Anti1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Bo Anti2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Gauche 1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Gauche 3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-align="center"
|colspan=4|'''Energy'''(a.u.)
|-align="center"
| -231.69260213||-231.69253506||-231.69166702||-231.69266120
|-align="center"
|colspan=4|'''Point Group'''
|-align="center"
|C2||Ci||C2||C1
|}

The energies here from the HF 3-21G are similarly close, but contrary to what was expected, it was found that ''gauche3'' was actually the lowest in energy.

Beneath this, the ''anti'' conformers are found to be lower in energy than the other ''gauche'' conformer. This may be rationalised as the minimalisation of steric hindrance lowering the energy of the overall molecule, as the antiperiplanar groups maintain the least steric strain.

However, the '''''cis'''''-''gauche'' conformer is lowest in energy due to stereoelectronic arguments.

The '''π'''-bonds of the C=C bonds are lying criss-crossed over each other in this conformer. This promotes effectual π-molecular orbital overlap and hence stabilises the molecule.

Next, the anti2 conformer with ci symmetry and the gauche1 conformer was taken and further optimised under the DFT B3LYP method with a more sophisticated basis set, 6-31G(d).

The resulting optimised molecule was then compared with the earlier attempt in terms of energy.

A further frequency analysis was conducted in order to check that the optimisation has proceeded to a minimum, being the case when all frequencies were real and positive.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=5|'''Comparison of the HF 3-21G and DFT B3LYP 6-31G(d) Optimisations'''
|-align="center"
|'''Method'''||colspan=2|'''HF 3-21G'''||colspan=2|'''DFT B3LYP 6-31G(d)'''
|-align="center"
|'''Conformer'''||'''anti2'''||'''gauche1'''||'''anti2'''||'''gauche1'''
|-align="center"
|'''Total Energy'''(a.u.)||-231.69253506||-231.69166702||-231.61170616||-231.61068821
|-align="center"
|1-2 C=C Bond Length (A)||1.3162||1.3156||1.3335||1.3331
|-align="center"
|2-3 C-C Bond Length (A)||1.5091||1.5041||1.5083||1.5044
|-align="center"
|3-4 C-C Bond Length (A)||1.5527||1.5481||1.5508||1.5482
|-align="center"
|4-5 C-C Bond Length (A)||1.5091||1.5041||1.5083||1.5044
|-align="center"
|5-6 C=C Bond Length (A)||1.3163||1.3335||1.3157||1.3331
|-align="center"
|1-2-3-4 Dihedral (o)||114.66||118.49||123.93||124.24
|-align="center"
|2-3-4-5 Dihedral (o)||180.00||180.00||-64.18||-65.01
|-align="center"
|3-4-5-6 Dihedral (o)||-114.66||-118.49||123.93||124.24
|}

As can be seen here, the energies returned differ in a noticeable manner. However, comparison of energies computed under different basis sets is unbalanced.

As such, comparisons of the geometry are notably more reliable in discerning the computed differences.

The bond lengths differ by negligible amounts. However, it can be seen that the various dihedral angles have been "tightened" more to their ideal figures (closer to 120o for the anticlinal dihedrals.

As such, using a more sophisticated basis set ensures a more accurate approximation to reality.

The log file yielded the following thermochemical data:-

'''1)Sum of electronic and zero-point Energies= -234.469195'''

'''2)Sum of electronic and thermal Energies= -234.461847'''

'''3)Sum of electronic and thermal Enthalpies= -234.460903'''

'''4)Sum of electronic and thermal Free Energies= -234.500782'''

Theses values concern the 1)Potential Energy at 0K that includes the zero-point vibrational energy term 2)Potential Energy at 298.15K and 1atm and adds contributions from translational, rotational and vibrational energy 3)Energy including a correction for RT(H=E+RT) for dissociation reactions and 4)The Entropic contribution.

==''Exercise 2'' - Optimising the "Chair" and "Boat" Transition Structures==

The Cope Rearrangement has been found to proceed via one of two possible transition states with differing conformation, as mentioned in the introduction:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=2|'''<big>Cope Rearrangement Transition State Conformations</big>'''
|-align="center"
|''Chair''||''Boat''
|-align="center"
|[[Image:pic2a.jpg]]||[[Image:pic2b.jpg]]
|}

When the rearrangment is thought as one that is concerted and proceeds with synchronicity of bond breaking and formation, these two transition states come to mind.

There is indeed a third type of transition state for this reaction, dubbed the '''''Dewar''''' transition state, that reportedly proceeds via a diradical tight-chair intermediate species that was discovered by Dupuis et al.<ref>The Cope Rearrangement Revisited, DOI:10.1021/ja00026a007</ref>, and is thus NOT concerted.

===The Chair Transition State===

The chair transition state was initally synthesised using two "resonant" '''allyl''' ('''C3H5''') fragments in gaussview, which were optimised under the 3-21G basis set, and subsequently placed in appropriate positions relative to one another to be optimised into the transition state.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=2|'''Allyl Fragment('''C3H5''')'''
|-align="center"
|colspan=2|[[Image:Allyl Frag.jpg|275px]]
|-align="center"
|'''sp3''' Centre Bond Angle (o)||109.5
|-align="center"
|'''Allyl''' C-C Bond Angle (o)||124.3
|-align="center"
|'''C-C''' Bond Length (A)||1.54
|-align="center"
|'''C=C''' Bond Length (A)||1.34
|-align="center"
|'''Allyl C-C''' Bond Length (A)||1.39
|}

The optimised allyl C-C bond angle is greater than the normal sp3 bond angle of '''109.5o''' due to resonance.

More importantly, the allyl C-C bond length is between that of a single bond and a double bond.

This optimised allyl fragment was then taken, duplicated and placed, facing opposite directions, with roughly a 2.2A distance between the terminal carbons of each fragment to mimic the shape of the chair transition state.

Thus, a frequency + optimisation was run that directed gaussian to reach a '''TS(Berny)''', with the elucidation of force constants.

The optimisation was then repeated but this time using the '''"frozen" coordinate method''', effectively fixing the two pairs of ends at a distance of 2.2A.

Clearly, this method requires that the general structure of the transition state in question be known. However, by fixing the positions in such a way, a more accurate optimisation can be done that avoids divergence.

Finally, the optimisation was performed a third time, using a normal guess '''Hessian''' method, which involves calculation of the force constant matrix.

After the three optimisations were complete, the following data were obtained:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=3|'''<big>Optimisation of Chair Transition State</big>'''
|-align="center"
|colspan=3|'''Structure'''
|-align="center"
|'''TS(Berny)'''||'''Frozen Coordinate'''||'''Hessian'''
|-align="center"
||<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Berny.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Frozen.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>red</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Hessian.mol</uploadedFileContents>
</jmolApplet></jmol>
|-align="center"
|colspan=3|'''Energies''' (a.u.)
|-align="center"
| -231.61932||-231.61932||-231.69167
|}

The structures returned from the normal force-constant calculation optimisation and the frozen coordinate method are indistinguishable from first glance, and their energies are virtually identical. However, the structure obtained from the Hessian is different, and so is its energy value.

'''FREQUENCY ANALYSIS'''

Vibration-wise, there was one negative, and thus "imaginary" frequency found with a wavenumber of '''-817.91cm-1'''.

When the stretch was visualised, the following was observed:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Imaginary Stretch at -817.91cm-1'''
|-
|[[Image:Imag Stretch.jpg|300px]]
|}

The appearance of this imaginary frequency resembles the motion that would ensue under the Cope Rearrangment - the action of bond breaking of the C-C σ bond in the middle of the 1,5-hexadiene with the subsequent formation of a new σ bond on across the terminals on the other side.

The fact that there is only '''one''' imaginary frequency value points to the notion that the cope rearrangment through this transition state is ''concerted'', and thus occurs in one ''synchronous'' step.

The number of imaginary frequencies produced from a frequency analysis for the midpoint of a reaction points towards the ''synchronicity'' of the reaction<ref>Cope Rearrangement Revisited, DOI:10.1021/ja00026a007</ref>.

===The Boat Transition State===

For the boat transition state, a different method was utilised, namely the '''QST2 method''', whereby reactant and product are defined, and the transition state is converged from these.

This required the previously optimised '''1,5-hexadiene''' molecule to be taken and numbered along its carbon skeleton.

The molecule was then duplicated and juxtaposed to the intial "reactant" molecule and renumbered in such a way to reflect the reacted, rearragement "product" as shown below:-

{|style="margin:1em auto 1em auto" border="1"
|-
|[[Image:Boat Method.JPG|400px]]
|}

The first instance of optimisation failed to produce an output, and thus the molecule was changed in the following way:-

The dihedral angle of both reactant and product were changed to 0 degrees, such that the two carbon groups either side of C3 and C4 are eclipsed, and the C2-C3-C4 and C3-C4-C5 bond angles were both changed to 100 degrees.

The modified structures were once again optimised under the QST2 method to yield the following structure:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''''QST2'' Optimised Boat Transition State'''
|-
|[[Image:FINAL Boat.jpg|400px]]
|}

'''DISCUSSION'''

Here, it is illustrated that the QST2 method is automated, and is thus very efficient, but is only effective if the shape provided of the reactants and products cloely mirror that of the transition state, otherwise convergence is difficult.

The '''QST3''' method was also attempted <ref>QST Calculation, http://hdl.handle.net/10042/to-2902</ref>, whereby an addition guess of the transition state geometry was implemented. If the guess of the transition state is sound, then the QST3 is by far more accurate than the QST2 method, and does not rely so much on the shape of the reactants and products.

initally, it would seem that in order for the cope rearrangement to occur, to two carbon terminals would be required to be close to one another. This is not the case with the "''anti''" conformers of the molecule, as the terminals are situated as far apart as they may possibly be, '''antiperplanar''' to one another.

Once free rotation moves the carbon groups closer together, into the gauche configuration, where the terminals are more poised to react, cope rearrangement would proceed.

The relative orientations of the vinyl groups would ultimately dictate which transition state would be traversed. For instance, if the vectors of the vinyl groups cross, such as the case for the gauche1 conformer, then as the terminals align into a bonding orientaton, the ''chair'' conformation would be adopted by the transition state.

Conversely, if the two vinyl groups lie in a pseudo-parallel fashion, where their vectors do NOT cross, such as is the case for gauche3, then the transition state would be of ''boat'' topography.

'''IRC'''

The Intrinsic Reaction Coordinate looks more cloesly at the potential surface of a reaction and works to pinpoint the highest point in a local area of the potential surface by creeping up the steepest slopes in the immediate area.

This calculation was attempted with 50 points, and did not converge

'''Energy Calculation'''

The two transition states were optimised a final time under DFT B3LYP 6-31G(d) conditions.

The geometries that were outputted were very similar to the previous examples, however

==The Diels-Alder Cycloaddition==

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Diels-Alder Reaction of Butadiene and Ethylene'''
|-align="center"
|[[Image:Mb_da3.jpg|400px]]
|}

The Diels-Alder Reaction is an example of a pericyclic cycloaddition, that usually involves the reaction between an electron rich diene and an electron poor dienophile.

The process is described as a [4s + 2s], 6 electron process that forms 2 new σ bonds.

When viewing the molecular orbital picture of butadiene and ethylene, the molecular orbitals are classified as either '''''s'''''(''symmetric'') or '''''a'''''(''antisymmetric'') in relation to the bisecting plane as shown below.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Bisecting Plane for Diels-alder'''
|-
|[[Image:Mb_da2.jpg|200px]]
|}

'''Investigation of Molecular Orbital Symmetries of Butadiene and Ethene'''

Butadiene and Ethylene were optimised and their molecular orbitals visualised:-

'''''Butadiene'''''

The HOMO and LUMO are considered with reference to the bisecting plane to determine their symmetry:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=4|'''HOMO and LUMO of Butadiene'''
|-align="center"
|'''Molecular Orbital'''||'''2D Representation'''||'''3D Representation'''||'''s/a'''
|-align="center"
|'''HOMO'''||[[Image:BDE HOMO.jpg]]||[[Image:3D BDE HOMO.jpg]]||a
|-align="center"
|'''LUMO'''||[[Image:BDE LUMO.jpg]]||[[Image:3D BDE LUMO.jpg]]||s
|}

File:3D BDE LUMO.jpg

2009-11-13T13:07:40Z

Byt07:

File:3D BDE HOMO.jpg

2009-11-13T13:07:24Z

Byt07:

File:BDE LUMO.png

2009-11-13T12:55:27Z

Byt07:

File:BDE HOMO.png

2009-11-13T12:55:17Z

Byt07:

Rep:Mod:parkbom

2009-11-13T12:46:05Z

Byt07:

=Module 3 - The Computation of the Transition State=

==Introduction==

The '''Transition State''' is an energetic maximum within a given reaction coordinate and represents the point at which bond breaking and bond forming is occurring simultaneously. This is represented by dotted lines in formal notation and depicts the transient nature of the bond(s) in question in the transition state.

The analysis of potential energy surfaces are effective in finding transition states; the transition states can be described graphically as "dams" that represent energy metastability, that the reactants must "cross" in order to form the products.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Potential Energy Surface Reaction Coordinate'''<ref>http://www.ch.ic.ac.uk/motm/porphyrins/TSdiscovery.html</ref>||'''Cope Rearrangement'''<ref>http://www.ch.ic.ac.uk/local/organic/pericyclic/</ref>
|-
|[[Image:PES.gif]]||[[Image:Cope Arr.gif|350px]]
|}

From the point of view of computational chemistry, the methods of molecular mechanical structure determination that employed imaginary force fields are ineffective in the calculations concerning the transition state in larger molecules, as they do not define the dynamics of bond making and bond forming that are apparent in the transient nature of the transition state.

As such, in combination with molecular orbital based calculations, the following of the potential energy surface of a molecule can be used to locate and analyse the transition state.

This computational exercise concerns the rearrangement of ''1,5-Hexadiene'' in the '''Cope Rearrangment'''(above).

The Cope Rearrangement proceeds via a [3,3]-Sigmatropic shift involving the motion of 6 electrons and is thermally-driven.

In terms of transition states, the cope rearrangement can reportedly proceed via two checkpoints, the '''boat''' or the '''chair''':-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Chair'''||'''Boat'''
|-
|[[Image:pic2a.jpg]]||[[Image:pic2b.jpg]]
|}

The purpose of this exercise is to ascertain which reaction pathway is preferred by computation of each transition state in the elucidation of geometry, repective energies and other thermochemical data.

==''Exercise 1'' - Optimising the Reactants and Products==

In this section, the "''reactant''" '''1,5-hexadiene''' and the "''product''" '''1,5-hexadiene''' are modelled in correspondence to the cope rearrangement.

The flexible nature of 1,5-hexadiene means that there is an initial dilemma of likely conformation at rest.

Several starting conformations were thus taken and their geometries optimised to find the lowest energy conformer.

Initially, 1,5-hexadiene was taken, ensuring a roughly "'''anti'''" linkage in the centre of the molecule, so that the central C atoms '''3''' and '''4''' are approximately '''antiperiplanar''' to one another. This molecule was cleaned in the gaussview builder interface and then taken and optimised using the '''Hartree-Fock''' method with basis set '''3-21G''' through Gaussian.

'''b)''' similarly, a synclinal, or "gauche" version of 1,5-hexadiene was taken and optimised at the HF 3-21G level.

The energies of both conformers and also their symmetry were noted.

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|'''"''Anti''" 1,5-Hexadiene'''||'''"''Gauche''" 1,5-Hexadiene'''
|-align="center"
|[[Image:Bo Anti.jpg|420px]]||[[Image:Bo Gauche.jpg|400px]]
|}

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|'''Confomer'''||'''''Anti''-1,5-Hexadiene'''||'''''Gauche''-1,5-Hexadiene'''
|-align="center"
|'''Total Energy'''(a.u.)||-231.69254||-231.69266
|-align="center"
|'''Point Group'''||ci||None
|}

From a purely steric-orientated argument, the ''anti'' configuration should have a lower energy than the ''gauche'', as there is less steric hindrance from the two ends of the carbon chain 180 degrees from one another than next to each other in the gauche conformation.

As it so happens, the energy for the ''anti'' conformer is reportedly lower than that of the ''gauche'' conformer, but by a minute amount of roughly a ten-thousandth of a hartree ('''1 hartree = 4.3597482E-18 J''').

Even so, it was appreciated that the central C-C-C-C linkage of 1,5-hexadiene lends itself to quite a high degree of rotational freedom, and thus, further examples of conformational isomerism. This is reinforced by the fact of the two remarkably different conformers having remarkably similar energies, and points to the fact that 1,5-hexadiene experiences conformational ambiguity often<ref>Cope Rearrangement of 1,5-Hexadiene: Full Geometry Optimizations Using Analytic MR-CISD and MR-AQCC Gradient Methods, doi:10.1021/jp0259014</ref>.

As such, 2 other conformers were taken and optimised:-

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|colspan=4|'''<big>Different Conformers of 1,5-Hexadiene</big>'''
|-align="center"
|(''g120,a,g120'')-'''Anti1'''||(''g120,a,g-120'')-'''Anti2'''||(''g120,g-60,g120'')-'''Gauche1'''||(''g120,g60,g-120'')-'''Gauche3'''
|-align="center"
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Bo Anti1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Bo Anti2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Gauche 1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Gauche 3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-align="center"
|colspan=4|'''Energy'''(a.u.)
|-align="center"
| -231.69260213||-231.69253506||-231.69166702||-231.69266120
|-align="center"
|colspan=4|'''Point Group'''
|-align="center"
|C2||Ci||C2||C1
|}

The energies here from the HF 3-21G are similarly close, but contrary to what was expected, it was found that ''gauche3'' was actually the lowest in energy.

Beneath this, the ''anti'' conformers are found to be lower in energy than the other ''gauche'' conformer. This may be rationalised as the minimalisation of steric hindrance lowering the energy of the overall molecule, as the antiperiplanar groups maintain the least steric strain.

However, the '''''cis'''''-''gauche'' conformer is lowest in energy due to stereoelectronic arguments.

The '''π'''-bonds of the C=C bonds are lying criss-crossed over each other in this conformer. This promotes effectual π-molecular orbital overlap and hence stabilises the molecule.

Next, the anti2 conformer with ci symmetry and the gauche1 conformer was taken and further optimised under the DFT B3LYP method with a more sophisticated basis set, 6-31G(d).

The resulting optimised molecule was then compared with the earlier attempt in terms of energy.

A further frequency analysis was conducted in order to check that the optimisation has proceeded to a minimum, being the case when all frequencies were real and positive.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=5|'''Comparison of the HF 3-21G and DFT B3LYP 6-31G(d) Optimisations'''
|-align="center"
|'''Method'''||colspan=2|'''HF 3-21G'''||colspan=2|'''DFT B3LYP 6-31G(d)'''
|-align="center"
|'''Conformer'''||'''anti2'''||'''gauche1'''||'''anti2'''||'''gauche1'''
|-align="center"
|'''Total Energy'''(a.u.)||-231.69253506||-231.69166702||-231.61170616||-231.61068821
|-align="center"
|1-2 C=C Bond Length (A)||1.3162||1.3156||1.3335||1.3331
|-align="center"
|2-3 C-C Bond Length (A)||1.5091||1.5041||1.5083||1.5044
|-align="center"
|3-4 C-C Bond Length (A)||1.5527||1.5481||1.5508||1.5482
|-align="center"
|4-5 C-C Bond Length (A)||1.5091||1.5041||1.5083||1.5044
|-align="center"
|5-6 C=C Bond Length (A)||1.3163||1.3335||1.3157||1.3331
|-align="center"
|1-2-3-4 Dihedral (o)||114.66||118.49||123.93||124.24
|-align="center"
|2-3-4-5 Dihedral (o)||180.00||180.00||-64.18||-65.01
|-align="center"
|3-4-5-6 Dihedral (o)||-114.66||-118.49||123.93||124.24
|}

As can be seen here, the energies returned differ in a noticeable manner. However, comparison of energies computed under different basis sets is unbalanced.

As such, comparisons of the geometry are notably more reliable in discerning the computed differences.

The bond lengths differ by negligible amounts. However, it can be seen that the various dihedral angles have been "tightened" more to their ideal figures (closer to 120o for the anticlinal dihedrals.

As such, using a more sophisticated basis set ensures a more accurate approximation to reality.

The log file yielded the following thermochemical data:-

'''1)Sum of electronic and zero-point Energies= -234.469195'''

'''2)Sum of electronic and thermal Energies= -234.461847'''

'''3)Sum of electronic and thermal Enthalpies= -234.460903'''

'''4)Sum of electronic and thermal Free Energies= -234.500782'''

Theses values concern the 1)Potential Energy at 0K that includes the zero-point vibrational energy term 2)Potential Energy at 298.15K and 1atm and adds contributions from translational, rotational and vibrational energy 3)Energy including a correction for RT(H=E+RT) for dissociation reactions and 4)The Entropic contribution.

==''Exercise 2'' - Optimising the "Chair" and "Boat" Transition Structures==

The Cope Rearrangement has been found to proceed via one of two possible transition states with differing conformation, as mentioned in the introduction:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=2|'''<big>Cope Rearrangement Transition State Conformations</big>'''
|-align="center"
|''Chair''||''Boat''
|-align="center"
|[[Image:pic2a.jpg]]||[[Image:pic2b.jpg]]
|}

When the rearrangment is thought as one that is concerted and proceeds with synchronicity of bond breaking and formation, these two transition states come to mind.

There is indeed a third type of transition state for this reaction, dubbed the '''''Dewar''''' transition state, that reportedly proceeds via a diradical tight-chair intermediate species that was discovered by Dupuis et al.<ref>The Cope Rearrangement Revisited, DOI:10.1021/ja00026a007</ref>, and is thus NOT concerted.

===The Chair Transition State===

The chair transition state was initally synthesised using two "resonant" '''allyl''' ('''C3H5''') fragments in gaussview, which were optimised under the 3-21G basis set, and subsequently placed in appropriate positions relative to one another to be optimised into the transition state.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=2|'''Allyl Fragment('''C3H5''')'''
|-align="center"
|colspan=2|[[Image:Allyl Frag.jpg|275px]]
|-align="center"
|'''sp3''' Centre Bond Angle (o)||109.5
|-align="center"
|'''Allyl''' C-C Bond Angle (o)||124.3
|-align="center"
|'''C-C''' Bond Length (A)||1.54
|-align="center"
|'''C=C''' Bond Length (A)||1.34
|-align="center"
|'''Allyl C-C''' Bond Length (A)||1.39
|}

The optimised allyl C-C bond angle is greater than the normal sp3 bond angle of '''109.5o''' due to resonance.

More importantly, the allyl C-C bond length is between that of a single bond and a double bond.

This optimised allyl fragment was then taken, duplicated and placed, facing opposite directions, with roughly a 2.2A distance between the terminal carbons of each fragment to mimic the shape of the chair transition state.

Thus, a frequency + optimisation was run that directed gaussian to reach a '''TS(Berny)''', with the elucidation of force constants.

The optimisation was then repeated but this time using the '''"frozen" coordinate method''', effectively fixing the two pairs of ends at a distance of 2.2A.

Clearly, this method requires that the general structure of the transition state in question be known. However, by fixing the positions in such a way, a more accurate optimisation can be done that avoids divergence.

Finally, the optimisation was performed a third time, using a normal guess '''Hessian''' method, which involves calculation of the force constant matrix.

After the three optimisations were complete, the following data were obtained:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=3|'''<big>Optimisation of Chair Transition State</big>'''
|-align="center"
|colspan=3|'''Structure'''
|-align="center"
|'''TS(Berny)'''||'''Frozen Coordinate'''||'''Hessian'''
|-align="center"
||<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Berny.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Frozen.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>red</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Hessian.mol</uploadedFileContents>
</jmolApplet></jmol>
|-align="center"
|colspan=3|'''Energies''' (a.u.)
|-align="center"
| -231.61932||-231.61932||-231.69167
|}

The structures returned from the normal force-constant calculation optimisation and the frozen coordinate method are indistinguishable from first glance, and their energies are virtually identical. However, the structure obtained from the Hessian is different, and so is its energy value.

'''FREQUENCY ANALYSIS'''

Vibration-wise, there was one negative, and thus "imaginary" frequency found with a wavenumber of '''-817.91cm-1'''.

When the stretch was visualised, the following was observed:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Imaginary Stretch at -817.91cm-1'''
|-
|[[Image:Imag Stretch.jpg|300px]]
|}

The appearance of this imaginary frequency resembles the motion that would ensue under the Cope Rearrangment - the action of bond breaking of the C-C σ bond in the middle of the 1,5-hexadiene with the subsequent formation of a new σ bond on across the terminals on the other side.

The fact that there is only '''one''' imaginary frequency value points to the notion that the cope rearrangment through this transition state is ''concerted'', and thus occurs in one ''synchronous'' step.

The number of imaginary frequencies produced from a frequency analysis for the midpoint of a reaction points towards the ''synchronicity'' of the reaction<ref>Cope Rearrangement Revisited, DOI:10.1021/ja00026a007</ref>.

===The Boat Transition State===

For the boat transition state, a different method was utilised, namely the '''QST2 method''', whereby reactant and product are defined, and the transition state is converged from these.

This required the previously optimised '''1,5-hexadiene''' molecule to be taken and numbered along its carbon skeleton.

The molecule was then duplicated and juxtaposed to the intial "reactant" molecule and renumbered in such a way to reflect the reacted, rearragement "product" as shown below:-

{|style="margin:1em auto 1em auto" border="1"
|-
|[[Image:Boat Method.JPG|400px]]
|}

The first instance of optimisation failed to produce an output, and thus the molecule was changed in the following way:-

The dihedral angle of both reactant and product were changed to 0 degrees, such that the two carbon groups either side of C3 and C4 are eclipsed, and the C2-C3-C4 and C3-C4-C5 bond angles were both changed to 100 degrees.

The modified structures were once again optimised under the QST2 method to yield the following structure:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''''QST2'' Optimised Boat Transition State'''
|-
|[[Image:FINAL Boat.jpg|400px]]
|}

'''DISCUSSION'''

Here, it is illustrated that the QST2 method is automated, and is thus very efficient, but is only effective if the shape provided of the reactants and products cloely mirror that of the transition state, otherwise convergence is difficult.

The '''QST3''' method was also attempted <ref>QST Calculation, http://hdl.handle.net/10042/to-2902</ref>, whereby an addition guess of the transition state geometry was implemented. If the guess of the transition state is sound, then the QST3 is by far more accurate than the QST2 method, and does not rely so much on the shape of the reactants and products.

initally, it would seem that in order for the cope rearrangement to occur, to two carbon terminals would be required to be close to one another. This is not the case with the "''anti''" conformers of the molecule, as the terminals are situated as far apart as they may possibly be, '''antiperplanar''' to one another.

Once free rotation moves the carbon groups closer together, into the gauche configuration, where the terminals are more poised to react, cope rearrangement would proceed.

The relative orientations of the vinyl groups would ultimately dictate which transition state would be traversed. For instance, if the vectors of the vinyl groups cross, such as the case for the gauche1 conformer, then as the terminals align into a bonding orientaton, the ''chair'' conformation would be adopted by the transition state.

Conversely, if the two vinyl groups lie in a pseudo-parallel fashion, where their vectors do NOT cross, such as is the case for gauche3, then the transition state would be of ''boat'' topography.

'''IRC'''

The Intrinsic Reaction Coordinate looks more cloesly at the potential surface of a reaction and works to pinpoint the highest point in a local area of the potential surface by creeping up the steepest slopes in the immediate area.

This calculation was attempted with 50 points, and did not converge

'''Energy Calculation'''

The two transition states were optimised a final time under DFT B3LYP 6-31G(d) conditions.

The geometries that were outputted were very similar to the previous examples, however

==The Diels-Alder Cycloaddition==

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Diels-Alder Reaction of Butadiene and Ethylene'''
|-
|[[Image:Mb_da3.jpg|200px]]
|}

The Diels-Alder Reaction is an example of a pericyclic cycloaddition, that usually involves the reaction between an electron rich diene and an electron poor dienophile.

The process is described as a [4s + 2s], 6 electron process that forms 2 new σ bonds.

When viewing the molecular orbital picture of butadiene and ethylene, the molecular orbitals are classified as either '''''s'''''(''symmetric'') or '''''a'''''(''antisymmetric'') in relation to the bisecting plane as shown below.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Bisecting Plane for Diels-alder'''
|-
|[[Image:Mb_da2.jpg|200px]]
|}

'''Investigation of Molecular Orbital Symmetries of Butadiene and Ethene'''

Butadiene and Ethylene were optimised and their molecular orbitals visualised:-

Rep:Mod:parkbom

2009-11-13T12:13:30Z

Byt07:

=Module 3 - The Computation of the Transition State=

==Introduction==

The '''Transition State''' is an energetic maximum within a given reaction coordinate and represents the point at which bond breaking and bond forming is occurring simultaneously. This is represented by dotted lines in formal notation and depicts the transient nature of the bond(s) in question in the transition state.

The analysis of potential energy surfaces are effective in finding transition states; the transition states can be described graphically as "dams" that represent energy metastability, that the reactants must "cross" in order to form the products.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Potential Energy Surface Reaction Coordinate'''<ref>http://www.ch.ic.ac.uk/motm/porphyrins/TSdiscovery.html</ref>||'''Cope Rearrangement'''<ref>http://www.ch.ic.ac.uk/local/organic/pericyclic/</ref>
|-
|[[Image:PES.gif]]||[[Image:Cope Arr.gif|350px]]
|}

From the point of view of computational chemistry, the methods of molecular mechanical structure determination that employed imaginary force fields are ineffective in the calculations concerning the transition state in larger molecules, as they do not define the dynamics of bond making and bond forming that are apparent in the transient nature of the transition state.

As such, in combination with molecular orbital based calculations, the following of the potential energy surface of a molecule can be used to locate and analyse the transition state.

This computational exercise concerns the rearrangement of ''1,5-Hexadiene'' in the '''Cope Rearrangment'''(above).

The Cope Rearrangement proceeds via a [3,3]-Sigmatropic shift involving the motion of 6 electrons and is thermally-driven.

In terms of transition states, the cope rearrangement can reportedly proceed via two checkpoints, the '''boat''' or the '''chair''':-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Chair'''||'''Boat'''
|-
|[[Image:pic2a.jpg]]||[[Image:pic2b.jpg]]
|}

The purpose of this exercise is to ascertain which reaction pathway is preferred by computation of each transition state in the elucidation of geometry, repective energies and other thermochemical data.

==''Exercise 1'' - Optimising the Reactants and Products==

In this section, the "''reactant''" '''1,5-hexadiene''' and the "''product''" '''1,5-hexadiene''' are modelled in correspondence to the cope rearrangement.

The flexible nature of 1,5-hexadiene means that there is an initial dilemma of likely conformation at rest.

Several starting conformations were thus taken and their geometries optimised to find the lowest energy conformer.

Initially, 1,5-hexadiene was taken, ensuring a roughly "'''anti'''" linkage in the centre of the molecule, so that the central C atoms '''3''' and '''4''' are approximately '''antiperiplanar''' to one another. This molecule was cleaned in the gaussview builder interface and then taken and optimised using the '''Hartree-Fock''' method with basis set '''3-21G''' through Gaussian.

'''b)''' similarly, a synclinal, or "gauche" version of 1,5-hexadiene was taken and optimised at the HF 3-21G level.

The energies of both conformers and also their symmetry were noted.

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|'''"''Anti''" 1,5-Hexadiene'''||'''"''Gauche''" 1,5-Hexadiene'''
|-align="center"
|[[Image:Bo Anti.jpg|420px]]||[[Image:Bo Gauche.jpg|400px]]
|}

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|'''Confomer'''||'''''Anti''-1,5-Hexadiene'''||'''''Gauche''-1,5-Hexadiene'''
|-align="center"
|'''Total Energy'''(a.u.)||-231.69254||-231.69266
|-align="center"
|'''Point Group'''||ci||None
|}

From a purely steric-orientated argument, the ''anti'' configuration should have a lower energy than the ''gauche'', as there is less steric hindrance from the two ends of the carbon chain 180 degrees from one another than next to each other in the gauche conformation.

As it so happens, the energy for the ''anti'' conformer is reportedly lower than that of the ''gauche'' conformer, but by a minute amount of roughly a ten-thousandth of a hartree ('''1 hartree = 4.3597482E-18 J''').

Even so, it was appreciated that the central C-C-C-C linkage of 1,5-hexadiene lends itself to quite a high degree of rotational freedom, and thus, further examples of conformational isomerism. This is reinforced by the fact of the two remarkably different conformers having remarkably similar energies, and points to the fact that 1,5-hexadiene experiences conformational ambiguity often<ref>Cope Rearrangement of 1,5-Hexadiene: Full Geometry Optimizations Using Analytic MR-CISD and MR-AQCC Gradient Methods, doi:10.1021/jp0259014</ref>.

As such, 2 other conformers were taken and optimised:-

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|colspan=4|'''<big>Different Conformers of 1,5-Hexadiene</big>'''
|-align="center"
|(''g120,a,g120'')-'''Anti1'''||(''g120,a,g-120'')-'''Anti2'''||(''g120,g-60,g120'')-'''Gauche1'''||(''g120,g60,g-120'')-'''Gauche3'''
|-align="center"
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Bo Anti1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Bo Anti2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Gauche 1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Gauche 3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-align="center"
|colspan=4|'''Energy'''(a.u.)
|-align="center"
| -231.69260213||-231.69253506||-231.69166702||-231.69266120
|-align="center"
|colspan=4|'''Point Group'''
|-align="center"
|C2||Ci||C2||C1
|}

The energies here from the HF 3-21G are similarly close, but contrary to what was expected, it was found that ''gauche3'' was actually the lowest in energy.

Beneath this, the ''anti'' conformers are found to be lower in energy than the other ''gauche'' conformer. This may be rationalised as the minimalisation of steric hindrance lowering the energy of the overall molecule, as the antiperiplanar groups maintain the least steric strain.

However, the '''''cis'''''-''gauche'' conformer is lowest in energy due to stereoelectronic arguments.

The '''π'''-bonds of the C=C bonds are lying criss-crossed over each other in this conformer. This promotes effectual π-molecular orbital overlap and hence stabilises the molecule.

Next, the anti2 conformer with ci symmetry and the gauche1 conformer was taken and further optimised under the DFT B3LYP method with a more sophisticated basis set, 6-31G(d).

The resulting optimised molecule was then compared with the earlier attempt in terms of energy.

A further frequency analysis was conducted in order to check that the optimisation has proceeded to a minimum, being the case when all frequencies were real and positive.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=5|'''Comparison of the HF 3-21G and DFT B3LYP 6-31G(d) Optimisations'''
|-align="center"
|'''Method'''||colspan=2|'''HF 3-21G'''||colspan=2|'''DFT B3LYP 6-31G(d)'''
|-align="center"
|'''Conformer'''||'''anti2'''||'''gauche1'''||'''anti2'''||'''gauche1'''
|-align="center"
|'''Total Energy'''(a.u.)||-231.69253506||-231.69166702||-231.61170616||-231.61068821
|-align="center"
|1-2 C=C Bond Length (A)||1.3162||1.3156||1.3335||1.3331
|-align="center"
|2-3 C-C Bond Length (A)||1.5091||1.5041||1.5083||1.5044
|-align="center"
|3-4 C-C Bond Length (A)||1.5527||1.5481||1.5508||1.5482
|-align="center"
|4-5 C-C Bond Length (A)||1.5091||1.5041||1.5083||1.5044
|-align="center"
|5-6 C=C Bond Length (A)||1.3163||1.3335||1.3157||1.3331
|-align="center"
|1-2-3-4 Dihedral (o)||114.66||118.49||123.93||124.24
|-align="center"
|2-3-4-5 Dihedral (o)||180.00||180.00||-64.18||-65.01
|-align="center"
|3-4-5-6 Dihedral (o)||-114.66||-118.49||123.93||124.24
|}

As can be seen here, the energies returned differ in a noticeable manner. However, comparison of energies computed under different basis sets is unbalanced.

As such, comparisons of the geometry are notably more reliable in discerning the computed differences.

The bond lengths differ by negligible amounts. However, it can be seen that the various dihedral angles have been "tightened" more to their ideal figures (closer to 120o for the anticlinal dihedrals.

As such, using a more sophisticated basis set ensures a more accurate approximation to reality.

The log file yielded the following thermochemical data:-

'''1)Sum of electronic and zero-point Energies= -234.469195'''

'''2)Sum of electronic and thermal Energies= -234.461847'''

'''3)Sum of electronic and thermal Enthalpies= -234.460903'''

'''4)Sum of electronic and thermal Free Energies= -234.500782'''

Theses values concern the 1)Potential Energy at 0K that includes the zero-point vibrational energy term 2)Potential Energy at 298.15K and 1atm and adds contributions from translational, rotational and vibrational energy 3)Energy including a correction for RT(H=E+RT) for dissociation reactions and 4)The Entropic contribution.

==''Exercise 2'' - Optimising the "Chair" and "Boat" Transition Structures==

The Cope Rearrangement has been found to proceed via one of two possible transition states with differing conformation, as mentioned in the introduction:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=2|'''<big>Cope Rearrangement Transition State Conformations</big>'''
|-align="center"
|''Chair''||''Boat''
|-align="center"
|[[Image:pic2a.jpg]]||[[Image:pic2b.jpg]]
|}

When the rearrangment is thought as one that is concerted and proceeds with synchronicity of bond breaking and formation, these two transition states come to mind.

There is indeed a third type of transition state for this reaction, dubbed the '''''Dewar''''' transition state, that reportedly proceeds via a diradical tight-chair intermediate species that was discovered by Dupuis et al.<ref>The Cope Rearrangement Revisited, DOI:10.1021/ja00026a007</ref>, and is thus NOT concerted.

===The Chair Transition State===

The chair transition state was initally synthesised using two "resonant" '''allyl''' ('''C3H5''') fragments in gaussview, which were optimised under the 3-21G basis set, and subsequently placed in appropriate positions relative to one another to be optimised into the transition state.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=2|'''Allyl Fragment('''C3H5''')'''
|-align="center"
|colspan=2|[[Image:Allyl Frag.jpg|275px]]
|-align="center"
|'''sp3''' Centre Bond Angle (o)||109.5
|-align="center"
|'''Allyl''' C-C Bond Angle (o)||124.3
|-align="center"
|'''C-C''' Bond Length (A)||1.54
|-align="center"
|'''C=C''' Bond Length (A)||1.34
|-align="center"
|'''Allyl C-C''' Bond Length (A)||1.39
|}

The optimised allyl C-C bond angle is greater than the normal sp3 bond angle of '''109.5o''' due to resonance.

More importantly, the allyl C-C bond length is between that of a single bond and a double bond.

This optimised allyl fragment was then taken, duplicated and placed, facing opposite directions, with roughly a 2.2A distance between the terminal carbons of each fragment to mimic the shape of the chair transition state.

Thus, a frequency + optimisation was run that directed gaussian to reach a '''TS(Berny)''', with the elucidation of force constants.

The optimisation was then repeated but this time using the '''"frozen" coordinate method''', effectively fixing the two pairs of ends at a distance of 2.2A.

Clearly, this method requires that the general structure of the transition state in question be known. However, by fixing the positions in such a way, a more accurate optimisation can be done that avoids divergence.

Finally, the optimisation was performed a third time, using a normal guess '''Hessian''' method, which involves calculation of the force constant matrix.

After the three optimisations were complete, the following data were obtained:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=3|'''<big>Optimisation of Chair Transition State</big>'''
|-align="center"
|colspan=3|'''Structure'''
|-align="center"
|'''TS(Berny)'''||'''Frozen Coordinate'''||'''Hessian'''
|-align="center"
||<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Berny.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Frozen.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>red</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Hessian.mol</uploadedFileContents>
</jmolApplet></jmol>
|-align="center"
|colspan=3|'''Energies''' (a.u.)
|-align="center"
| -231.61932||-231.61932||-231.69167
|}

The structures returned from the normal force-constant calculation optimisation and the frozen coordinate method are indistinguishable from first glance, and their energies are virtually identical. However, the structure obtained from the Hessian is different, and so is its energy value.

'''FREQUENCY ANALYSIS'''

Vibration-wise, there was one negative, and thus "imaginary" frequency found with a wavenumber of '''-817.91cm-1'''.

When the stretch was visualised, the following was observed:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Imaginary Stretch at -817.91cm-1'''
|-
|[[Image:Imag Stretch.jpg|300px]]
|}

The appearance of this imaginary frequency resembles the motion that would ensue under the Cope Rearrangment - the action of bond breaking of the C-C σ bond in the middle of the 1,5-hexadiene with the subsequent formation of a new σ bond on across the terminals on the other side.

The fact that there is only '''one''' imaginary frequency value points to the notion that the cope rearrangment through this transition state is ''concerted'', and thus occurs in one ''synchronous'' step.

The number of imaginary frequencies produced from a frequency analysis for the midpoint of a reaction points towards the ''synchronicity'' of the reaction<ref>Cope Rearrangement Revisited, DOI:10.1021/ja00026a007</ref>.

===The Boat Transition State===

For the boat transition state, a different method was utilised, namely the '''QST2 method''', whereby reactant and product are defined, and the transition state is converged from these.

This required the previously optimised '''1,5-hexadiene''' molecule to be taken and numbered along its carbon skeleton.

The molecule was then duplicated and juxtaposed to the intial "reactant" molecule and renumbered in such a way to reflect the reacted, rearragement "product" as shown below:-

{|style="margin:1em auto 1em auto" border="1"
|-
|[[Image:Boat Method.JPG|400px]]
|}

The first instance of optimisation failed to produce an output, and thus the molecule was changed in the following way:-

The dihedral angle of both reactant and product were changed to 0 degrees, such that the two carbon groups either side of C3 and C4 are eclipsed, and the C2-C3-C4 and C3-C4-C5 bond angles were both changed to 100 degrees.

The modified structures were once again optimised under the QST2 method to yield the following structure:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''''QST2'' Optimised Boat Transition State'''
|-
|[[Image:FINAL Boat.jpg|400px]]
|}

'''DISCUSSION'''

Here, it is illustrated that the QST2 method is automated, and is thus very efficient, but is only effective if the shape provided of the reactants and products cloely mirror that of the transition state, otherwise convergence is difficult.

The '''QST3''' method was also attempted <ref>QST Calculation, http://hdl.handle.net/10042/to-2902</ref>, whereby an addition guess of the transition state geometry was implemented. If the guess of the transition state is sound, then the QST3 is by far more accurate than the QST2 method, and does not rely so much on the shape of the reactants and products.

initally, it would seem that in order for the cope rearrangement to occur, to two carbon terminals would be required to be close to one another. This is not the case with the "''anti''" conformers of the molecule, as the terminals are situated as far apart as they may possibly be, '''antiperplanar''' to one another.

Once free rotation moves the carbon groups closer together, into the gauche configuration, where the terminals are more poised to react, cope rearrangement would proceed.

The relative orientations of the vinyl groups would ultimately dictate which transition state would be traversed. For instance, if the vectors of the vinyl groups cross, such as the case for the gauche1 conformer, then as the terminals align into a bonding orientaton, the ''chair'' conformation would be adopted by the transition state.

Conversely, if the two vinyl groups lie in a pseudo-parallel fashion, where their vectors do NOT cross, such as is the case for gauche3, then the transition state would be of ''boat'' topography.

'''IRC'''

The Intrinsic Reaction Coordinate looks more cloesly at the potential surface of a reaction and works to pinpoint the highest point in a local area of the potential surface by creeping up the steepest slopes in the immediate area.

This calculation was attempted with 50 points, and did not converge

'''Energy Calculation'''

The two transition states were optimised a final time under DFT B3LYP 6-31G(d) conditions.

The geometries that were outputted were very similar to the previous examples, however

==The Diels-Alder Cycloaddition==

The Diels-Alder Reaction is an example of a pericyclic cycloaddition, that usually involves the reaction between an electron rich diene and an electron poor dienophile.

The process is described as a [4s + 2s], 6 electron process that forms 2 new σ bonds.

When viewing the molecular orbital picture of butadiene and ethylene, the molecular orbitals are classified as either '''''s'''''(''symmetric'') or '''''a'''''(''antisymmetric'') in relation to the bisecting plane as shown below.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Bisecting Plane for Diels-alder'''
|-
|[[Image:Mb_da2.jpg||200px]]
|}

Rep:Mod:parkbom

2009-11-13T11:59:54Z

Byt07:

=Module 3 - The Computation of the Transition State=

==Introduction==

The '''Transition State''' is an energetic maximum within a given reaction coordinate and represents the point at which bond breaking and bond forming is occurring simultaneously. This is represented by dotted lines in formal notation and depicts the transient nature of the bond(s) in question in the transition state.

The analysis of potential energy surfaces are effective in finding transition states; the transition states can be described graphically as "dams" that represent energy metastability, that the reactants must "cross" in order to form the products.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Potential Energy Surface Reaction Coordinate'''<ref>http://www.ch.ic.ac.uk/motm/porphyrins/TSdiscovery.html</ref>||'''Cope Rearrangement'''<ref>http://www.ch.ic.ac.uk/local/organic/pericyclic/</ref>
|-
|[[Image:PES.gif]]||[[Image:Cope Arr.gif|350px]]
|}

From the point of view of computational chemistry, the methods of molecular mechanical structure determination that employed imaginary force fields are ineffective in the calculations concerning the transition state in larger molecules, as they do not define the dynamics of bond making and bond forming that are apparent in the transient nature of the transition state.

As such, in combination with molecular orbital based calculations, the following of the potential energy surface of a molecule can be used to locate and analyse the transition state.

This computational exercise concerns the rearrangement of ''1,5-Hexadiene'' in the '''Cope Rearrangment'''(above).

The Cope Rearrangement proceeds via a [3,3]-Sigmatropic shift involving the motion of 6 electrons and is thermally-driven.

In terms of transition states, the cope rearrangement can reportedly proceed via two checkpoints, the '''boat''' or the '''chair''':-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Chair'''||'''Boat'''
|-
|[[Image:pic2a.jpg]]||[[Image:pic2b.jpg]]
|}

The purpose of this exercise is to ascertain which reaction pathway is preferred by computation of each transition state in the elucidation of geometry, repective energies and other thermochemical data.

==''Exercise 1'' - Optimising the Reactants and Products==

In this section, the "''reactant''" '''1,5-hexadiene''' and the "''product''" '''1,5-hexadiene''' are modelled in correspondence to the cope rearrangement.

The flexible nature of 1,5-hexadiene means that there is an initial dilemma of likely conformation at rest.

Several starting conformations were thus taken and their geometries optimised to find the lowest energy conformer.

Initially, 1,5-hexadiene was taken, ensuring a roughly "'''anti'''" linkage in the centre of the molecule, so that the central C atoms '''3''' and '''4''' are approximately '''antiperiplanar''' to one another. This molecule was cleaned in the gaussview builder interface and then taken and optimised using the '''Hartree-Fock''' method with basis set '''3-21G''' through Gaussian.

'''b)''' similarly, a synclinal, or "gauche" version of 1,5-hexadiene was taken and optimised at the HF 3-21G level.

The energies of both conformers and also their symmetry were noted.

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|'''"''Anti''" 1,5-Hexadiene'''||'''"''Gauche''" 1,5-Hexadiene'''
|-align="center"
|[[Image:Bo Anti.jpg|420px]]||[[Image:Bo Gauche.jpg|400px]]
|}

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|'''Confomer'''||'''''Anti''-1,5-Hexadiene'''||'''''Gauche''-1,5-Hexadiene'''
|-align="center"
|'''Total Energy'''(a.u.)||-231.69254||-231.69266
|-align="center"
|'''Point Group'''||ci||None
|}

From a purely steric-orientated argument, the ''anti'' configuration should have a lower energy than the ''gauche'', as there is less steric hindrance from the two ends of the carbon chain 180 degrees from one another than next to each other in the gauche conformation.

As it so happens, the energy for the ''anti'' conformer is reportedly lower than that of the ''gauche'' conformer, but by a minute amount of roughly a ten-thousandth of a hartree ('''1 hartree = 4.3597482E-18 J''').

Even so, it was appreciated that the central C-C-C-C linkage of 1,5-hexadiene lends itself to quite a high degree of rotational freedom, and thus, further examples of conformational isomerism. This is reinforced by the fact of the two remarkably different conformers having remarkably similar energies, and points to the fact that 1,5-hexadiene experiences conformational ambiguity often<ref>Cope Rearrangement of 1,5-Hexadiene: Full Geometry Optimizations Using Analytic MR-CISD and MR-AQCC Gradient Methods, doi:10.1021/jp0259014</ref>.

As such, 2 other conformers were taken and optimised:-

{|style="margin:1em auto 1em auto;" border="1"
|-align="center"
|colspan=4|'''<big>Different Conformers of 1,5-Hexadiene</big>'''
|-align="center"
|(''g120,a,g120'')-'''Anti1'''||(''g120,a,g-120'')-'''Anti2'''||(''g120,g-60,g120'')-'''Gauche1'''||(''g120,g60,g-120'')-'''Gauche3'''
|-align="center"
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Bo Anti1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Bo Anti2.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Gauche 1.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Gauche 3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-align="center"
|colspan=4|'''Energy'''(a.u.)
|-align="center"
| -231.69260213||-231.69253506||-231.69166702||-231.69266120
|-align="center"
|colspan=4|'''Point Group'''
|-align="center"
|C2||Ci||C2||C1
|}

The energies here from the HF 3-21G are similarly close, but contrary to what was expected, it was found that ''gauche3'' was actually the lowest in energy.

Beneath this, the ''anti'' conformers are found to be lower in energy than the other ''gauche'' conformer. This may be rationalised as the minimalisation of steric hindrance lowering the energy of the overall molecule, as the antiperiplanar groups maintain the least steric strain.

However, the '''''cis'''''-''gauche'' conformer is lowest in energy due to stereoelectronic arguments.

The '''π'''-bonds of the C=C bonds are lying criss-crossed over each other in this conformer. This promotes effectual π-molecular orbital overlap and hence stabilises the molecule.

Next, the anti2 conformer with ci symmetry and the gauche1 conformer was taken and further optimised under the DFT B3LYP method with a more sophisticated basis set, 6-31G(d).

The resulting optimised molecule was then compared with the earlier attempt in terms of energy.

A further frequency analysis was conducted in order to check that the optimisation has proceeded to a minimum, being the case when all frequencies were real and positive.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=5|'''Comparison of the HF 3-21G and DFT B3LYP 6-31G(d) Optimisations'''
|-align="center"
|'''Method'''||colspan=2|'''HF 3-21G'''||colspan=2|'''DFT B3LYP 6-31G(d)'''
|-align="center"
|'''Conformer'''||'''anti2'''||'''gauche1'''||'''anti2'''||'''gauche1'''
|-align="center"
|'''Total Energy'''(a.u.)||-231.69253506||-231.69166702||-231.61170616||-231.61068821
|-align="center"
|1-2 C=C Bond Length (A)||1.3162||1.3156||1.3335||1.3331
|-align="center"
|2-3 C-C Bond Length (A)||1.5091||1.5041||1.5083||1.5044
|-align="center"
|3-4 C-C Bond Length (A)||1.5527||1.5481||1.5508||1.5482
|-align="center"
|4-5 C-C Bond Length (A)||1.5091||1.5041||1.5083||1.5044
|-align="center"
|5-6 C=C Bond Length (A)||1.3163||1.3335||1.3157||1.3331
|-align="center"
|1-2-3-4 Dihedral (o)||114.66||118.49||123.93||124.24
|-align="center"
|2-3-4-5 Dihedral (o)||180.00||180.00||-64.18||-65.01
|-align="center"
|3-4-5-6 Dihedral (o)||-114.66||-118.49||123.93||124.24
|}

As can be seen here, the energies returned differ in a noticeable manner. However, comparison of energies computed under different basis sets is unbalanced.

As such, comparisons of the geometry are notably more reliable in discerning the computed differences.

The bond lengths differ by negligible amounts. However, it can be seen that the various dihedral angles have been "tightened" more to their ideal figures (closer to 120o for the anticlinal dihedrals.

As such, using a more sophisticated basis set ensures a more accurate approximation to reality.

The log file yielded the following thermochemical data:-

'''1)Sum of electronic and zero-point Energies= -234.469195'''

'''2)Sum of electronic and thermal Energies= -234.461847'''

'''3)Sum of electronic and thermal Enthalpies= -234.460903'''

'''4)Sum of electronic and thermal Free Energies= -234.500782'''

Theses values concern the 1)Potential Energy at 0K that includes the zero-point vibrational energy term 2)Potential Energy at 298.15K and 1atm and adds contributions from translational, rotational and vibrational energy 3)Energy including a correction for RT(H=E+RT) for dissociation reactions and 4)The Entropic contribution.

==''Exercise 2'' - Optimising the "Chair" and "Boat" Transition Structures==

The Cope Rearrangement has been found to proceed via one of two possible transition states with differing conformation, as mentioned in the introduction:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=2|'''<big>Cope Rearrangement Transition State Conformations</big>'''
|-align="center"
|''Chair''||''Boat''
|-align="center"
|[[Image:pic2a.jpg]]||[[Image:pic2b.jpg]]
|}

When the rearrangment is thought as one that is concerted and proceeds with synchronicity of bond breaking and formation, these two transition states come to mind.

There is indeed a third type of transition state for this reaction, dubbed the '''''Dewar''''' transition state, that reportedly proceeds via a diradical tight-chair intermediate species that was discovered by Dupuis et al.<ref>The Cope Rearrangement Revisited, DOI:10.1021/ja00026a007</ref>, and is thus NOT concerted.

===The Chair Transition State===

The chair transition state was initally synthesised using two "resonant" '''allyl''' ('''C3H5''') fragments in gaussview, which were optimised under the 3-21G basis set, and subsequently placed in appropriate positions relative to one another to be optimised into the transition state.

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=2|'''Allyl Fragment('''C3H5''')'''
|-align="center"
|colspan=2|[[Image:Allyl Frag.jpg|275px]]
|-align="center"
|'''sp3''' Centre Bond Angle (o)||109.5
|-align="center"
|'''Allyl''' C-C Bond Angle (o)||124.3
|-align="center"
|'''C-C''' Bond Length (A)||1.54
|-align="center"
|'''C=C''' Bond Length (A)||1.34
|-align="center"
|'''Allyl C-C''' Bond Length (A)||1.39
|}

The optimised allyl C-C bond angle is greater than the normal sp3 bond angle of '''109.5o''' due to resonance.

More importantly, the allyl C-C bond length is between that of a single bond and a double bond.

This optimised allyl fragment was then taken, duplicated and placed, facing opposite directions, with roughly a 2.2A distance between the terminal carbons of each fragment to mimic the shape of the chair transition state.

Thus, a frequency + optimisation was run that directed gaussian to reach a '''TS(Berny)''', with the elucidation of force constants.

The optimisation was then repeated but this time using the '''"frozen" coordinate method''', effectively fixing the two pairs of ends at a distance of 2.2A.

Clearly, this method requires that the general structure of the transition state in question be known. However, by fixing the positions in such a way, a more accurate optimisation can be done that avoids divergence.

Finally, the optimisation was performed a third time, using a normal guess '''Hessian''' method, which involves calculation of the force constant matrix.

After the three optimisations were complete, the following data were obtained:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|colspan=3|'''<big>Optimisation of Chair Transition State</big>'''
|-align="center"
|colspan=3|'''Structure'''
|-align="center"
|'''TS(Berny)'''||'''Frozen Coordinate'''||'''Hessian'''
|-align="center"
||<jmol><jmolApplet><title>Anti1</title><color>blue</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Berny.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>purple</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Frozen.mol</uploadedFileContents>
</jmolApplet></jmol>
|<jmol><jmolApplet><title>Anti1</title><color>red</color>
<size>150</size><script>zoom 200; cpk -20;</script>
<uploadedFileContents>Hessian.mol</uploadedFileContents>
</jmolApplet></jmol>
|-align="center"
|colspan=3|'''Energies''' (a.u.)
|-align="center"
| -231.61932||-231.61932||-231.69167
|}

The structures returned from the normal force-constant calculation optimisation and the frozen coordinate method are indistinguishable from first glance, and their energies are virtually identical. However, the structure obtained from the Hessian is different, and so is its energy value.

'''FREQUENCY ANALYSIS'''

Vibration-wise, there was one negative, and thus "imaginary" frequency found with a wavenumber of '''-817.91cm-1'''.

When the stretch was visualised, the following was observed:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''Imaginary Stretch at -817.91cm-1'''
|-
|[[Image:Imag Stretch.jpg|300px]]
|}

The appearance of this imaginary frequency resembles the motion that would ensue under the Cope Rearrangment - the action of bond breaking of the C-C σ bond in the middle of the 1,5-hexadiene with the subsequent formation of a new σ bond on across the terminals on the other side.

The fact that there is only '''one''' imaginary frequency value points to the notion that the cope rearrangment through this transition state is ''concerted'', and thus occurs in one ''synchronous'' step.

The number of imaginary frequencies produced from a frequency analysis for the midpoint of a reaction points towards the ''synchronicity'' of the reaction<ref>Cope Rearrangement Revisited, DOI:10.1021/ja00026a007</ref>.

===The Boat Transition State===

For the boat transition state, a different method was utilised, namely the '''QST2 method''', whereby reactant and product are defined, and the transition state is converged from these.

This required the previously optimised '''1,5-hexadiene''' molecule to be taken and numbered along its carbon skeleton.

The molecule was then duplicated and juxtaposed to the intial "reactant" molecule and renumbered in such a way to reflect the reacted, rearragement "product" as shown below:-

{|style="margin:1em auto 1em auto" border="1"
|-
|[[Image:Boat Method.JPG|400px]]
|}

The first instance of optimisation failed to produce an output, and thus the molecule was changed in the following way:-

The dihedral angle of both reactant and product were changed to 0 degrees, such that the two carbon groups either side of C3 and C4 are eclipsed, and the C2-C3-C4 and C3-C4-C5 bond angles were both changed to 100 degrees.

The modified structures were once again optimised under the QST2 method to yield the following structure:-

{|style="margin:1em auto 1em auto" border="1"
|-align="center"
|'''''QST2'' Optimised Boat Transition State'''
|-
|[[Image:FINAL Boat.jpg|400px]]
|}

'''DISCUSSION'''

Here, it is illustrated that the QST2 method is automated, and is thus very efficient, but is only effective if the shape provided of the reactants and products cloely mirror that of the transition state, otherwise convergence is difficult.

The '''QST3''' method was also attempted <ref>QST Calculation, http://hdl.handle.net/10042/to-2902</ref>, whereby an addition guess of the transition state geometry was implemented. If the guess of the transition state is sound, then the QST3 is by far more accurate than the QST2 method, and does not rely so much on the shape of the reactants and products.

initally, it would seem that in order for the cope rearrangement to occur, to two carbon terminals would be required to be close to one another. This is not the case with the "''anti''" conformers of the molecule, as the terminals are situated as far apart as they may possibly be, '''antiperplanar''' to one another.

Once free rotation moves the carbon groups closer together, into the gauche configuration, where the terminals are more poised to react, cope rearrangement would proceed.

The relative orientations of the vinyl groups would ultimately dictate which transition state would be traversed. For instance, if the vectors of the vinyl groups cross, such as the case for the gauche1 conformer, then as the terminals align into a bonding orientaton, the ''chair'' conformation would be adopted by the transition state.

Conversely, if the two vinyl groups lie in a pseudo-parallel fashion, where their vectors do NOT cross, such as is the case for gauche3, then the transition state would be of ''boat'' topography.

'''IRC'''

The Intrinsic Reaction Coordinate looks more cloesly at the potential surface of a reaction and works to pinpoint the highest point in a local area of the potential surface by creeping up the steepest slopes in the immediate area.

This calculation was attempted with 50 points, and did not converge

'''Energy Calculation'''

The two transition states were optimised a final time under DFT B3LYP 6-31G(d) conditions.

The geometries that were outputted were very similar to the previous examples, however

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