==Cope Rearrangement==
The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic cycloaddition. This study will focus on the optimisation reactant, transition state and product structure of the above reaction.

===Optimise Starting Material and Product===
[[File:Gauche-321g.PNG|220px|thumb|left|Optimised Gauche 1,5-hexanediene, C2 symmetry]]
First the starting 1,5-hexadiene with a ‘gauche’ linkage has also be optimised at the same theory level and using the same method. Total Energy was determined to be -231.68771610a.u, which shows agreement with appendix1 <ref name=ic>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3</ref>. Gauche configuration. This molecule has a C2 symmetry.

Then the starting material, 1,5-hexadiene with an ‘anti’ linkage was optimised at HF/3-21G level of theory using Hartree Fock method. Total energy of -231.69253528a.u. was obtained. By comparing to Appendix1<ref name=ic></ref>, it was determined that this structure has the same energy as anti-2 configuration. The point group for this molecule was determined to be Ci.

[[File:Anti2-631g.PNG|220px|thumb|right|Optimised Anti2 1,5-hexanediene, Ci symmetry]]

Reoptimise anti-2 structure at B3LYP/6-31G* level of theory. 6-31G* is a higher level of theory because apart from s and p polarisation, it also involves d-type polarisation on carbon atoms. <ref name="ja00060a048">W.L.Jorgensen, D. Lim, J.F.Blake, "Ab Initio Study of Diels-Alder Reactions of Cyclopentadiene with Ethylene Isoprene, Cyclopentadiene, Acrylonitrile, and Methyl Vinyl Ketone", ''J. Am. Chem. Soc.'', '''1993''', ''115'', 2936-2942.{{DOI|10.1021/ja00060a048}}</ref> Therefore, it is a higher level of theory and would give us more accurate result. Now the energy was shown to be -234.61170458a.u., which is lower than the energy calculated at HF/3-21G* level of theory.The symmetry from both calculations maintained remains the same(Ci). Therefore, there is not much change in the overall geometry. However, slight change in dihededral angle and bond length were noticed.

Some more optimisations have been done and 8 structures out of 10 in Appendix 1<ref name=ic></ref> has been found. The other 2 was not computed due to the shortage of time. 'Gauche3' appears to be the lowest energy conformer. A study carried by Gung and Zhu<ref>B.W.Gong, Z.H.Zhu, "Conformational Study of 1,5-Hexadiene And 1,5-Diene-3,4-Diols", ''J. Am. Chem. Soc.'', '''1995''', ''117'', 1783-1788.{{DOI|10.1021/ja00111a016}}</ref> did the same energy calculation at 6-31G* level of theory and suggested that gauche comformers are in general, lower in Energy. This can be explained by a favourable hyper-conjugation interaction between C-H sigma orbital and C=C pi* orbital that only presents in gauche conformers. Because this trend cannot be observed in the calculation we've done at 3-21G level of theory, it is very likely that 3-21G* theory did not take this interaction into account.

{| border="1" class="wikitable"
|+ Table1. Energy of Other Conformers
!
! Gauche2
! Gauche3
! Gauche4
! Gauche6
! Anti3
! Anti4
|-
! Appearance
| [[File:Gauche2xc.PNG|150px]]
| [[File:Gauche3xc.PNG|150px]]
| [[File:Gauche4xc.PNG|150px]]
| [[File:Gauche6xc.PNG|150px]]
| [[File:Anti3xc.PNG|150px]]
| [[File:Anti4xc.PNG|150px]]
|-
! Energy
| -231.69166701a.u.
| -231.69266122a.u.
| -231.69153035a.u.
| -231.68961573a.u.
| -231.68907066a.u.
| -231.69097055a.u.
|-
!Point Group
| C2
| C1
| C2
| C1
| C2h
| C1
|}

A frequency calculation was then done on the B3LYP/6-31G* optimised anti-2 stucture. It confirms this structure is a minimum because all the vibrational frequencies are real. Thermal energies at different conditions were recorded in the table below.

{| class="wikitable" border="1"
|+ Table2. Summary of Thermalchemistry Data from Frequency Calculation
|-
| Sum of electronic and zero-point Energies || -234.469259 a.u. || 0K, E = Eelec + ZPE
|-
| Sum of electronic and thermal Energies || -234.461965a.u. || 298K, 1atm, E = E + Evib + Erot + Etrans
|-
| Sum of electronic and thermal Enthalpies || -234.461021a.u. || 298K, 1atm, H = E + RT
|-
| Sum of electronic and thermal Free Energies || -234.507881a.u. || 298K, 1atm G = H - TS
|}

===Optimisation of 'Chair' and 'Boat' Transition Structure===
[[File:Chairtsanime.gif|600px|thumb|right|Animation of Transition State Vibration]]
All calculation in this section were done at HF/3-21G level of theory unless otherwise stated.

A fragment of CH2CHCH2 was optimised then pasted on a new file twice. They were arranged in a state that is approximate a chair transition state.

If the guessed structure is very close to the actual transition state structure, a direct optimisation to transition state can be performed. One thing to be noted is whenever we are calculating frequency, ‘Opt=NoEigen’ should be entered in the Additional keyword box to avoid calculation crash. If structure guessed is on a point far away from transition structure, the curvature would be different. If the calculation can be done successfully, it means the guessed structure is a good approximation and actual transition structure could therefore be calculated this way. This calculation takes approx.5 minutes. From frequency calculation, an imaginary frequency of 818.08cm-1 was observed. It represents a negative curvature on the potential energy surface, i.e.a transition state.The animation on the right hand side proves that the transition state is corresponding to a Cope rearrangement.

Otherwise, the transition state is usually optimised using one of these two methods, frozen bond or quadratic synchronous transit(QST).

====Frozen bond Method====
[[File:Chairtsfrozen.png|220px|thumb|right|Frozen Bond]]
[[File:Chairtsnonfrozen.png|220px|thumb|left|Optimised Transition Structure Directly From Guess]]

This can be done by frozen the distance between terminal atoms and optimise the rest of the molecule. The minimum obtained here has a very similar shape as the transition structure calculated directly from reactants as can be seen in the figures below. But there is one significant difference, is that bond forming/breaking distance are fixed.

From here, we unfrozen reaction coordinate and optimise this structure to a transition state.The very similar result as direct optimisation was obtained. The advantage of this method is that it can save time calculating the entire Hessian space.Also, it is much simpler to operate than the next method(QST2) we are going to discuss.

====QST Method====
Common QST methods are QST2 and QST3.

In QST2, we have to know the optimised structures for both starting material and product. One should pay particular attention on the numbering of both reactant and product because they have to be the same in order for the calculation to work.

However, QST2 calculation does not allow rotation around bonds and this is the reason why the calculation would fail if either of the structures provided are not close enough to the transition state. Frequency calculation show an imaginary frequency at -817.90cm-1. This shows although the structure obtained is not the transition state we want, it is still a transition state.

{|style="margin: 0 auto;"
|[[File:Reactantqst2.PNG|220px|thumb|right|Reactant,with atom label]]
|[[File:Productqst2.PNG|220px|thumb|center|Product,with atom label]]
|[[File:Wrongtsqst2.PNG|220px|thumb|left|Wrong Transition State]]
|}

For the specific example we calculated, the problem can be easily fixed by define the dihedral angle between C2-C3-C4-C5 and the inside C2-C3-C4 and C3-C4-C5 angles to make both structure become a closer approximation of the transition state. The computed transition state is confirmed by the observation an imaginary vibrational frequency at 818.37cm-1.

{|style="margin: 0 auto;"
|[[File:Reactantangle.PNG|220px|thumb|right|Reactant, with angle correction]]
|[[File:Productangle.PNG|220px|thumb|center|Product,with angle correction]]
|[[File:Rightts_qst2.PNG|220px|thumb|left|Correct Transition State]]
|}

Another method, known as QST3, requires three structures, reactant, initial transition structure and the product, to do the calculation. Same as QST2, the numbering for all molecules has to be consistent for the calculation to work. The advantage of this calculation is that it quicker and the structural requirement for reactant and product are not as strict. A calculation using QST3 was not performed due to the shortage of time.

====Intrinsic Reaction Coordiante====
Predict Reactant and product from a transition state can be done using a method called Intrinsic Reaction Coordinate(IRC). In this exercise, because the transition state structure is symmetrical, only forward IRC was done. Force constant can be calculated at every step to give us thermochemistry information. 50 points were calculated along the IRC.

[[File:IRC!!!!!!!!!!!!!!!.PNG|400px]]

The resulting structure is closest to a Gauche2 structure(Energy -231.691382a.u.) but clearly, it has not reached a minimum yet. There are three ways to optimised this structure. The simplest method is just run a minimum optimisation on this structure.This requires the structure to be very close to the desired local minimum. Because all ten 1,5-hexdiene conformers are very close in energy(i.e.They are close to each other on the potential energy surface). Here, the optimisation gives gauche2(Energy -231.6916704a.u.). Repeating IRC with more steps is another solution, but too many steps can let it go across the local minimum and ending up in a wrong minimum structure or even somewhere in between. When it was set for 100 steps, instead of Gauche2, Gauche4 structure was obtained. It is also possible to run IRC and compute force constant at every step. This is the most accurate method, but it can be very time consuming and not always possible for large, complex system. This was not performed due to the shortage of time.

====Activation Energy====
Chair Transition State(obtained by frozen bond method) and Boat Transition State(Obtained by QST2) at HF/3-21G* level of theory were re-optimised at DFT/6-31G* level of theory. Thermochemistry data was obtained by operating a frequency(DFT/6-31G*) on the re-optimised structures.

{| border="1" class="wikitable"
|+ Table3. Summary of Energies
!
! HF/3-21G
! HF/3-21G
! HF/3-21G
! B3LYP/6-31G*
! B3LYP/6-31G*
! B3LYP/6-31G*
|-
!
! Electronic Energy
! Sum of Electronic and Zero-Point Energies
! Sum of Electronic and Thermal Energies
! Electronic Energy
! Sum of Electronic and Zero-Point Energies
! Sum of Electronic and Thermal Energies
|-
!
!
! at 0K
! at 298K
!
! at 0K
! at 298K
|-
! Chair TS
| 1 || 2 || 3 || 4 || 5 || 6 ||
|-
! Boat TS
| 1 || 2 || 3 || 4 || 5 || 6 ||
|-
! Anti2
| 1 || 2 || 3 || 4 || 5 || 6 ||
|}

Activation energy was calculated using Anti2 as reactant molecule therefore the result obtained can be compared with appendix1.

{| border="1" class="wikitable"
|+ Table4. Summary of Activation Energies(In kcal/mol)
!
! HF/3-21G
! HF/3-21G
! B3LYP/6-31G*
! B3LYP/6-31G*
! Expt.
|-
!
! at 0K
! at 298K
! at 0K
! at 298K
! at 0K
|-
! EactChair |
| 45.70148153 || 44.6924572 || 34.0643162 || 34.1755895 || 33.5+/-0.5
|-
! EactBoat
| 55.60350142 || 54.7598723 || 41.9580267 || 41.32519426 || 44.7+/-0.5
|}

It can be seen that activation energy calculated at 6-31G* level of theory is closer to experimental data.

==Diels Alder Cycloaddition==
===Optimisation of cis-butadiene===
The structure of cis-butandiene is optimised to a minimum before we view its HOMO and LUMO molecular orbital.HOMO MO is antisymmetric with respect to the plane while LUMO MO is symmetric.
{|style="margin: 0 auto;"
|[[File:Butandienehomo.PNG|220px|thumb|right|HOMO MO Antisymmetric]]
|[[File:Butandienelumo.PNG|220px|thumb|center|LUMO MO Symmetric]]
|}

===Transition state===

[[File:DAiianime.gif|1000px|thumb|right|Animation of Transition State Vibration]]
The first temptation to optimise butadiene, ethylene cycloaddition transition state was direct optimisation. However,the calculation failed. This means that the structure I generated was far away from the actual transition structure. Then frozen bond method was used and a transition state was obtained. This is confirmed by the observation of an imaginary frequency at -832.01cm-1. The transition state structure belongs to the point group Cs, the new partly formed sigma C-C bond's bond length is 2.11962A.

[[File:Lowestpositivefrequency.gif|600px|thumb|left|Vibration corresponds to the lowest positive frequency]]

Typical sp3-sp3 and sp2=sp2 C-C bondlengths are 1.53A a and 1.45A respectively.<ref name="jaP298700000S1">F.H.Allen, O.Kennard, D.G.Watson, "Tables of Bond Lengths determined by X-ray and Neutron Diffraction. Part1. Bond Lengths in Organic Compounds", ''J. Am. Chem. Soc.PERKIN TRANS.'', '''1987''', ''ii'', s1-s19.{{DOI|10.1039/P298700000S1}}</ref> The van der Waals radius of the C atom is 1.7A<ref name="ja953141">R.S.Rowland, R.Taylor "Intermolecular nonbonded contact distances expected from van der Waals radii", ''J. Phys. Chem.'', '''1996''', ''100(18)'', 7384-7391.{{DOI|10.1021/jp953141}}</ref> Although the partly formed sigma C-C bond is longer than sp3-sp3 C-C single bond, but it is within twice of the carbon van der Waals radii(3.4A). This means that there is an attractive interaction between those two carbon atoms.This attraction is likely to be the main driving force for the transition state-product transition.

Animation on the right hand side shows the transition state found corresponds to Diels-Alder transition state. It also shows that the formation of those two bonds are sychronous.The lowest positive frequency(146.94cm-1) corresponds to a rotation mode of ethlyene and cis-butandiene.This vibration is in the direction perpendicular to the reaction coordinate. Therefore, it does not contribute to the structure of transition state.

HOMO and LUMO of the transition state are shown in the figure below.

{|style="margin: 0 auto;"
|[[File:DA2homo.PNG|220px|thumb|right|HOMO MO Antisymmetric]]
|[[File:DA2lumo.PNG|220px|thumb|center|LUMO MO Symmetric]]
|}

Compare to the HOMO and LUMO MO of cis-butandiene, we can conclude that the HOMO of this transition structure is the combination of cis-butandiene HOMO and ethylene LUMO; LUMO of this transition state is the combination of cis-butandiene LUMO and ethylene HOMO. From the figure above, good overlap between molecular orbitals can be observed. This means the reaction is allowed.

The selection rule of percyclic reaction was discovered bDewar and Zimmerma independently<ref name="percyclic">H.Rezepa "http://www.ch.imperial.ac.uk/local/organic/pericyclic/p1_rules.html'{{DOI|10042.a3uxp}}</ref> and it is summarised in the table below.

{| border="1" class="wikitable"
|+ Table2. Pericyclic Reaction Selection Rule<ref name="percyclic"></ref>
! Condition
! Electron Count
! Stereochemistry
! Topology
|-
! Heat
| 4n+2 || Suprafacial || Hückel
|-
! Light
| 4n
| Suprafacial
| Hückel
|-
! Heat
| 4n
| Odd antarafacial
| Möbius
|-
! Light
| 4n+2
| Odd antarafacial
| Möbius
|}

Since the condition for Diels-Alder reaction is thermal and involves only suprafacial components, the electron count must be 4n+2 to ensure the product is of the same electronic excitation as the reactants. <ref name="percyclic"></ref> This reaction is a [4n+2] cycloaddition, which full fills the electron count condition. Therefore, it is thermally allowed provided the fact that interacting MOs of the reactants are close in energy.

===Regioselectivity===
The gap between interacting HOMO(diene) and LUMO(dienophile) MO can be narrowed by using suitable reactants with substituents. Electronwithdrawing groups can lower the LUMO in dienophile and Electrondonating groups can rises the HOMO in diene. A smaller energy gap leads to a lower activation energy and therefore, a more facile reaction.

Both transition states were calculated using frozen bond method. The distance between the bond forming carbons were set to be 3A to optimise the structure to a minimum. Afterwards, we unfrozen reaction coordinate and optimise this structure to a transition state. Optimised bond forming distance for endo transition state was determined to be 2.17045A. An imaginary frequency at -812.21cm-1 confirms the fact that a transition structure was obtained. For endo transition state, bond forming distance of 2.16239A and an imaginary frequency at -806.40cm-1 was recorded. The relative energy for endo and exo were calculated to be -0.05150480a.u. and -0.05041985a.u. respectively.

[[File:Exosteric.PNG|220px|left|thumb|Exo isomer: Steric Clash]]

In order to form endo product, carbonyl groups of maleic anhydride and -CH=CH-CH=CH- of cyclopentadiene must be syn to the reaction coordinate, while in the exo product, they should be in an anti arrangement.

In the exo isomer, the -CH2CH2- bridge staggeres the anhydride ring causing more steric hindrance. <ref>Organic Chemistry (2nd ed., J. Clayden, N. Greeves and S. Warren) - pages 880 and 886-888</ref>
This steric clash also affected the bond forming distance in the transition state. In endo isomer this distance is 0.00806A shower than in the exo isomer. Apart from steric effect, endo isomer is also stabilised by a phenomena called secondary orbital orverlap. Secondary orbital overlap, defined as a positive overlap of a non-interactive frame in the frontier molecular orbitals of a pericyclic reaction. This is only possible for endo isomer as demonstrated in the diagram below.<ref name=fleming>{{cite book
| last = Fleming
| first = Ian
| authorlink = Ian Fleming (chemist)
| title = Frontier Orbitals and Organic Chemical Reactions
| publisher = Wiley
| year = 1978
| location = London
| pages = 106–109
| isbn = 0-471-01819-8}}
</ref>Although exo transition state is more stained and not stabilised by secondary orbital overlap, it is still lower in energy and therefore, exo product would be the thermaldynamic product.

{|style="margin: 0 auto;"
|[[File:Endo1st2ndoverlap.PNG|400px|centre|thumb|Endo Frontier MO interaction]]
|[[File:1stoverlap.PNG|400px|centre|thumb|Exo Frontier MO interaction]]
|}

One thing to be noted is that this interaction is not affected by the orbitals which take part in bonding. And it would not be affected by other intermolecular interaction(i.e. steric hindrance) neither.<ref name=fleming></ref>

HOMO and LUMO MO of both endo and exo transition states are shown below.According to the discussion above, we would expect to see an orbital overlap between carbonyl carbons and diene backbone. However, the orbital diagrams did not show an interaction as such. The reason of this may be the limitation of the base set(AM1) we are using. AM1 belongs to a family called NDDO(Neglect of Diatomic Differential Overlap),in which overlap matrix is replaced by a unit matrix. . In these methods, overlapping of atomic orbitals of different elements are ignored. Although interaction with hydrogen was considered, there is no correction regarding to other atoms. Therefore, oxygen atomic orbital was not take account into the MO calculation. As can be seen in the MOs below, in any case, it appears no overlap between oxygen electron density with other part of MO.

{|style="margin: 0 auto;"
|[[File:DA3endohomo.PNG|220px|centre|thumb|Endo HOMO MO Antisymmetric]]
|[[File:DA3endolumo.PNG|220px|centre|thumb|Endo LUMO MO Antisymmetric]]
|[[File:DA3exohomo.PNG|220px|centre|thumb|Exo HOMO MO Antisymmetric]]
|[[File:DA3exolumo.PNG|220px|centre|thumb|Exo LUMO MO Antisymmetric]]
|}

==References==

<references/>

2014-12-07T22:29:03Z

Xc1412: /* Optimise Starting Material and Product */

==Cope Rearrangement==
The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic cycloaddition. This study will focus on the optimisation reactant, transition state and product structure of the above reaction.

===Optimise Starting Material and Product===
[[File:Gauche-321g.PNG|220px|thumb|left|Optimised Gauche 1,5-hexanediene, C2 symmetry]]
First the starting 1,5-hexadiene with a ‘gauche’ linkage has also be optimised at the same theory level and using the same method. Total Energy was determined to be -231.68771610a.u, which shows agreement with appendix1 <ref>https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3</ref>. Gauche configuration. This molecule has a C2 symmetry.

Then the starting material, 1,5-hexadiene with an ‘anti’ linkage was optimised at HF/3-21G level of theory using Hartree Fock method. Total energy of -231.69253528a.u. was obtained. By comparing to Appendix1, it was determined that this structure has the same energy as anti-2 configuration. The point group for this molecule was determined to be Ci.

[[File:Anti2-631g.PNG|220px|thumb|right|Optimised Anti2 1,5-hexanediene, Ci symmetry]]

Reoptimise anti-2 structure at B3LYP/6-31G* level of theory. 6-31G* is a higher level of theory because apart from s and p polarisation, it also involves d-type polarisation on carbon atoms. <ref name="ja00060a048">W.L.Jorgensen, D. Lim, J.F.Blake, "Ab Initio Study of Diels-Alder Reactions of Cyclopentadiene with Ethylene Isoprene, Cyclopentadiene, Acrylonitrile, and Methyl Vinyl Ketone", ''J. Am. Chem. Soc.'', '''1993''', ''115'', 2936-2942.{{DOI|10.1021/ja00060a048}}</ref> Therefore, it is a higher level of theory and would give us more accurate result. Now the energy was shown to be -234.55970458a.u., which is lower than the energy calculated at HF/3-21G* level of theory.The symmetry from both calculations maintained remains the same(Ci). Therefore, there is not much change in the overall geometry. However, slight change in dihededral angle and bond length were noticed.

Some more optimisations have been done and 8 structures out of 10 in Appendix 1 has been found. The other 2was not computed due to the shortage of time. 'Gauche3'and 'Anti2' are the lowest energy gauche and anti conformers respectively. Their energy will be used to calculate activation energy in the next few steps. A study carried by Gung and Zhu<ref>B.W.Gong, Z.H.Zhu, "Conformational Study of 1,5-Hexadiene And 1,5-Diene-3,4-Diols", ''J. Am. Chem. Soc.'', '''1995''', ''117'', 1783-1788.{{DOI|10.1021/ja00111a016}}</ref> did the same energy calculation at 6-31G* level of theory and suggested that gauche comformers are in general, lower in Energy. This can be explained by a favourable hyper-conjugation interaction between C-H sigma orbital and C=C pi* orbital that only presents in gauche conformers. Because this trend cannot be observed in the calculation we've done at 3-21G level of theory, it is very likely that 3-21G* theory did not take this interaction into account.

{| border="1" class="wikitable"
|+ Table1. Energy of Other Conformers
!
! Gauche2
! Gauche3
! Gauche4
! Gauche6
! Anti3
! Anti4
|-
! Appearance
| [[File:Gauche2xc.PNG|150px]]
| [[File:Gauche3xc.PNG|150px]]
| [[File:Gauche4xc.PNG|150px]]
| [[File:Gauche6xc.PNG|150px]]
| [[File:Anti3xc.PNG|150px]]
| [[File:Anti4xc.PNG|150px]]
|-
! Energy
| -231.69166701a.u.
| -231.69266122a.u.
| -231.69153035a.u.
| -231.68961573a.u.
| -231.68907066a.u.
| -231.69097055a.u.
|-
!Point Group
| C2
| C1
| C2
| C1
| C2h
| C1
|}

A frequency calculation was then done on the B3LYP/6-31G* optimised anti-2 stucture. It confirms this structure is a minimum because all the vibrational frequencies are real. Thermal energies at different conditions were recorded in the table below.

{| class="wikitable" border="1"
|+ Table2. Summary of Thermalchemistry Data from Frequency Calculation
|-
| Sum of electronic and zero-point Energies || -234.416259a.u. || 0K, E = Eelec + ZPE
|-
| Sum of electronic and thermal Energies || -234.408965a.u. || 298K, 1atm, E = E + Evib + Erot + Etrans
|-
| Sum of electronic and thermal Enthalpies || -234.408021a.u. || H = E + RT
|-
| Sum of electronic and thermal Free Energies || -234.447881a.u. || G = H - TS
|}

===Optimisation of 'Chair' and 'Boat' Transition Structure===
[[File:Chairtsanime.gif|600px|thumb|right|Animation of Transition State Vibration]]
All calculation in this section were done at HF/3-21G level of theory unless otherwise stated.

A fragment of CH2CHCH2 was optimised then pasted on a new file twice. They were arranged in a state that is approximate a chair transition state.

If the guessed structure is very close to the actual transition state structure, a direct optimisation to transition state can be performed. One thing to be noted is whenever we are calculating frequency, ‘Opt=NoEigen’ should be entered in the Additional keyword box to avoid calculation crash. If structure guessed is on a point far away from transition structure, the curvature would be different. If the calculation can be done successfully, it means the guessed structure is a good approximation and actual transition structure could therefore be calculated this way. This calculation takes approx.5 minutes. From frequency calculation, an imaginary frequency of 818.08cm-1 was observed. It represents a negative curvature on the potential energy surface, i.e.a transition state.The animation on the right hand side proves that the transition state is corresponding to a Cope rearrangement.

Otherwise, the transition state is usually optimised using one of these two methods, frozen bond or quadratic synchronous transit(QST).

====Frozen bond Method====
[[File:Chairtsfrozen.png|220px|thumb|right|Frozen Bond]]
[[File:Chairtsnonfrozen.png|220px|thumb|left|Optimised Transition Structure Directly From Guess]]

This can be done by frozen the distance between terminal atoms and optimise the rest of the molecule. The minimum obtained here has a very similar shape as the transition structure calculated directly from reactants as can be seen in the figures below. But there is one significant difference, is that bond forming/breaking distance are fixed.

From here, we unfrozen reaction coordinate and optimise this structure to a transition state.The very similar result as direct optimisation was obtained.(Energy Summary in Table2, in 'Activation Energy Section') The advantage of this method is that it can save time calculating the entire Hessian space.Also, it is much simpler to operate than the next method(QST2) we are going to discuss.

====QST Method====
Common QST methods are QST2 and QST3.

In QST2, we have to know the optimised structures for both starting material and product. One should pay particular attention on the numbering of both reactant and product because they have to be the same in order for the calculation to work.

However, QST2 calculation does not allow rotation around bonds and this is the reason why the calculation would fail if either of the structures provided are not close enough to the transition state. Frequency calculation show an imaginary frequency at -817.90cm-1. This shows although the structure obtained is not the transition state we want, it is still a transition state.

{|style="margin: 0 auto;"
|[[File:Reactantqst2.PNG|220px|thumb|right|Reactant,with atom label]]
|[[File:Productqst2.PNG|220px|thumb|center|Product,with atom label]]
|[[File:Wrongtsqst2.PNG|220px|thumb|left|Wrong Transition State]]
|}

For the specific example we calculated, the problem can be easily fixed by define the dihedral angle between C2-C3-C4-C5 and the inside C2-C3-C4 and C3-C4-C5 angles to make both structure become a closer approximation of the transition state. The computed transition state is confirmed by the observation an imaginary vibrational frequency at 818.37cm-1.

{|style="margin: 0 auto;"
|[[File:Reactantangle.PNG|220px|thumb|right|Reactant, with angle correction]]
|[[File:Productangle.PNG|220px|thumb|center|Product,with angle correction]]
|[[File:Rightts_qst2.PNG|220px|thumb|left|Correct Transition State]]
|}

Another method, known as QST3, requires three structures, reactant, initial transition structure and the product, to do the calculation. Same as QST2, the numbering for all molecules has to be consistent for the calculation to work. The advantage of this calculation is that it quicker and the structural requirement for reactant and product are not as strict. A calculation using QST3 was not performed due to the shortage of time.

====Intrinsic Reaction Coordiante====
Predict Reactant and product from a transition state can be done using a method called Intrinsic Reaction Coordinate(IRC). In this exercise, because the reaction coordination is symmetrical, only compute to one direction would give us enough information about the whole reaction. Force constant can be calculated at every step to give us information regards thermal energy. 50 points were calculated along the IRC.

[[File:IRC!!!!!!!!!!!!!!!.PNG|600px]]

The resulting structure is close to a ….. Because all gauche conformations are very close in energy, a simple optimise to minimum calculation of this resulting structure may resulting in a wrong minimum. Which is the case here, the optimisation gives gauche… instead of gauche … Repeating IRC with more steps is another solution, but still, it can go across local minimum and ending up in another minimum structure or somewhere in between. It is also possible to run IRC and compute force constant at every step.Although it is accurate, it can be very time consuming and not always possible for large, complex system.

====Activation Energy====

==Diels Alder Cycloaddition==
===Optimisation of cis-butadiene===
The structure of cis-butandiene is optimised to a minimum before we view its HOMO and LUMO molecular orbital.HOMO MO is antisymmetric with respect to the plane while LUMO MO is symmetric.
{|style="margin: 0 auto;"
|[[File:Butandienehomo.PNG|220px|thumb|right|HOMO MO Antisymmetric]]
|[[File:Butandienelumo.PNG|220px|thumb|center|LUMO MO Symmetric]]
|}

===Transition state===

[[File:DAiianime.gif|1000px|thumb|right|Animation of Transition State Vibration]]
The first temptation to optimise butadiene, ethylene cycloaddition transition state was direct optimisation. However,the calculation failed. This means that the structure I generated was far away from the actual transition structure. Then frozen bond method was used and a transition state was obtained. This is confirmed by the observation of an imaginary frequency at -832.01cm-1. The transition state structure belongs to the point group Cs, the new partly formed sigma C-C bond's bond length is 2.11962A.

[[File:Lowestpositivefrequency.gif|600px|thumb|left|Vibration corresponds to the lowest positive frequency]]

Typical sp3-sp3 and sp2=sp2 C-C bondlengths are 1.53A a and 1.45A respectively.<ref name="jaP298700000S1">F.H.Allen, O.Kennard, D.G.Watson, "Tables of Bond Lengths determined by X-ray and Neutron Diffraction. Part1. Bond Lengths in Organic Compounds", ''J. Am. Chem. Soc.PERKIN TRANS.'', '''1987''', ''ii'', s1-s19.{{DOI|10.1039/P298700000S1}}</ref> The van der Waals radius of the C atom is 1.7A<ref name="ja953141">R.S.Rowland, R.Taylor "Intermolecular nonbonded contact distances expected from van der Waals radii", ''J. Phys. Chem.'', '''1996''', ''100(18)'', 7384-7391.{{DOI|10.1021/jp953141}}</ref> Although the partly formed sigma C-C bond is longer than sp3-sp3 C-C single bond, but it is within twice of the carbon van der Waals radii(3.4A). This means that there is an attractive interaction between those two carbon atoms.This attraction is likely to be the main driving force for the transition state-product transition.

Animation on the right hand side shows the transition state found corresponds to Diels-Alder transition state. It also shows that the formation of those two bonds are sychronous.The lowest positive frequency(146.94cm-1) corresponds to a rotation mode of ethlyene and cis-butandiene.This vibration is in the direction perpendicular to the reaction coordinate. Therefore, it does not contribute to the structure of transition state.

HOMO and LUMO of the transition state are shown in the figure below.

{|style="margin: 0 auto;"
|[[File:DA2homo.PNG|220px|thumb|right|HOMO MO Antisymmetric]]
|[[File:DA2lumo.PNG|220px|thumb|center|LUMO MO Symmetric]]
|}

Compare to the HOMO and LUMO MO of cis-butandiene, we can conclude that the HOMO of this transition structure is the combination of cis-butandiene HOMO and ethylene LUMO; LUMO of this transition state is the combination of cis-butandiene LUMO and ethylene HOMO. From the figure above, good overlap between molecular orbitals can be observed. This means the reaction is allowed.

The selection rule of percyclic reaction was discovered bDewar and Zimmerma independently<ref name="percyclic">H.Rezepa "http://www.ch.imperial.ac.uk/local/organic/pericyclic/p1_rules.html'{{DOI|10042.a3uxp}}</ref> and it is summarised in the table below.

{| border="1" class="wikitable"
|+ Table2. Pericyclic Reaction Selection Rule<ref name="percyclic"></ref>
! Condition
! Electron Count
! Stereochemistry
! Topology
|-
! Heat
| 4n+2 || Suprafacial || Hückel
|-
! Light
| 4n
| Suprafacial
| Hückel
|-
! Heat
| 4n
| Odd antarafacial
| Möbius
|-
! Light
| 4n+2
| Odd antarafacial
| Möbius
|}

Since the condition for Diels-Alder reaction is thermal and involves only suprafacial components, the electron count must be 4n+2 to ensure the product is of the same electronic excitation as the reactants. <ref name="percyclic"></ref> This reaction is a [4n+2] cycloaddition, which full fills the electron count condition. Therefore, it is thermally allowed provided the fact that interacting MOs of the reactants are close in energy.

===Regioselectivity===
The gap between interacting HOMO(diene) and LUMO(dienophile) MO can be narrowed by using suitable reactants with substituents. Electronwithdrawing groups can lower the LUMO in dienophile and Electrondonating groups can rises the HOMO in diene. A smaller energy gap leads to a lower activation energy and therefore, a more facile reaction.

Both transition states were calculated using frozen bond method. The distance between the bond forming carbons were set to be 3A to optimise the structure to a minimum. Afterwards, we unfrozen reaction coordinate and optimise this structure to a transition state. Optimised bond forming distance for endo transition state was determined to be 2.17045A. An imaginary frequency at -812.21cm-1 confirms the fact that a transition structure was obtained. For endo transition state, bond forming distance of 2.16239A and an imaginary frequency at -806.40cm-1 was recorded. The relative energy for endo and exo were calculated to be -0.05150480a.u. and -0.05041985a.u. respectively.

[[File:Exosteric.PNG|220px|left|thumb|Exo isomer: Steric Clash]]

In order to form endo product, carbonyl groups of maleic anhydride and -CH=CH-CH=CH- of cyclopentadiene must be syn to the reaction coordinate, while in the exo product, they should be in an anti arrangement.

In the exo isomer, the -CH2CH2- bridge staggeres the anhydride ring causing more steric hindrance. <ref>Organic Chemistry (2nd ed., J. Clayden, N. Greeves and S. Warren) - pages 880 and 886-888</ref>
This steric clash also affected the bond forming distance in the transition state. In endo isomer this distance is 0.00806A shower than in the exo isomer. Apart from steric effect, endo isomer is also stabilised by a phenomena called secondary orbital orverlap. Secondary orbital overlap, defined as a positive overlap of a non-interactive frame in the frontier molecular orbitals of a pericyclic reaction. This is only possible for endo isomer as demonstrated in the diagram below.<ref name=fleming>{{cite book
| last = Fleming
| first = Ian
| authorlink = Ian Fleming (chemist)
| title = Frontier Orbitals and Organic Chemical Reactions
| publisher = Wiley
| year = 1978
| location = London
| pages = 106–109
| isbn = 0-471-01819-8}}
</ref>Although exo transition state is more stained and not stabilised by secondary orbital overlap, it is still lower in energy and therefore, exo product would be the thermaldynamic product.

{|style="margin: 0 auto;"
|[[File:Endo1st2ndoverlap.PNG|400px|centre|thumb|Endo Frontier MO interaction]]
|[[File:1stoverlap.PNG|400px|centre|thumb|Exo Frontier MO interaction]]
|}

One thing to be noted is that this interaction is not affected by the orbitals which take part in bonding. And it would not be affected by other intermolecular interaction(i.e. steric hindrance) neither.<ref name=fleming></ref>

HOMO and LUMO MO of both endo and exo transition states are shown below.According to the discussion above, we would expect to see an orbital overlap between carbonyl carbons and diene backbone. However, the orbital diagrams did not show an interaction as such. The reason of this may be the limitation of the base set(AM1) we are using. AM1 belongs to a family called NDDO(Neglect of Diatomic Differential Overlap),in which overlap matrix is replaced by a unit matrix. . In these methods, overlapping of atomic orbitals of different elements are ignored. Although interaction with hydrogen was considered, there is no correction regarding to other atoms. Therefore, oxygen atomic orbital was not take account into the MO calculation. As can be seen in the MOs below, in any case, it appears no overlap between oxygen electron density with other part of MO.

{|style="margin: 0 auto;"
|[[File:DA3endohomo.PNG|220px|centre|thumb|Endo HOMO MO Antisymmetric]]
|[[File:DA3endolumo.PNG|220px|centre|thumb|Endo LUMO MO Antisymmetric]]
|[[File:DA3exohomo.PNG|220px|centre|thumb|Exo HOMO MO Antisymmetric]]
|[[File:DA3exolumo.PNG|220px|centre|thumb|Exo LUMO MO Antisymmetric]]
|}

==References==

<references/>