Rep:Mod:hlj298

2015-12-17T08:16:27Z

Yll113: /* Ethylene and Cis-Butadiene */

== '''Study of the reaction profiles of the Cope Rearrangement and the Diels-Alder Cycloadditions''' ==
'''Y. L. J. Lam'''

''Department of Chemistry, South Kensington Campus, Imperial College London, London SW7 2AZ, United Kingdom''

''Received 18 December, 2015''

=== Abstract ===
The reactants, products and transition states of the Cope
Rearrangement of 1,5-hexadiene were investigated by ''GaussView 5.0'' at the '''HF/3-21G''' and '''B3LYP/6-31G*'''levels
of theories respectively. With that, the point groups, vibrational frequencies and different energies at different temperatures of the reactants, products and transition states were calculated. Also, by optimizing the transition structures with different methods, i.e. computing the force constants at the
beginning of the calculations, using the redundant coordinate editor and '''QST2''', at the '''HF/3-21G''' level of theory, closer views of the geometries of the transition states can be observed. Furthermore, by using the '''IRC''' method, the reaction profiles can be
obtained and the activation energies can therefore be calculated. Plus, using '''IRC''' method, all reaction intermediates
can now be observed, which helps us to understand the mechanism of the Cope Rearrangement. Similarly, for Diels-Alder Cycloadditions between ethene and
butadiene and Diels-Alder Cycloadditions between cyclohexa-1,3-diene and maleic anhydride, the reactants, products and transition states were optimized and
their respective energies were calculated at '''AM1 semi-empirical molecular orbital method'''. Furthermore, the symmetries of the
molecular orbitals were visualized and the reaction profiles calculated by '''IRC''' method were obtained.

=== '''Introduction''' ===
Chemical reactions are happening around the world in every second. Some reactions are fast, whilst some are slow. The most common and general reason for that is on the kinetic and thermodynamic aspects. On the kinetic aspect, we might argue that the energy barrier(s) form the reactant(s) to the product(s) is/are huge, and therefore, the reactant(s) cannot overcome the barrier(s) and the reaction is slow or does not proceed. The transformation between crude carbon and diamond is a good example. The energy difference between crude carbon and diamond is just few kcal/mol, however, the energy barrier for the transformation is huge. Hence, the transformation is extremely slow, or even does not proceed. With that, diamond symbolizes eternity. On the other hand, on the thermodynamic aspect, we might argue that the reaction is endothermic, i.e. absorbing/requiring heat from the surroundings in order to proceed. In fact, these two aspects just provide us with a little bit of the story and therefore, chemists, or scientists in general, study the mechanism of the reactions to find out the full story. Unfortunately, some reactions are spontaneous, such as the thiocyanation of the iron complex. Also, some intermediates of the reactions are unstable, which cannot be separated or detected even using very advanced analytical instruments, such as nuclear magnetic resonance (NMR) spectromenter. Therefore, scientists devised some programs and computational methods to find out the mechanism of the reactions. Here we use ''GaussView 5.0'' for our investigation.

==== Computational Theory ====
[[File:Yll113 AM1 and HF.jpg|thumb|463x463px|'''Figure 1.''' HOMO and LUMO (highlighted in yellow) of cis-butadiene under the basis of calculation '''HF/3-21G '''(left) and '''AM1''' (right)]]
In ''GaussView 5.0'', there are numerous methods for calculation, such as '''HF/3-21G''', '''B3LYP/6-31G*''', '''semiempirical AM1''', '''MP4 '''and '''MP2'''. Here, the first two calculation method, namely, '''HF/3-21G''' and '''B3LYP/6-31G* '''were applied for calculation of the Cope Rearrangement Reaction, while '''semiempirical AM1''' was used for the investigation of the two Diels-Alder reactions.

N.B. No matter which method applied, the RMS Gradient Norm in hartress would also be computed. This is a measure of how well does the optimisation go during the calculation of the
structure drawn. The closer to zero, the better the structure is optimised.

===== Hartree-Fock ('''HF''') Method =====
Hartree-Fock theory ('''HF''') is the fundamentals of electronic structure theory. It gives a good starting point for more elaborate theoretical methods which can approximate the electronic Schrödinger equation better. It is the basis of the molecular orbital (MO) theory that assumes the motion of each electron can be described by a single-particle function/orbital and it does not depend on/interact with the instantaneous motions of the other electrons.<ref>C. D. Sherrill, ''An Introduction to Hartree-Fock Molecular Orbital Theory'', 2000</ref>

===== Becke, 3-parameter, Lee-Yeang-Parr ('''B3LYP''') Method =====
Beeke, 3-parameter, Lee-Yang-Parr ('''B3LYP''') is one of the most commonly used hybrid functionals. Hybrid functionals are a class of approximation of the exchange-correlation energy functional in density functional theory.<ref>What is B3LYP?, https://www.quora.com/What-is-B3LYP (accessed December 2015)</ref> '''B3LYP''' contains an '''HF''' exchange with the weight of 0.2, which can be regarded as a uniform screening of
exchange by 80 %.<ref>C. H. Patterson, ''International Journal of Quantum Chemistry'', 2006, '''106 '''(15), 3383</ref> '''B3LYP''' also takes a set of atomization
and ionization energies, proton affinities and total atomic energies into account.<ref>A. D. Becke, ''The Journal of Chemical Physics'', 1993, '''98''', 5648</ref>

===== Semiempirical Austin Model 1 ('''AM1''') =====
Semiempirical Austin Model 1 ('''AM1''') based on the Neglect of Differential Diatomic Overlap (NDDO) integral approximation.<ref>M.
J. S. Dewar, E. G. Zoebisch, E. F. Healy and J. J. P. Stewart, ''J. Am. Chem. Soc.'', 1985, '''107''' (13), 3902</ref> Therefore, when taking the same molecule for '''AM1''' and '''HF/3-21G''', you would find that the numbers of HOMO and LUMO are different, which '''AM1''' gives smaller numbers as shown in Figure 1. This is due to the neglect of the low-lying orbitals overlapping. With that, '''AM1''' proceeds much faster than '''HF/3-21G''' for the sake of time.

==== The Cope Rearrangement ====
The Cope Rearrangement is an organic reaction involving [3,3]-sigmatropic rearrangement of 1,5-dienes, which resembles the Claisen Rearrangement.<ref>A. C. Cope and E. M. Hardy, ''J. Am. Chem. Soc.'', 1940, '''62''' (2), 441</ref> The mechanism of the Rearrangement has sparked a controversy – whether it is concerted, dissociative or stepwise.<ref>O. Wiest, K. A. Black and K. N. Houk, ''J. Am. Chem. Soc.'', 1994, '''116''', 10336</ref> With that, first, each conformer of the reactant, 1,5-hexadiene, was optimised by '''HF/3-21G'''. The lowest energy conformer of 1,5-hexadiene was found. Then, as we know, the Rearrangement undergoes either a chair or boat transition state. So, each transition state was optimised by '''HF/3-21G '''as well. By looking into the energy difference between the transition states and the reactant, the activation energy of the Cope Rearrangement with 1,5-hexadiene was found. In order to find the reaction profile and see how the 1,5-diene rearranges, i.e. the mechanism, the transition state was optimised again with
mainly two methods. The coordinate of the chair transition state was first frozen, with the bond to be made set as 2.20000 Å. 2.20000 Å is a good bond length for partially C-C bond as suggested by the chemists’ observations in the literature. <ref>N. H. Kendall, Y. Li and J. D. Evanseck, ''Angew. Chem. Int. Ed. Engl.'', 1992, '''31''' (6), 682</ref> Then, after the optimization of the frozen coordinate, the partly form 2.20000 Å can be relaxed and the structure was then reoptimised. This methods skips the process of computing the whole force constant matrix i.e. Hessian, which saves time and costs. Furthermore, the boat transition state was optimised by '''QST2'''. '''QST2''' has a higher constrains in which requires a more accurate transition state structure to be put in. In this case, the dihedral angle plays an important role in order to be calculated by ''GaussView'' 5.0. Hence, this method is more expensive and time-consuming. From the optimised transition states, an '''IRC''' can be run for the optimised structure to see the full reaction profile. Also, the intermediates of the reaction can be observed. And finally, the reactant and two transition states
were optimised with '''B3LYP/6-31G*''' similarly. Hence, the two calculation methods can be compared by looking into the numbers obtained. Also, the numbers can be compared against the
experimental values. As explained above, '''B3LYP''' takes a more in-depth consideration, the numbers got from this method should be closer to the reality.

==== The Diels-Alder Cycloaddition ====
The Diels-Alder cycloaddition is a [4+2] cycloaddition between a dienophile and a conjugated alkene to give a cyclohexane system. Here, calculations on two Diels-Alder cycloaddition reactions are reported. They are (1) ethylene and butadiene and (2) cyclohexa-1,3-diene and maleic anhydride.

For Diels-Alder cycloaddition reaction, it is well-known that the reaction gives exo and/or endo product. Exo product implies the reaction pathway is thermodynamically controlled to give more stable product; endo product implies
the reaction pathway is kinetically controlled to give a relatively less stable product. In other words, the activation energy to form the exo product is higher than that of endo, however, the endo product is higher in energy than exo. This can usually be explained by the secondary orbital effects. In our cases, both the exo and endo products were investigated undoubtedly. This time, as you may notice, the molecule is more large in size and there are two reactants instead of just one reactant in the Cope Rearrangement, a simpler method of calculation was implemented, which is '''AM1'''. Also, the electronic distributions and orbitals of the HOMO and LUMO of the transition states were computed and visualised.

=== '''Computational Method''' ===
''All calculations were performed by GaussView 5.0. Relevant JSmol files were uploaded here, however, due to some technique glitches, some bonds, especially double bonds, might not come up properly. Yet, the structures of the molecules are generally correctly shown.''

==== The Cope Rearrangement ====
[[File:Yll113 CR.png|thumb|'''Figure 2.''' The Cope Rearrangement of 1,5-hexadiene]]
An anti and gauche conformation of the 1,5-hexadiene were drawn respectively. The drawn structures were first optimised by a not very accurate technique, i.e. '''Clean'''. Then, the '''clean'''ed structure were optimised by '''HF/3-21G'''. The point group and the energy of each conformer were found and compared to locate the low-energy minima. The optimised structures from '''HF/3-21G''' were then reoptimised by '''B3LYP/6-31G*'''. The point group of each conformer was checked and confirmed. Also, the comparison of the same conformer under different calculation method '''HF/3-21G''' and '''B3LYP/6-31G''' was carried out by looking into energy, bond lengths and bond angles. Furthermore, the '''Thermochemistry''' using job type '''Frequency''' was found in both '''HF/3-21G '''and''' B3LYP/6-31G* '''optimised anti conformers.

The boat and chair transition structures were also drawn and '''clean'''ed. The point group of each transition state was found.

Firstly, the chair transition structure was '''optimised to TS (Berny)''' in '''Opt + Freq '''using the basis of '''HF/3-21G'''. Force constant was calculated '''once'''. The frequency of vibration was checked to make sure there is one imaginary vibrational frequency. Then, '''freeze''' '''coordinate''' of the molecule by freezing the carbon-carbon bond to be made as 2.20000 Å. After that, the frozen coordinate was relaxed so the carbon-carbon bond to be made no longer be restricted to 2.20000 Å. The geometry of the transition state was then compared.

Secondly, at the same time, the boat transition structure was optimised by '''QST2''' method by specifying the reactants and products of the reaction under the basis of '''HF/3-21G'''. Labelling the atoms in
the reactant and product, and adjusting the central '''C-C-C-C '''dihedral angle to 0o plus the two inside '''C-C-C''' angles to 100o, the reactant and product could now be optimised by '''QST2''' method.

Comparing the optimised chair and boat transition structures, the connecting conformer of 1,5-hexadiene was found. The reaction energy profile was then calculated by '''IRC''' under '''HF/3-21G''' with 50 points and force constant as always for every small steps. With that, the mechanism of the reaction, as well as the whole reaction energy profile, could be observed clearly. Take the last point on the '''IRC''' and run a normal '''optimisation''' under '''HF/3-21G''' to obtain a minimized geometry.

Eventually, re'''optimise''' the structures of the two transition states with '''Opt + Freq '''under the basis of '''B3LYP/6-31G*'''. The geometries and energies of the transition structure under two different basis were compared. With that, these computed values were also compared against experimental values.

==== The Diels-Alder Cycloadditions ====
[[File:Yll113DA1.jpg|thumb|'''Figure 3. '''The Diels-Alder Cycloadditions between ethylene and butadiene]]

===== Ethylene and butadiene =====
The structure of cis-butadiene was first optimised under the basis of '''semiempirical AM1'''. The HOMO and LUMO of cis butadiene were visualised and its symmetry was determined.

The transition state of the reaction was drawn and optimised under the basis of '''semiempirical AM1'''. The '''IRC''' was carried out in order to confirm the transition structure postulated is of the correct one. Furthermore, the HOMO of the transition structure was visualised and the nodal
planes and properties of the system were interpreted.

===== Cyclohexa-1,3-diene and maleic anhydride =====
[[File:Yll113DA2.jpg|thumb|'''Figure 4. '''The Diels-Alder Cycloadditions between cyclohexa-1,3-diene and maleic anhydride]]
The transition states of the exo and endo products were drawn and optimised under the basis of '''semiempirical AM1'''. The '''IRC''' was carried out in order to confirm the transition structure postulated is of the correct one. For the number of points, 21 points were used for exo transition states and 24 for endo. This is because the energy was too shallow and the slopes tend to zero after the number of points specified above and ''GaussView 5.0'' cannot predict which direction should it goes on to calculate. Furthermore,
the bond lengths, orientation and the HOMO of the transition structures were investigated.

=== '''Results and Discussion''' ===

==== The Cope Rearrangement ====

===== Optimisation of Reactant =====
1,5-hexadiene has three free rotating carbon-carbon bonds. Each of them has three rotational minima. This gives 27 conformations of the 1,5-hexadiene molecule. Yet, only ten of them were energetically distinct due to symmetry and enantiomeric relationships.<ref>B. G. Rocque, J. M. Gonzales and H. F. Schaefer, ''Molecular Physics'', 2002, '''100''' (4), 441</ref> Two of them, the ''Ci anti ''and ''C1 gauche ''structure in here'' ''were drawn and optimizied as shown in Figure A and B and their energies were calculated as shown in Table 1.
{| class="wikitable"
!Conformation
!Molecule
!Point Group
!Energy/ Hartrees*
HF/3-21G
!RMS Gradient Norm/Hartrees*
HF/3-21G
!Relative Energy#/ kcal/mol
!Newman Projections
|-
|Gauche3
|<jmol>
<jmolApplet>
<title>Figure A: Gauge3 conformer of 1,5-hexadiene</title><color>white</color>
<size>250</size>
<uploadedFileContents> yll113CR_GAUGE_PART1.LOG </uploadedFileContents>
</jmolApplet>
</jmol>
|C1
|<nowiki>-231.69266120</nowiki>
|0.00001556
|0.00
|[[File:Yll113 torsion gauche.jpg|centre|frame|'''Figure 5.''' Newman Projection of C1 gauche3 1,5-hexadiene]]
|-
|Anti2
|<jmol>
<jmolApplet>
<title>Figure B: Anti2 conformer of 1,5-hexadiene</title><color>white</color>
<size>250</size>
<uploadedFileContents>YLL113CR ANTI PART1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|Ci
|<nowiki>-231.69253528</nowiki>
|0.00001891
|0.08
|[[File:Yll113 torsion anti.jpg|centre|frame|'''Figure 6.''' Newman Projection of Ci anti2 1,5-hexadiene]]
|}
<nowiki>*</nowiki>1 hartree = 627.509 kcal/mol

#The difference in energy between the conformer and the lowest energy conformer, in here, which is Gauche3. Then convert Hartree to kcal/mol by *

'''Table 1. '''Conformational analysis of anti2 and gauche3 of 1,5-hexadiene

As shown in Table 1, the energy of Gauche3 is surprisingly lower than the anti2 conformation of 1,5-hexadiene. In most cases, the antiperiplanar conformation of a molecule, such as anti2, is more favourable as it has the least steric clashes. Therefore, usually the antiperiplanar conformation is of the lowest energy. However, here, apart from sterics, the stereoelectroncs concept has also been taken into account. The vinyl proton, in a through space manner, can interact with the π or π* orbital on the sp2 carbon which is separated by four bonds from it.<ref>M. Nishio and M. Hirota, ''Tetrahedron'', 1989, '''45 '''(23), 7201</ref> This is so-called CH-π interaction. The Newman Projection in Figure 5 gives us a closer look on how they are close in space and interact; and the Newman projection in Figure 6 tells us why the vinyl proton cannot interact with the π or π* system through space. Therefore, the gauche3 conformation is more stable than anti2 and of lower energy in 1,5-hexadiene.

Focusing on anti2 conformer of the 1,5-hexadiene, the '''HF/3-21G''' optimised molecule was reoptimised by '''B3LYP/6-31G'''. The energies and the geometries of the anti2 1,5-hexadiene under two basis of calculation method were compared and shown in Table 2.
[[File:Yll113Anti2.png|thumb|'''Figure 7. '''Anti2 conformer of 1,5-hexadiene with labelled atoms]]
{| class="wikitable"
! rowspan="2" |Calculation Method
! rowspan="2" |Point Group
! rowspan="2" |Energy/ Hartrees
! rowspan="2" |RMS Gradient Norm/Hartrees
!Bond length between/ Å
!Bond length between/ Å
!Bond length between/ Å
!Bond angle between
!Bond angle between
|-
|C14-C12 and C4-C1
|C12-C9 and C6-C4
|C6-C9
|C14-C12-C9 and
C1-C4-C6
|C12-C9-C6 and
C4-C6-C9
|-

|'''HF/3-21G'''
|Ci
|<nowiki>-231.69253528</nowiki>
|0.00001891
|1.31613
|1.50891
|1.55275
|124.80579o
|111.34878o
|-
|'''B3LYP/6-31G*'''
|Ci
|<nowiki>-234.61171063</nowiki>
|0.00001249
|1.33350
|1.50419
|1.54816
|125.29968o
|112.67081o
|}
'''Table 2. '''Comparison with the anti2 conformer of 1,5-hexadiene with two calculation method basis ('''HF/3-21G '''and '''B3LYP/6-31G*''')

As seen in Table 2, the point group of the same conformer does not change no matter which method of calculation was carried out. The energy of the conformer with '''B3LYP/6-31G*''' is lower than that of '''HF/3-21G'''. Also, the bond lengths and bond angles changed with a certain amount as shown.

Furthermore, by submitting job type as '''Frequency''' in the optimised structure of anti2 conformer of 1,5-hexadiene under both calculation basis '''HF/3-21G '''and '''B3LYP/6-31G*''', the electronic energy, sum of electronic and zero-point energies at 0 K and sum of electronic and thermal energies at 298.15 K could be determined as shown in Table 3.

{| border="1" cellspacing="1" cellpadding="10"
|-
|rowspan="3"| ''' '''
!colspan="4" |'''HF/3-21G a'''
! colspan="4" |'''B3LYP/6-31G* b'''
|-
! rowspan="2" width="125" align="center" | '''Electronic energy'''
! width="125" align="center" | '''Sum of electronic and zero-point energies'''
! width="125" align="center" | '''Sum of electronic and thermal energies'''
! rowspan="2" width="125" align="center" | '''IR spectrum'''
! rowspan="2" width="125" align="center" | '''Electronic energy'''
! width="125" align="center" | '''Sum of electronic and zero-point energies'''
! width="125" align="center" | '''Sum of electronic and thermal energies'''
! rowspan="2" width="125" align="center" | '''IR spectrum'''
|-
! width="125" align="center" | '''at 0 K'''
! width="125" align="center" | '''at 298.15 K'''
! width="125" align="center" | '''at 0 K'''
! width="125" align="center" | '''at 298.15 K'''
|-
| '''Reactant (anti2)'''
| align="center" | -231.692534
| align="center" |-231.539539
| align="center" | -231.532565
|[[File:Yll113ANTI3-21IR.png|thumb|'''Figure 8. '''IR spectrum of anti2 reactant 1,5-hexadiene calculated by '''HF/3-21G''']]
| align="center" | -234.611703
| align="center" | -234.469213
| align="center" | -234.461857
|[[File:Yll113ANTI6-31IR.png|thumb|'''Figure 9. '''IR spectrum of anti2 reactant 1,5-hexadiene calculated by '''B3LYP/6-31G*''']]
|}

a [https://wiki.ch.ic.ac.uk/wiki/images/5/52/Yll113CR_ANTI_PART4.LOG File]; b [https://wiki.ch.ic.ac.uk/wiki/images/5/54/Yll113_CR_ANTI_PART3.LOG File]

'''Table 3.''' Summary of energies (in hartree) of the reactant (anti2) Comparing Figure 8 and 9, both IR spectra are similar which shows that they have similar stretching/bending mode even under different methods of calculations. However, the strongest peak in Figure 8 is at 1112 cm-1 and that in Figure 9 is 940 cm-1. Yet, they correspond to the same mode of vibration, which is the =C-H bending. Therefore, according to the equation, the wavenumber of absorbance, ν can be calculated:

:<math>\nu = \frac{1}{2 \pi c} \sqrt{\frac{k}{\mu}}</math>

where ''μ'' is the reduced mass, ''k'' is the spring constant for the bond and ''c'' is the light speed,

we can therefore know that the spring constant of the molecule under '''B3LYP/6-31G* '''basis reduced.

Now, focusing on gauche3 conformer of the 1,5-hexadiene, similarly, the '''HF/3-21G''' optimised molecule was reoptimised by '''B3LYP/6-31G'''. The energies and the geometries of the gauche3 1,5-hexadiene under two basis of calculation method were compared and shown in Table 4.
[[File:Yll113Gauche3.png|thumb|'''Figure 10. '''Gauche3 conformer of 1,5-hexadiene with labelled atoms]]
{| class="wikitable"
! rowspan="2" |Calculation Method
! rowspan="2" |Point Group
! rowspan="2" |Energy/ Hartrees
! rowspan="2" |RMS Gradient Norm/Hartrees
! colspan="3" |Bond length between/ Å
! colspan="2" |Bond angle between
|-
|C14-C12 and C4-C1
|C12-C9 and C6-C4
|C6-C9
|C14-C12-C9 and
C1-C4-C6
|C12-C9-C6 and
C4-C6-C9
|-

|'''HF/3-21G'''
|C1
|<nowiki>-231.69266120</nowiki>
|0.00001555
|1.31646
|1.50929
|1.55314
|125.02428o
|111.80728o
|-
|'''B3LYP/6-31G*'''
|C1
|<nowiki>-234.61132605</nowiki>
|0.00000360
|1.33382
|1.50491
|1.55007
|125.49464o
|113.46225o
|}
'''Table 4. '''Comparison with the anti2 conformer of 1,5-hexadiene with two calculation method basis ('''HF/3-21G '''and '''B3LYP/6-31G*''')

As seen in Table 4, the point group of the same conformer, again, does not change no matter which method of calculation was carried out. The energy of the conformer with '''B3LYP/6-31G*''' is lower than that of '''HF/3-21G'''. Also, the bond lengths and bond angles changed with a certain amount as shown.

Furthermore, by submitting job type as '''Frequency''' in the optimised structure of gauche3 conformer of 1,5-hexadiene under both calculation basis '''HF/3-21G '''and '''B3LYP/6-31G*''', the electronic energy, sum of electronic and zero-point energies at 0 K and sum of electronic and thermal energies at 298.15 K could be determined as shown in Table 5.

{| border="1" cellspacing="1" cellpadding="10"
|-
| rowspan="3" | ''' '''
! colspan="4" |'''HF/3-21G c'''
! colspan="4" |'''B3LYP/6-31G* d'''
|-
! rowspan="2" width="125" align="center" | '''Electronic energy'''
! width="125" align="center" | '''Sum of electronic and zero-point energies'''
! width="125" align="center" | '''Sum of electronic and thermal energies'''
! rowspan="2" |IR spectrum
! rowspan="2" width="125" align="center" | '''Electronic energy'''
! width="125" align="center" | '''Sum of electronic and zero-point energies'''
! width="125" align="center" | '''Sum of electronic and thermal energies'''
! rowspan="2" |IR spectrum
|-
! width="125" align="center" | '''at 0 K'''
! width="125" align="center" | '''at 298.15 K'''
! width="125" align="center" | '''at 0 K'''
! width="125" align="center" | '''at 298.15 K'''
|-
| '''Reactant (Gauche 3)'''
| align="center" |-231.692692
| align="center" |-231.539486
| align="center" |-231.532646
|[[File:Yll113GAUCHE3-21IR.png|thumb|'''Figure 11. '''IR spectrum of gauche3 conformer of 1,5-hexadiene calculated by '''HF/3-21G''']]
| align="center" | -234.611326
| align="center" |-234.468719
| align="center" |-234.461477
|[[File:Yll113GAUCHE6-31IR.png|thumb|'''Figure 12.''' IR spectrum of gauche3 conformer of 1,5-hexadiene calculated by '''B3LYP/6-31G*''']]
|}

c [https://wiki.ch.ic.ac.uk/wiki/images/9/98/Yll113CR_GAUGE_PART4.LOG File] ; d [https://wiki.ch.ic.ac.uk/wiki/images/c/ca/Yll113CR_GAUGE_PART3.LOG File]

'''Table 5.''' Summary of energies (in hartree) of the reactant (Gauche3) Comparing Figure 11 and 12, both IR spectra are similar which shows that they have similar stretching/bending mode even under different methods of calculations. However, the strongest peak in Figure 12 is at 939 cm-1 and that in Figure 11 is 1111 cm-1. Yet, they correspond to the same mode of vibration, which is also the =C-H bending. Therefore, similar to the anti2 conformer's case as mentioned above, we can therefore know that the spring constant of the molecule under '''B3LYP/6-31G* '''basis reduced.

===== Optimisation of transition state =====

After optimising the reactants, the chair and boat transition states were optimised accordingly using mainly two different methods. But before that, an allyl fragment (CH2CHCH2) was optimised using '''HF/3-21G''' level of theory for the sake of convenience in constructing the chair and boat transition states. A brief summary was shown in Table 6.
{| class="wikitable"
!
! colspan="2" |Molecule
!Point Group
!Energy/Hartree
!RMS Gradient Norm/ Hartrees
|-
|Allyl fragment
CH2CHCH2
|<jmol>
<jmolApplet>
<title>Figure C: Allyl Fragment</title><color>white</color>
<size>250</size>
<uploadedFileContents> Yll113CR TS 1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|[[File:Yll11313.jpg|thumb|'''Figure 13. '''Optimised Structure of the allyl fragment]]
|C2v
|<nowiki>-115.82304010</nowiki>
|0.00002945
|}
'''Table 6. '''Summary of the optimised allyl fragment

Then, both chair and boat transition state were drawn and optimised using the '''optimisation to TS (Berny)''' under '''HF/3-21G'''. Figure 14 and Figure C show the optimized structure of the chair transition state while Figure 15 and Figure D show the optimized structure of the boat transition state. Table 7 shows the summary of results.
{| class="wikitable"
! rowspan="2" |
! colspan="2" rowspan="2" |Molecule
! rowspan="2" |Point Group
! rowspan="2" |Energy/ Hartree
! rowspan="2" |RMS Gradient Norm/Hartrees
! colspan="6" |Partial C-C bond length between/ Å
|-
!C14-C6
!C11-C3
!C14-C9
!C6-C1
!C9-C11
!C1-C3
|-
|Optimised Chair Transition State
|<jmol>
<jmolApplet>
<title>Figure D: Optimised Chair transition state</title><color>white</color>
<size>250</size>
<uploadedFileContents>Yll113CHAIR PART1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|[[File:Yll113CHAIR3-21.png|thumb|'''Figure 14. '''Optimised Chair Structure by '''HF/3-21G''' with atoms labelled ]]
|C2h
|<nowiki>-231.61932238</nowiki>
|0.00002645
|2.02016
|2.02016
|1.38929
|1.38930
|1.38930
|1.38929
|-
|Optimised Boat Transition State
|<jmol>
<jmolApplet>
<title>Figure E: Optimised Boat transition state</title><color>white</color>
<size>250</size>
<uploadedFileContents>Yll113BOAT PART1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|[[File:Yll11315.jpg|thumb|'''Figure 15. '''Optimised Boat Structure by '''HF/3-21G''' with atoms labelled]]
|C2v
|<nowiki>-231.60280235</nowiki>
|0.00003872
|2.14060
|2.14060
|1.38138
|1.38138
|1.38138
|1.38138
|}

'''Table 7. '''Summary of the optimised chair and boat transition states by '''optimisation to TS (Berny) '''under '''HF/3-21G''' basis

Furthermore, the transition structures’ '''Frequencies''' were calculated as shown in Table 8.
{| class="wikitable"
!
!Molecule
!IR spectrum
!Imaginary Frequency/ cm-1
|-
|Boat Transition State
|[[File:Yll113Boat Part 1 imag freq.gif]]
|[[File:Yll11317.jpg|thumb|'''Figure 16. '''IR spectrum of the optimised Boat Structure by '''HF/3-21G''']]
|<nowiki>-840</nowiki>
|-
|Chair Transition State
|[[File: Yll113Chair Part 1 imag freq.gif]]
|[[File:Yll11316.jpg|thumb|'''Figure 17. '''IR spectrum of the optimised Chair Structure by '''HF/3-21G''']]
|<nowiki>-818</nowiki>
|}
'''Table 8.''' IR spectra and imaginary frequencies of the boat and chair transition states

As you may notice that, the
imaginary frequency comes up when calculating with the transition states. This
is common, in other words, this should appear to let us know the transition
structure we postulated is correct.

A transition state is the first
order saddle point on the potential energy surface. Therefore, the force
applied to the saddle point against to the displacement. As force and
displacement are vectors, the force constant will be a negative number.Therefore, according to
:<math>\nu = \frac{1}{2 \pi c} \sqrt{\frac{k}{\mu}}</math>

where ''μ'' is the reduced mass, ''k'' is the spring constant for the bond and ''c'' is the light speed,

the square root of a negative
force constant k gives an imaginary wave number/frequency v. In other words,
the appearance of an imaginary frequency tells us that the structure is a
saddle point of the reaction pathway.

The chair transition state
was followed by first 'frozen' then 'relaxed'. The boat transition structure
was followed by '''QST2''' calculation method.

====== Chair Transition State ======
After the above '''optimisation''', the chair transition
state was reoptimised again with another method. This method first freezes the
coordinate of the molecule, in this case, freeze the bond to be made in the
Cope Rearrangement of 1,5-hexadiene as 2.20000 Å. The molecule then optimised with the frozen
coordinate. Details of this optimisation was summarized in Table 9.
{| class="wikitable"
! rowspan="2" |
! colspan="2" rowspan="2" |Molecule
! rowspan="2" |Point Group
! rowspan="2" |
{| class="MsoTableGrid"

|
<nowiki> </nowiki>Energy/ Hartree

|}
! rowspan="2" |RMS Gradient Norm/ Hartree
! colspan="3" |Bond length between/ Å
! rowspan="2" |Imaginary Frequency/ cm-1
! rowspan="2" |IR Spectrum
|-
!C6-C14 and C3-11
!C1-C3 and C9-C14
!C1-C6 and C9-C11
|-
|Optimised Chair Transition Structure with frozen coordinate
|<jmol>
<jmolApplet>
<title>Figure F: Optimised Chair transition state with frozen coordinate</title><color>white</color>
<size>250</size>
<uploadedFileContents>Yll113CHAIR PART2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
[[File: Yll113Chair frozen.gif]]
|[[File:Yll11318.jpg|thumb|'''Figure 18. '''The optimised chair transition structure with frozen coordinate and atoms labelling]]
|
{| class="MsoTableGrid"

|
<nowiki> </nowiki>C2h

|}
|<nowiki>-231.61518510</nowiki>
|0.00325573
|2.20000
|1.38135
|1.38128
|<nowiki>-765</nowiki>
|[[File:Yll11319.jpg|thumb|'''Figure 19. '''IR spectrum of the optimised chair transition structure with frozen coordinate]]
|}

'''Table 9. '''Summary of the optimisation of the chair transition structure with
frozen coordinate(s)

From Table 9, we may notice
that the RMS Gradient Norm value is quite far off from zero. Also, the
imaginary frequency becomes much higher than -818 cm-1 (Shown in
Table 8). With these two pieces of information, we can deduce that the frozen
coordinate(s) affect(s) the force constant of the transition state which does
not give a good optimisation of transition structure. With that, after applying
the frozen coordinate to the molecule, the molecule was reoptimised again with
a degree of '''Derivative '''to the '''Bond'''. Details of the reoptimisation
were presented in Table 10.
{| class="wikitable"
! rowspan="2" |
! colspan="2" rowspan="2" |Molecule
! rowspan="2" |Point Group
! rowspan="2" |Energy/ Hartree
! rowspan="2" |RMS Gradient Norm/ Hartree
! colspan="4" |Bond length between/ Å
! rowspan="2" |Imaginary Frequency/ cm-1
! rowspan="2" |
{| class="MsoTableGrid"

|
<nowiki> </nowiki>IR Spectrum

|}
|-
!C14-C6
!C11-C3
!C14-C9 and C6-C1
!C9-C11 and C1-C3
|-
|Optimised Chair Transition
Structure with a degree of '''Derivative'''
to the '''Bond'''
|<jmol>
<jmolApplet>
<title>Figure G: Optimised Chair transition state with a degree of Derivative to the Bond</title><color>white</color>
<size>250</size>
<uploadedFileContents>Yll113CHAIR PART3.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
[[File: Yll113Chair relax.gif]]
|[[File:Yll11320.jpg|thumb|'''Figure 20. '''The optimised chair transition structure with a degree of '''Derivative''' to the '''Bond''' and atoms labelling]]
|C2h
|<nowiki>-231.61932233</nowiki>
|0.00002127
|2.02075
|2.02071
|1.38929
|1.38930
|<nowiki>-818</nowiki>
|[[File:Yll11321.jpg|thumb|'''Figure 21. '''IR spectrum of the optimised chair transition structure with a degree of '''Derivative''' to the '''Bond''']]
|}
'''Table 10. '''Summary of the reoptimisation of the chair transition
structure with a degree of '''Derivative''' to the '''Bond'''

Now, in Table 10, the RMS
Gradient Norm value is close to zero. Also, the imaginary frequency goes back
to -818 cm-1, indicating that the coordinates no longer be frozen
and the stretching/bending mode of the transition state is able to undergo
freely.

Comparing the bond lengths
in Table 7 and 10, we can see that the difference between bond lengths of the
single bond to be made/ broken calculated in two methods is just less than
0.0006 Å. And also, there is no difference in bond length of the double bond to be make/broken ‘inside’ the system. This tells us that the two optimisation
methods are rather similar under the consideration on the Cope Rearrangement
Reaction.

====== Boat Transition State ======
Instead of using the frozen
coordinate method as for the chair transition state above, another method, '''QST2''', was applied to the boat
transition state under the '''HF/3-21G'''
basis. In order to use this method, without any ‘Link died’, the reactant and
product have to be drawn and labelled carefully. '''QST2''' is a method which interpolates the reactant and product to
give a transition state. Therefore, it will fall if the structure of the
reactant and product are not close to the transition state. And therefore, all
molecules have to be carefully labelled and adjusted.

[[File:Yll11322.png|500px|center]]

'''Figure 22. '''The drawings and adjustments of angles of the reactant (left)
and product (right) for '''QST2''' Method,
i.e. the central C-C-C-C dihedral angle was changed to 0o and inside
C-C-C were reduced to 100o

After the adjustment, the job was run and the optimized molecule converge to the boat transition structure. Summary was shown in Table 11.
{| class="wikitable"
! rowspan="2" |
! colspan="2" rowspan="2" |Molecule
! rowspan="2" |Point Group
! rowspan="2" |Energy/Hartree
! rowspan="2" |RMS Gradient Norm/ Hartree
! colspan="3" |Bond length between/ Å
! rowspan="2" |Imaginary Frequency/ cm-1
! rowspan="2" |IR spectrum
|-
!C1-C6
!C3-C4
!C5-C6, C4-C5, C3-C2 and C1-C2
|-
|Boat transition structure
under '''QST2''' method
|[[File:Yll113Boat qst2.gif]]
|[[File:Yll11323.jpg|thumb|'''Figure 23. '''The optimised Boat transition structure with atom labelling]]
|C2v
|<nowiki>-231.60280241</nowiki>
|0.00002436
|2.13994
|2.14019
|1.38149
|<nowiki>-840</nowiki>
|[[File:Yll11324.jpg|thumb|'''Figure 24. '''IR spectrum of the optimised boat transition structure]]
|}
'''Table 11. '''Summary of the boat transition structure under '''QST2 '''method

====== Intrinsic Reaction Coordinate''' '''('''IRC''') ======
In order to confirm that our transition state is of the
correct one, '''Intrinsic Reaction
Coordinate '''('''IRC''') will be carried
out.

As mentioned above, transition state is the first order
saddle point of the reaction pathway. Therefore, it will start to go to the
product or back to the reactant with it falls off. It resembles that a ball is
at the tip of the mountain, which starts to roll off the mountain on the side
with the steepest slope. Also, when we are doing '''IRC''', we can determine whether the reaction goes forward, backward
or both sides. Also, the number of points, which means the number of little
steps that the geometry of the molecule changes, can be adjusted. A low number
of points will just give us a very rough idea that tell us a little bit about
our transition state. Also, the last point on the '''IRC''' is far from the minimum geometry. A high number of points gives
us more accurate results, however two problems could be raised. First, the time
for calculation will be long and Most importantly, as it goes down the slope
and reaches the minimum geometry, i.e. the plateau of energy, the slope will
become very small or even zero again. However, as the energy difference of the
next or previous geometry compared to the geometry of itself is too small, ''GaussView 5.0'' may not able to know which
direction the molecule should proceed to. And this, therefore, results in ‘Link
died’. Therefore, the most common technique is to have a good number of points,
then take the last point on the IRC and run it with a normal optimisation.

Here, as we know that the
Cope Rearrangement has a symmetric reaction pathway, taking the chair
transition structure, we will run '''IRC'''
on it with 50 points.
{| class="wikitable"
![[File:Yll113hlj29825.jpg|thumb|'''Figure 25. '''Total energy along '''IRC '''of the chair transition structure]]
![[File:Yll11326.jpg|thumb|'''Figure 26. '''RMS Gradient Norm of '''IRC '''of the chair transition structure]]
![[File: Yll113Chair irc.gif]]
|}

[[File:Yll11327.jpg|500px]]

'''Figure 27. '''The product of the Cope Rearrangement after optimisation

The first point on Figure 25 is -231.61932233 Hartree and the last point is -231.69157881 Hartree. Then, we take the last point and optimise it, we get the structure shown in Figure 27.

The structure is of C2
symmetry and the energy calculated is -231.69166702 Hartree. This matches with
Gauche2 C2 on Appendix 1. And therefore, this is how the conformer
of 1,5-hexadiene connects with the chair transition structure.

====== Activation Energy of the Cope Rearrangement ======
Finally, we optimise the chair and boat transition states we got from above, reoptimise it with job Opt + Freq
under a more advanced calculation '''B3LYP/6-31G*'''. And from that, the thermochemistry data were given and we can know the
activation energy of the reaction by comparing to Table 3, which anti2 is used
as a local minimum rather than gauche3 as a global minimum.
{| border="1" cellspacing="1" cellpadding="10"
|-
| rowspan="3" | ''' '''
! colspan="3" |'''HF/3-21G'''
! colspan="3" |'''B3LYP/6-31G*'''
|-
| rowspan="2" width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| rowspan="2" width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619323
| align="center" | -231.466698
| align="center" | -231.461339
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409009
|-
| '''Boat TS'''
| align="center" | -231.602803
| align="center" | -231.450932
| align="center" | -231.445303
| align="center" | -234.543094
| align="center" | -234.402342
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692534
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.611703
| align="center" | -234.469213
| align="center" | -234.461857
|}
'''Table 11'''. Summary of energies of chair, boat and reactant (anti2) structure
{| border="1" cellspacing="1" cellpadding="10"
|-
| rowspan="2" | ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.69
| align="center" | 34.06
| align="center" | 33.16
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.96
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
'''Table 12'''. Summary of activation energies in kcal/mol

==== The Diels-Alder Cycloadditions ====

===== Ethylene and Cis-Butadiene =====
First, the structures of the reactants, ethylene and cis-butadiene, were '''optimised to minimum''' under '''semiempirical AM1''' method. For the butadiene, in order to be in the cis conformer, the dihedral angle was adjusted to be 0o. Details are listed in Table 13.
{| class="wikitable"
!
!Molecule
!Point Group
!HOMO
!LUMO
!Energy/Hartree
!RMS Gradient Norm/ Hartree
!IR spectrum
|-
|Ethylene
|<jmol>
<jmolApplet>
<title>Figure H: Optimised Ethylene</title><color>white</color>
<size>250</size>
<uploadedFileContents>Yll113ETHENE OPTAM1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|D2h
|[[File:Yll11331.jpg|thumb|'''Figure 31. '''HOMO of Ethylene]]
|[[File:Yll11330.jpg|thumb|'''Figure 30.''' LUMO of ethylene]]
|0.02619029
|0.00008755
|[[File:Yll11328.jpg|thumb|'''Figure 28. '''IR spectrum of Ethylene]]
|-
|Cis-Butadiene
|<jmol>
<jmolApplet>
<title>Figure I: Optimised Cis-Butadiene</title><color>white</color>
<size>250</size>
<uploadedFileContents>Yll113CISBUTADIENE OPTAM1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|C2v
|[[File:Yll11332.jpg|thumb|'''Figure 32. '''HOMO of cis-butadiene]]
|[[File:Yll11333.jpg|thumb|'''Figure 33. '''LUMO of cis-butadiene]]
|0.04878534
|0.00000087
|[[File:Yll11329.jpg|thumb|'''Figure 29.''' IR spectrum of cis-butadiene]]
|}
'''Table 13.''' Summary of optimised ethylene and cis-butadiene

Looking into Figure 30-33, as we know that the plane is perpendicular to the molecule, the HOMO of Ethylene is symmetric while that of LUMO is antisymmetric.

Also, the HOMO of cis-butadiene is antisymmetric and that of LUMO is symmetric.

Then, the transition state of the reaction was able to constructed using the optimised structure of the reactants made above. By making the bond lengths of the bonds to be made in Diels-Alder [4+2] cycloaddition to 2.20000 Å in the transition state, which is the 'dash bond', the transition structure was optimized to '''Opt + Freq minimisation to''' '''TS (Berny)''' under the same method of calculation - '''semiempitical AM1'''. Details are listed in Table 14.
{| class="wikitable"
!
! colspan="2" |Molecule
!Point Group
!HOMO
!LUMO
!Energy/Hartree
!RMS Gradient Norm/ Hartree
!Imaginary Frequency/ cm-1
!IR spectrum
|-
|Transition state
|<jmol>
<jmolApplet>
<title>Figure J: Optimised Transition State of Cis-butadiene and Ethylene</title><color>white</color>
<size>250</size>
<uploadedFileContents>Yll113DA1 TS(BERRY).LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|[[File:Yll11334.jpg|thumb|'''Figure 34. '''Transition state of ethylene + cis-butadiene]]
|C1
|[[File:Yll11336.jpg|thumb|'''Figure 36. '''HOMO of transition state]]
|[[File:Yll11337.jpg|thumb|'''Figure 37. '''LUMO of transition state]]
|0.11165467
|0.00002792
|<nowiki>-956</nowiki>
|[[File:Yll11335.jpg|thumb|'''Figure 35. '''IR spectrum of the transition state]]
|}
'''Table 14.''' Summary of optimised transition state

From Figure 36, we can see that the HOMO of the transition state is antisymmetric whilst the LUMO of the transition state is symmetric. By making very careful comparison between Figure 36, Figure 37 and Figure 30-33, we can see that the HOMO of the transition state in Figure 36 is a combination of Figure 32 and 30; the LUMO of the transition state in Figure 37 is a combination of Figure 31 and 33. We can clearly see that the HOMO and LUMO of the transition state have a complementary combination of HOMO and LUMO of the reactants.

Taking a closer look to HOMO of the transition state. Recalling Woodward Hoffmann’s Rule, (4q+2)s+(4r)a = odd for thermally allowed reaction, we have both π2s and π4s. Therefore, the reaction is thermally allowed by letting q = 0, which gives the value of 1 which is odd.

Furthermore, from Table 14, we notice that there is an imaginary frequency reported at -956 cm-1. As explained above, the transition state should have one imaginary frequency to account for the negative force constant. With that, this imaginary frequency confirms that the transition structure we postulated from the optimised reactants is valid, i.e. it is really a transition state. The animation of where the imaginary frequency originates from, which shows the motion of the transition state - how the two reactants approach to each other and bonds are formed, is shown below.

JSMOL

From the above figure, we can see that the bond formation from the reactant to the product happens at the same time, i.e. synchronous, on both sides of the transition structure. Therefore, we can say that this Diels-Alder cycloaddition is a concerted [4+2] pericyclic cycloaddition, which matches with what we learnt in Pericyclic Reaction course.

On top of that, the geometry of the transition structure was investigated by looking into the optimised bond lengths between carbon atoms Details are shown in Figure 38 and Table 15.[[File:Yll11338.jpg|thumb|'''Figure 38. '''Transition state of ethylene + cis-butadiene with atoms labelled]]
{| class="wikitable"
!
!Bond length/ Å
|-
|C7-C9
|2.11938
|-
|C12-C5
|2.11944
|-
|C12-C9
|1.38284
|-
|C3-C7
|1.38187
|-
|C3-C1
|1.39750
|-
|C5-C1
|1.38175
|}
'''Table 15. '''Geometry analysis of the transition state

According to the literature <ref>M. A. Fox and J. K. Whitesell, ''Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen'', Springer, 1995</ref>, C-C carbon-carbon single bond is 1.54 Å, and C=C carbon-carbon double bond is 1.34 Å. Also, the Van der Waals radius of carbon is 1.70 Å,<ref>A. Bondi,(1964), ''J. Phys. Chem.'', 1964, '''68''' (3), 441</ref>
According to the reaction scheme shown in Figure 3, a single bond is forming between C7 and C9, also another single bond is forming between C12-C5. Comparing the data in Table 15 with the literature, we can see that the bond length of two bonds to be made is longer than C-C, but shorter than the twice of carbon's Van der Waals radius. This tells us some hints that the terminal carbons between two reactants are attracting with each other to a certain degree. Furthermore, taking a look to other carbon-carbon bonds within the system, we notice that they are somewhere between C-C and C=C. This tells us that either C-C is becoming C=C or C=C is becoming C-C in the transition state.
After that, the '''IRC''' of the above optimised transition state was carried out with both direction and force constant calculated always for 50 points to see the reaction profile.
{| class="wikitable"
|[[File:Yll11339.jpg|thumb|'''Figure 39.''' IRC of the transition state of ethylene + cis-butadiene]]
|[[File:Yll11340.jpg|thumb|'''Figure 40. '''RMS Gradient Norm of transition state of ethylene + cis-butadiene]]
|JSMOL
|}
In Figure 39, we can clearly see that the reactants was first passed through the energy barrier to get the transition state and it went down the slope to give the product.
Finally, the activation energy for this reaction was calculated in Table 16.
{| class="wikitable"
!
!Ethylene
!Cis-butadiene
!Transition state
!Activation Energy
|-
|Energy/Hartree
|0.02619029
|0.04878534
|0.11165467
|0.03667904
(23.02 kcal/mol)
|}
'''Table 16. '''Activation energy analysis of Diels-Alder Reaction between ethylene and cis-butadiene

===== Cyclohexa-1,3-diene and Maleic Anhydride =====
First, the structures of
the reactants, ethylene and cis-butadiene, were '''optimised to minimum''' under '''semiempirical AM1''' method. Details are listed in Table 17.
{| class="wikitable"
!Optimised Structure
!Molecule
!Point Group
!Energy/Hartree
|-
|Cyclohexa-1,3-diene
|[[File:Yll113Cycloopt.jpg|thumb|'''Figure 41. '''Optimised structure of cyclohexa-1,3-diene]]
|C2
|0.02771130
|-
|Maleic Anhydride
|[[File:Yll113Maopt.jpg|thumb|'''Figure 42. '''Optimised structure of Maleic Anhydride]]
|Cs
|<nowiki>-0.12182418</nowiki>
|}
'''Table 17. '''Summary
of the optimized reactant

Then, the two transition states, exo and endo, of the reaction were able to construct. By making the bond lengths of the bonds to be made in Diels-Alder [4+2] cycloaddition to 2.20000 Å in the transition state, which is the 'dash bond', the transition structure was optimized to '''Opt + Freq minimisation to''' '''TS (Berny)''' under the same method of calculation - '''semiempitical AM1'''. Details are listed in Table 18.
{| class="wikitable"
!Transition Structure
!Molecule
!Point Group
!HOMO
!LUMO
!Energy/Hartree
!'''RMS Gradient Norm/ Hartree'''
!'''Imaginary Frequency/ cm-1'''
!IR spectrum
|-
|Exo
|[[File:Yll113Exots.jpg|thumb|'''Figure 43.''' Optimised Structure of Exo Transition State]]
|C1
|[[File:Yll113Exo homo.jpg|thumb|'''Figure 45. '''HOMO of Exo Transition State]]
|[[File:Yll113Exo lumo.jpg|thumb|'''Figure 47.''' LUMO of Exo Transition State]]
|<nowiki>-0.05041984</nowiki>
|0.00000883
|<nowiki>-812</nowiki>
|[[File:Yll113Exo ir.jpg|thumb|'''Figure 49. '''Exo Transition State IR spectrum]]
|-
|Endo
|[[File:Yll113Endots.jpg|thumb|'''Figure 44. '''Optimised Structure of Enfo Transition State]]
|C1
|[[File:Yll113Endo homo.jpg|thumb|'''Figure 46'''. HOMO of Endo Transition State]]
|[[File:Yll113Endo lumo.jpg|thumb|'''Figure 48.''' LUMO of Endo Transition States]]
|<nowiki>-0.05150469</nowiki>
|0.00004308
|<nowiki>-807</nowiki>
|[[File:Yll113Endo ir.jpg|thumb|'''Figure 50.''' Endo Transition State IR Spectrum]]
|}
'''Table 18. '''Summary of the optimized transition structures

In Table 18, we can see that both HOMO and LUMO of the exo
transition state are antisymmetric with respect to the plane. Also, both HOMO and LUMO of the endo transition state are antisymmetric as well.

Also, we notice that the energy of exo is higher than that of endo. This can be explained by the poorer overlap between the C=C π and C=O π* compared to that of endo. This is called secondary orbital effect, which will be further discussed below.

There is one imaginary frequency in both transition structure. This confirms that both of them are valid postulated transition structures.
{| class="wikitable"
|JSMOL
'''Imaginary frequency of exo transition state animation'''
|JSMOL
'''Imaginary frequency of exo transition state animation'''
|}
Again, from the above animations, we can see that the bonds are formed in a synchronous fashion, which means the Diels-Alder Cycloaddition reaction is concerted.

Furthermore, the geometries of both transition states were examined carefully in Table 19.
{| class="wikitable"
!Geometry summary of Exo Transition State (Please refer to Figure 43 for atom labelling)
!Geometry summary of Endo Transition State (Please refer to Figure 44 for atom labelling)
|-
|
{| class="wikitable"
!
!Bond length/ Å
!
!Bond length/ Å
|-
|C5-C13
|2.17032
|C6-C14
|2.17035
|-
|C5-C3
|1.39440
|C3-C1
|1.39674
|-
|C1-C6
|1.39440
|C14-C13
|1.4612
|-
|C7-C6
|1.48976
|C7-C10
|1.52208
|-
|C10-C5
|1.48976
|C13-C15
|1.48822
|-
|C14-C16
|1.48822
| colspan="2" |N/A
|-
|C7-C16
(Through space)
|2.94507
|C10-C15
(Through space)
|2.94484
|-
|C1-C16
(Through space)
|3.78172
|C3-C15
(Through Space)
|3.78155
|}
|
{| class="wikitable"
!
!Bond length/ Å
!
!Bond length/ Å
|-
|C15-C7
|2.16230
|C16-C5
|2.16229
|-
|C1-C3
|1.39726
|C3-C7
|1.39296
|-
|C1-C5
|1.39308
|C7-C9
|1.49503
|-
|C9-C12
|1.52300
|C5-C12
|1.49054
|-
|C16-C18
|1.48918
|C15-C17
|1.48903
|-
|C15-C16
|1.40863
| colspan="2" |N/A
|-
|C1-C18
(Through space)
|2.89232
|C3-C17
(Through space)
|2.89203
|}
|}
'''Table 19.''' Geometry analysis of exo and endo transition states

According to the reaction scheme shown in Figure 4, a single bond is forming between C5 and C13, also another single bond is forming between C6-C14 for exo; C15 and C7 plus C16 and C5 for endo, which is what the first row in the two tables in the left and right in Table 19 shows. the single bond to be made Comparing these values with literature, we find that they are longer than C-C but shorter than twice of carbon's Van der Waals' radius. This tells us some hints that these pairs of carbons between two reactants are attracting with each other to a certain degree. Furthermore, taking a look to other carbon-carbon bonds within the system, i.e. except row 1 and those labelled with (through space), we notice that they are somewhere between C-C and C=C. This tells us that either C-C is becoming C=C or C=C is becoming C-C in the transition state.

Now, looking at the through space bond length. In the exo transition structure, the through space distance -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH2-CH2- is smaller than the twice of the Van der Waals radius but larger than C-C which gives us a hint that they might interact. In the endo transition structure, the through space distance -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH=CH- is smaller than the twice of the Van der Waals radius but larger than C-C which gives us a hint that they might interact. However, according to the definition of secondary orbital effect, it is looking for the interaction between the C=C π of the diene and C=O π* of the dienophile. Endo clearly shows that as explained, but exo seems to just demonstrate the sterics clash between the -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH2-CH2- fragment of diene. In order to further confirm that exo has no secondary orbital effect, a measurement of bond length was carried out between -(C=O)-O-(C=O)- fragment of the maleic anhydride and the -CH=CH- in diene in the exo transition state. The result was shown in the last row on the left table in Table 19. This shows that they are too far away which means they are not possible to interact.

Now, looking back to the HOMO of exo and endo transition states in Figure 45 and 46 respectively. We can definitely see that the overlap between the two reactants is relatively smaller in exo. From these two pieces of information, we can conclude that the endo is kinetically controlled, while exo is thermodynamically controlled.

After that, the '''IRC''' of the both optimised transition state was carried out with both direction and force constant calculated always for the reaction profile. 21 points were used for exo transition states and 24 for endo (reasons explained under '''Introduction)''' to see the reaction profiles.
{| class="wikitable"
! colspan="3" |Exo Transition State
! colspan="3" |Endo Transition State
|-
|[[File:Yll113Exo irc.jpg|thumb|'''Figure 51.''' IRC of the exo transtion state]]
|[[File:Yll113Exo rms.jpg|thumb|'''Figure 52. '''RMS of the exo transition structure]]
|JSMOL
|[[File:Yll113Endo irc.jpg|thumb|'''Figure 53. '''IRC of the endo transition state]]
|[[File:Yll113Endo rms.jpg|thumb|'''Figure 54.''' RMS of the endo transition state]]
|JSMOL
|}
And eventually, the activation energies of the reaction via different transition structures were summarised in Table 20.
{| class="wikitable"
!
!Cyclohexa-1,3-diene
!Maleic Anhydride
!Endo Transition State
!ExoTransition State
!Activation Energy via endo
!Activation Energy via exo
|-
|Energy/Hartree
|0.02771130
|<nowiki>-0.12182418</nowiki>
|<nowiki>-0.05150469</nowiki>
|<nowiki>-0.05041984</nowiki>
|0.04260819
(26.74 kcal/mol)
|0.04369304
(27.42 kcal/mol)
|}
'''Table 20.''' Activation energy analysis

=== '''Conclusion''' ===

=== '''Reference''' ===

<gallery>
File:
</gallery>

File:Yll113DA1 TS(BERRY).LOG

2015-12-17T08:14:52Z

Yll113:

Rep:Mod:hlj298

2015-12-17T08:09:09Z

Yll113: /* Ethylene and Cis-Butadiene */

== '''Study of the reaction profiles of the Cope Rearrangement and the Diels-Alder Cycloadditions''' ==
'''Y. L. J. Lam'''

''Department of Chemistry, South Kensington Campus, Imperial College London, London SW7 2AZ, United Kingdom''

''Received 18 December, 2015''

=== Abstract ===
The reactants, products and transition states of the Cope
Rearrangement of 1,5-hexadiene were investigated by ''GaussView 5.0'' at the '''HF/3-21G''' and '''B3LYP/6-31G*'''levels
of theories respectively. With that, the point groups, vibrational frequencies and different energies at different temperatures of the reactants, products and transition states were calculated. Also, by optimizing the transition structures with different methods, i.e. computing the force constants at the
beginning of the calculations, using the redundant coordinate editor and '''QST2''', at the '''HF/3-21G''' level of theory, closer views of the geometries of the transition states can be observed. Furthermore, by using the '''IRC''' method, the reaction profiles can be
obtained and the activation energies can therefore be calculated. Plus, using '''IRC''' method, all reaction intermediates
can now be observed, which helps us to understand the mechanism of the Cope Rearrangement. Similarly, for Diels-Alder Cycloadditions between ethene and
butadiene and Diels-Alder Cycloadditions between cyclohexa-1,3-diene and maleic anhydride, the reactants, products and transition states were optimized and
their respective energies were calculated at '''AM1 semi-empirical molecular orbital method'''. Furthermore, the symmetries of the
molecular orbitals were visualized and the reaction profiles calculated by '''IRC''' method were obtained.

=== '''Introduction''' ===
Chemical reactions are happening around the world in every second. Some reactions are fast, whilst some are slow. The most common and general reason for that is on the kinetic and thermodynamic aspects. On the kinetic aspect, we might argue that the energy barrier(s) form the reactant(s) to the product(s) is/are huge, and therefore, the reactant(s) cannot overcome the barrier(s) and the reaction is slow or does not proceed. The transformation between crude carbon and diamond is a good example. The energy difference between crude carbon and diamond is just few kcal/mol, however, the energy barrier for the transformation is huge. Hence, the transformation is extremely slow, or even does not proceed. With that, diamond symbolizes eternity. On the other hand, on the thermodynamic aspect, we might argue that the reaction is endothermic, i.e. absorbing/requiring heat from the surroundings in order to proceed. In fact, these two aspects just provide us with a little bit of the story and therefore, chemists, or scientists in general, study the mechanism of the reactions to find out the full story. Unfortunately, some reactions are spontaneous, such as the thiocyanation of the iron complex. Also, some intermediates of the reactions are unstable, which cannot be separated or detected even using very advanced analytical instruments, such as nuclear magnetic resonance (NMR) spectromenter. Therefore, scientists devised some programs and computational methods to find out the mechanism of the reactions. Here we use ''GaussView 5.0'' for our investigation.

==== Computational Theory ====
[[File:Yll113 AM1 and HF.jpg|thumb|463x463px|'''Figure 1.''' HOMO and LUMO (highlighted in yellow) of cis-butadiene under the basis of calculation '''HF/3-21G '''(left) and '''AM1''' (right)]]
In ''GaussView 5.0'', there are numerous methods for calculation, such as '''HF/3-21G''', '''B3LYP/6-31G*''', '''semiempirical AM1''', '''MP4 '''and '''MP2'''. Here, the first two calculation method, namely, '''HF/3-21G''' and '''B3LYP/6-31G* '''were applied for calculation of the Cope Rearrangement Reaction, while '''semiempirical AM1''' was used for the investigation of the two Diels-Alder reactions.

N.B. No matter which method applied, the RMS Gradient Norm in hartress would also be computed. This is a measure of how well does the optimisation go during the calculation of the
structure drawn. The closer to zero, the better the structure is optimised.

===== Hartree-Fock ('''HF''') Method =====
Hartree-Fock theory ('''HF''') is the fundamentals of electronic structure theory. It gives a good starting point for more elaborate theoretical methods which can approximate the electronic Schrödinger equation better. It is the basis of the molecular orbital (MO) theory that assumes the motion of each electron can be described by a single-particle function/orbital and it does not depend on/interact with the instantaneous motions of the other electrons.<ref>C. D. Sherrill, ''An Introduction to Hartree-Fock Molecular Orbital Theory'', 2000</ref>

===== Becke, 3-parameter, Lee-Yeang-Parr ('''B3LYP''') Method =====
Beeke, 3-parameter, Lee-Yang-Parr ('''B3LYP''') is one of the most commonly used hybrid functionals. Hybrid functionals are a class of approximation of the exchange-correlation energy functional in density functional theory.<ref>What is B3LYP?, https://www.quora.com/What-is-B3LYP (accessed December 2015)</ref> '''B3LYP''' contains an '''HF''' exchange with the weight of 0.2, which can be regarded as a uniform screening of
exchange by 80 %.<ref>C. H. Patterson, ''International Journal of Quantum Chemistry'', 2006, '''106 '''(15), 3383</ref> '''B3LYP''' also takes a set of atomization
and ionization energies, proton affinities and total atomic energies into account.<ref>A. D. Becke, ''The Journal of Chemical Physics'', 1993, '''98''', 5648</ref>

===== Semiempirical Austin Model 1 ('''AM1''') =====
Semiempirical Austin Model 1 ('''AM1''') based on the Neglect of Differential Diatomic Overlap (NDDO) integral approximation.<ref>M.
J. S. Dewar, E. G. Zoebisch, E. F. Healy and J. J. P. Stewart, ''J. Am. Chem. Soc.'', 1985, '''107''' (13), 3902</ref> Therefore, when taking the same molecule for '''AM1''' and '''HF/3-21G''', you would find that the numbers of HOMO and LUMO are different, which '''AM1''' gives smaller numbers as shown in Figure 1. This is due to the neglect of the low-lying orbitals overlapping. With that, '''AM1''' proceeds much faster than '''HF/3-21G''' for the sake of time.

==== The Cope Rearrangement ====
The Cope Rearrangement is an organic reaction involving [3,3]-sigmatropic rearrangement of 1,5-dienes, which resembles the Claisen Rearrangement.<ref>A. C. Cope and E. M. Hardy, ''J. Am. Chem. Soc.'', 1940, '''62''' (2), 441</ref> The mechanism of the Rearrangement has sparked a controversy – whether it is concerted, dissociative or stepwise.<ref>O. Wiest, K. A. Black and K. N. Houk, ''J. Am. Chem. Soc.'', 1994, '''116''', 10336</ref> With that, first, each conformer of the reactant, 1,5-hexadiene, was optimised by '''HF/3-21G'''. The lowest energy conformer of 1,5-hexadiene was found. Then, as we know, the Rearrangement undergoes either a chair or boat transition state. So, each transition state was optimised by '''HF/3-21G '''as well. By looking into the energy difference between the transition states and the reactant, the activation energy of the Cope Rearrangement with 1,5-hexadiene was found. In order to find the reaction profile and see how the 1,5-diene rearranges, i.e. the mechanism, the transition state was optimised again with
mainly two methods. The coordinate of the chair transition state was first frozen, with the bond to be made set as 2.20000 Å. 2.20000 Å is a good bond length for partially C-C bond as suggested by the chemists’ observations in the literature. <ref>N. H. Kendall, Y. Li and J. D. Evanseck, ''Angew. Chem. Int. Ed. Engl.'', 1992, '''31''' (6), 682</ref> Then, after the optimization of the frozen coordinate, the partly form 2.20000 Å can be relaxed and the structure was then reoptimised. This methods skips the process of computing the whole force constant matrix i.e. Hessian, which saves time and costs. Furthermore, the boat transition state was optimised by '''QST2'''. '''QST2''' has a higher constrains in which requires a more accurate transition state structure to be put in. In this case, the dihedral angle plays an important role in order to be calculated by ''GaussView'' 5.0. Hence, this method is more expensive and time-consuming. From the optimised transition states, an '''IRC''' can be run for the optimised structure to see the full reaction profile. Also, the intermediates of the reaction can be observed. And finally, the reactant and two transition states
were optimised with '''B3LYP/6-31G*''' similarly. Hence, the two calculation methods can be compared by looking into the numbers obtained. Also, the numbers can be compared against the
experimental values. As explained above, '''B3LYP''' takes a more in-depth consideration, the numbers got from this method should be closer to the reality.

==== The Diels-Alder Cycloaddition ====
The Diels-Alder cycloaddition is a [4+2] cycloaddition between a dienophile and a conjugated alkene to give a cyclohexane system. Here, calculations on two Diels-Alder cycloaddition reactions are reported. They are (1) ethylene and butadiene and (2) cyclohexa-1,3-diene and maleic anhydride.

For Diels-Alder cycloaddition reaction, it is well-known that the reaction gives exo and/or endo product. Exo product implies the reaction pathway is thermodynamically controlled to give more stable product; endo product implies
the reaction pathway is kinetically controlled to give a relatively less stable product. In other words, the activation energy to form the exo product is higher than that of endo, however, the endo product is higher in energy than exo. This can usually be explained by the secondary orbital effects. In our cases, both the exo and endo products were investigated undoubtedly. This time, as you may notice, the molecule is more large in size and there are two reactants instead of just one reactant in the Cope Rearrangement, a simpler method of calculation was implemented, which is '''AM1'''. Also, the electronic distributions and orbitals of the HOMO and LUMO of the transition states were computed and visualised.

=== '''Computational Method''' ===
''All calculations were performed by GaussView 5.0. Relevant JSmol files were uploaded here, however, due to some technique glitches, some bonds, especially double bonds, might not come up properly. Yet, the structures of the molecules are generally correctly shown.''

==== The Cope Rearrangement ====
[[File:Yll113 CR.png|thumb|'''Figure 2.''' The Cope Rearrangement of 1,5-hexadiene]]
An anti and gauche conformation of the 1,5-hexadiene were drawn respectively. The drawn structures were first optimised by a not very accurate technique, i.e. '''Clean'''. Then, the '''clean'''ed structure were optimised by '''HF/3-21G'''. The point group and the energy of each conformer were found and compared to locate the low-energy minima. The optimised structures from '''HF/3-21G''' were then reoptimised by '''B3LYP/6-31G*'''. The point group of each conformer was checked and confirmed. Also, the comparison of the same conformer under different calculation method '''HF/3-21G''' and '''B3LYP/6-31G''' was carried out by looking into energy, bond lengths and bond angles. Furthermore, the '''Thermochemistry''' using job type '''Frequency''' was found in both '''HF/3-21G '''and''' B3LYP/6-31G* '''optimised anti conformers.

The boat and chair transition structures were also drawn and '''clean'''ed. The point group of each transition state was found.

Firstly, the chair transition structure was '''optimised to TS (Berny)''' in '''Opt + Freq '''using the basis of '''HF/3-21G'''. Force constant was calculated '''once'''. The frequency of vibration was checked to make sure there is one imaginary vibrational frequency. Then, '''freeze''' '''coordinate''' of the molecule by freezing the carbon-carbon bond to be made as 2.20000 Å. After that, the frozen coordinate was relaxed so the carbon-carbon bond to be made no longer be restricted to 2.20000 Å. The geometry of the transition state was then compared.

Secondly, at the same time, the boat transition structure was optimised by '''QST2''' method by specifying the reactants and products of the reaction under the basis of '''HF/3-21G'''. Labelling the atoms in
the reactant and product, and adjusting the central '''C-C-C-C '''dihedral angle to 0o plus the two inside '''C-C-C''' angles to 100o, the reactant and product could now be optimised by '''QST2''' method.

Comparing the optimised chair and boat transition structures, the connecting conformer of 1,5-hexadiene was found. The reaction energy profile was then calculated by '''IRC''' under '''HF/3-21G''' with 50 points and force constant as always for every small steps. With that, the mechanism of the reaction, as well as the whole reaction energy profile, could be observed clearly. Take the last point on the '''IRC''' and run a normal '''optimisation''' under '''HF/3-21G''' to obtain a minimized geometry.

Eventually, re'''optimise''' the structures of the two transition states with '''Opt + Freq '''under the basis of '''B3LYP/6-31G*'''. The geometries and energies of the transition structure under two different basis were compared. With that, these computed values were also compared against experimental values.

==== The Diels-Alder Cycloadditions ====
[[File:Yll113DA1.jpg|thumb|'''Figure 3. '''The Diels-Alder Cycloadditions between ethylene and butadiene]]

===== Ethylene and butadiene =====
The structure of cis-butadiene was first optimised under the basis of '''semiempirical AM1'''. The HOMO and LUMO of cis butadiene were visualised and its symmetry was determined.

The transition state of the reaction was drawn and optimised under the basis of '''semiempirical AM1'''. The '''IRC''' was carried out in order to confirm the transition structure postulated is of the correct one. Furthermore, the HOMO of the transition structure was visualised and the nodal
planes and properties of the system were interpreted.

===== Cyclohexa-1,3-diene and maleic anhydride =====
[[File:Yll113DA2.jpg|thumb|'''Figure 4. '''The Diels-Alder Cycloadditions between cyclohexa-1,3-diene and maleic anhydride]]
The transition states of the exo and endo products were drawn and optimised under the basis of '''semiempirical AM1'''. The '''IRC''' was carried out in order to confirm the transition structure postulated is of the correct one. For the number of points, 21 points were used for exo transition states and 24 for endo. This is because the energy was too shallow and the slopes tend to zero after the number of points specified above and ''GaussView 5.0'' cannot predict which direction should it goes on to calculate. Furthermore,
the bond lengths, orientation and the HOMO of the transition structures were investigated.

=== '''Results and Discussion''' ===

==== The Cope Rearrangement ====

===== Optimisation of Reactant =====
1,5-hexadiene has three free rotating carbon-carbon bonds. Each of them has three rotational minima. This gives 27 conformations of the 1,5-hexadiene molecule. Yet, only ten of them were energetically distinct due to symmetry and enantiomeric relationships.<ref>B. G. Rocque, J. M. Gonzales and H. F. Schaefer, ''Molecular Physics'', 2002, '''100''' (4), 441</ref> Two of them, the ''Ci anti ''and ''C1 gauche ''structure in here'' ''were drawn and optimizied as shown in Figure A and B and their energies were calculated as shown in Table 1.
{| class="wikitable"
!Conformation
!Molecule
!Point Group
!Energy/ Hartrees*
HF/3-21G
!RMS Gradient Norm/Hartrees*
HF/3-21G
!Relative Energy#/ kcal/mol
!Newman Projections
|-
|Gauche3
|<jmol>
<jmolApplet>
<title>Figure A: Gauge3 conformer of 1,5-hexadiene</title><color>white</color>
<size>250</size>
<uploadedFileContents> yll113CR_GAUGE_PART1.LOG </uploadedFileContents>
</jmolApplet>
</jmol>
|C1
|<nowiki>-231.69266120</nowiki>
|0.00001556
|0.00
|[[File:Yll113 torsion gauche.jpg|centre|frame|'''Figure 5.''' Newman Projection of C1 gauche3 1,5-hexadiene]]
|-
|Anti2
|<jmol>
<jmolApplet>
<title>Figure B: Anti2 conformer of 1,5-hexadiene</title><color>white</color>
<size>250</size>
<uploadedFileContents>YLL113CR ANTI PART1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|Ci
|<nowiki>-231.69253528</nowiki>
|0.00001891
|0.08
|[[File:Yll113 torsion anti.jpg|centre|frame|'''Figure 6.''' Newman Projection of Ci anti2 1,5-hexadiene]]
|}
<nowiki>*</nowiki>1 hartree = 627.509 kcal/mol

#The difference in energy between the conformer and the lowest energy conformer, in here, which is Gauche3. Then convert Hartree to kcal/mol by *

'''Table 1. '''Conformational analysis of anti2 and gauche3 of 1,5-hexadiene

As shown in Table 1, the energy of Gauche3 is surprisingly lower than the anti2 conformation of 1,5-hexadiene. In most cases, the antiperiplanar conformation of a molecule, such as anti2, is more favourable as it has the least steric clashes. Therefore, usually the antiperiplanar conformation is of the lowest energy. However, here, apart from sterics, the stereoelectroncs concept has also been taken into account. The vinyl proton, in a through space manner, can interact with the π or π* orbital on the sp2 carbon which is separated by four bonds from it.<ref>M. Nishio and M. Hirota, ''Tetrahedron'', 1989, '''45 '''(23), 7201</ref> This is so-called CH-π interaction. The Newman Projection in Figure 5 gives us a closer look on how they are close in space and interact; and the Newman projection in Figure 6 tells us why the vinyl proton cannot interact with the π or π* system through space. Therefore, the gauche3 conformation is more stable than anti2 and of lower energy in 1,5-hexadiene.

Focusing on anti2 conformer of the 1,5-hexadiene, the '''HF/3-21G''' optimised molecule was reoptimised by '''B3LYP/6-31G'''. The energies and the geometries of the anti2 1,5-hexadiene under two basis of calculation method were compared and shown in Table 2.
[[File:Yll113Anti2.png|thumb|'''Figure 7. '''Anti2 conformer of 1,5-hexadiene with labelled atoms]]
{| class="wikitable"
! rowspan="2" |Calculation Method
! rowspan="2" |Point Group
! rowspan="2" |Energy/ Hartrees
! rowspan="2" |RMS Gradient Norm/Hartrees
!Bond length between/ Å
!Bond length between/ Å
!Bond length between/ Å
!Bond angle between
!Bond angle between
|-
|C14-C12 and C4-C1
|C12-C9 and C6-C4
|C6-C9
|C14-C12-C9 and
C1-C4-C6
|C12-C9-C6 and
C4-C6-C9
|-

|'''HF/3-21G'''
|Ci
|<nowiki>-231.69253528</nowiki>
|0.00001891
|1.31613
|1.50891
|1.55275
|124.80579o
|111.34878o
|-
|'''B3LYP/6-31G*'''
|Ci
|<nowiki>-234.61171063</nowiki>
|0.00001249
|1.33350
|1.50419
|1.54816
|125.29968o
|112.67081o
|}
'''Table 2. '''Comparison with the anti2 conformer of 1,5-hexadiene with two calculation method basis ('''HF/3-21G '''and '''B3LYP/6-31G*''')

As seen in Table 2, the point group of the same conformer does not change no matter which method of calculation was carried out. The energy of the conformer with '''B3LYP/6-31G*''' is lower than that of '''HF/3-21G'''. Also, the bond lengths and bond angles changed with a certain amount as shown.

Furthermore, by submitting job type as '''Frequency''' in the optimised structure of anti2 conformer of 1,5-hexadiene under both calculation basis '''HF/3-21G '''and '''B3LYP/6-31G*''', the electronic energy, sum of electronic and zero-point energies at 0 K and sum of electronic and thermal energies at 298.15 K could be determined as shown in Table 3.

{| border="1" cellspacing="1" cellpadding="10"
|-
|rowspan="3"| ''' '''
!colspan="4" |'''HF/3-21G a'''
! colspan="4" |'''B3LYP/6-31G* b'''
|-
! rowspan="2" width="125" align="center" | '''Electronic energy'''
! width="125" align="center" | '''Sum of electronic and zero-point energies'''
! width="125" align="center" | '''Sum of electronic and thermal energies'''
! rowspan="2" width="125" align="center" | '''IR spectrum'''
! rowspan="2" width="125" align="center" | '''Electronic energy'''
! width="125" align="center" | '''Sum of electronic and zero-point energies'''
! width="125" align="center" | '''Sum of electronic and thermal energies'''
! rowspan="2" width="125" align="center" | '''IR spectrum'''
|-
! width="125" align="center" | '''at 0 K'''
! width="125" align="center" | '''at 298.15 K'''
! width="125" align="center" | '''at 0 K'''
! width="125" align="center" | '''at 298.15 K'''
|-
| '''Reactant (anti2)'''
| align="center" | -231.692534
| align="center" |-231.539539
| align="center" | -231.532565
|[[File:Yll113ANTI3-21IR.png|thumb|'''Figure 8. '''IR spectrum of anti2 reactant 1,5-hexadiene calculated by '''HF/3-21G''']]
| align="center" | -234.611703
| align="center" | -234.469213
| align="center" | -234.461857
|[[File:Yll113ANTI6-31IR.png|thumb|'''Figure 9. '''IR spectrum of anti2 reactant 1,5-hexadiene calculated by '''B3LYP/6-31G*''']]
|}

a [https://wiki.ch.ic.ac.uk/wiki/images/5/52/Yll113CR_ANTI_PART4.LOG File]; b [https://wiki.ch.ic.ac.uk/wiki/images/5/54/Yll113_CR_ANTI_PART3.LOG File]

'''Table 3.''' Summary of energies (in hartree) of the reactant (anti2) Comparing Figure 8 and 9, both IR spectra are similar which shows that they have similar stretching/bending mode even under different methods of calculations. However, the strongest peak in Figure 8 is at 1112 cm-1 and that in Figure 9 is 940 cm-1. Yet, they correspond to the same mode of vibration, which is the =C-H bending. Therefore, according to the equation, the wavenumber of absorbance, ν can be calculated:

:<math>\nu = \frac{1}{2 \pi c} \sqrt{\frac{k}{\mu}}</math>

where ''μ'' is the reduced mass, ''k'' is the spring constant for the bond and ''c'' is the light speed,

we can therefore know that the spring constant of the molecule under '''B3LYP/6-31G* '''basis reduced.

Now, focusing on gauche3 conformer of the 1,5-hexadiene, similarly, the '''HF/3-21G''' optimised molecule was reoptimised by '''B3LYP/6-31G'''. The energies and the geometries of the gauche3 1,5-hexadiene under two basis of calculation method were compared and shown in Table 4.
[[File:Yll113Gauche3.png|thumb|'''Figure 10. '''Gauche3 conformer of 1,5-hexadiene with labelled atoms]]
{| class="wikitable"
! rowspan="2" |Calculation Method
! rowspan="2" |Point Group
! rowspan="2" |Energy/ Hartrees
! rowspan="2" |RMS Gradient Norm/Hartrees
! colspan="3" |Bond length between/ Å
! colspan="2" |Bond angle between
|-
|C14-C12 and C4-C1
|C12-C9 and C6-C4
|C6-C9
|C14-C12-C9 and
C1-C4-C6
|C12-C9-C6 and
C4-C6-C9
|-

|'''HF/3-21G'''
|C1
|<nowiki>-231.69266120</nowiki>
|0.00001555
|1.31646
|1.50929
|1.55314
|125.02428o
|111.80728o
|-
|'''B3LYP/6-31G*'''
|C1
|<nowiki>-234.61132605</nowiki>
|0.00000360
|1.33382
|1.50491
|1.55007
|125.49464o
|113.46225o
|}
'''Table 4. '''Comparison with the anti2 conformer of 1,5-hexadiene with two calculation method basis ('''HF/3-21G '''and '''B3LYP/6-31G*''')

As seen in Table 4, the point group of the same conformer, again, does not change no matter which method of calculation was carried out. The energy of the conformer with '''B3LYP/6-31G*''' is lower than that of '''HF/3-21G'''. Also, the bond lengths and bond angles changed with a certain amount as shown.

Furthermore, by submitting job type as '''Frequency''' in the optimised structure of gauche3 conformer of 1,5-hexadiene under both calculation basis '''HF/3-21G '''and '''B3LYP/6-31G*''', the electronic energy, sum of electronic and zero-point energies at 0 K and sum of electronic and thermal energies at 298.15 K could be determined as shown in Table 5.

{| border="1" cellspacing="1" cellpadding="10"
|-
| rowspan="3" | ''' '''
! colspan="4" |'''HF/3-21G c'''
! colspan="4" |'''B3LYP/6-31G* d'''
|-
! rowspan="2" width="125" align="center" | '''Electronic energy'''
! width="125" align="center" | '''Sum of electronic and zero-point energies'''
! width="125" align="center" | '''Sum of electronic and thermal energies'''
! rowspan="2" |IR spectrum
! rowspan="2" width="125" align="center" | '''Electronic energy'''
! width="125" align="center" | '''Sum of electronic and zero-point energies'''
! width="125" align="center" | '''Sum of electronic and thermal energies'''
! rowspan="2" |IR spectrum
|-
! width="125" align="center" | '''at 0 K'''
! width="125" align="center" | '''at 298.15 K'''
! width="125" align="center" | '''at 0 K'''
! width="125" align="center" | '''at 298.15 K'''
|-
| '''Reactant (Gauche 3)'''
| align="center" |-231.692692
| align="center" |-231.539486
| align="center" |-231.532646
|[[File:Yll113GAUCHE3-21IR.png|thumb|'''Figure 11. '''IR spectrum of gauche3 conformer of 1,5-hexadiene calculated by '''HF/3-21G''']]
| align="center" | -234.611326
| align="center" |-234.468719
| align="center" |-234.461477
|[[File:Yll113GAUCHE6-31IR.png|thumb|'''Figure 12.''' IR spectrum of gauche3 conformer of 1,5-hexadiene calculated by '''B3LYP/6-31G*''']]
|}

c [https://wiki.ch.ic.ac.uk/wiki/images/9/98/Yll113CR_GAUGE_PART4.LOG File] ; d [https://wiki.ch.ic.ac.uk/wiki/images/c/ca/Yll113CR_GAUGE_PART3.LOG File]

'''Table 5.''' Summary of energies (in hartree) of the reactant (Gauche3) Comparing Figure 11 and 12, both IR spectra are similar which shows that they have similar stretching/bending mode even under different methods of calculations. However, the strongest peak in Figure 12 is at 939 cm-1 and that in Figure 11 is 1111 cm-1. Yet, they correspond to the same mode of vibration, which is also the =C-H bending. Therefore, similar to the anti2 conformer's case as mentioned above, we can therefore know that the spring constant of the molecule under '''B3LYP/6-31G* '''basis reduced.

===== Optimisation of transition state =====

After optimising the reactants, the chair and boat transition states were optimised accordingly using mainly two different methods. But before that, an allyl fragment (CH2CHCH2) was optimised using '''HF/3-21G''' level of theory for the sake of convenience in constructing the chair and boat transition states. A brief summary was shown in Table 6.
{| class="wikitable"
!
! colspan="2" |Molecule
!Point Group
!Energy/Hartree
!RMS Gradient Norm/ Hartrees
|-
|Allyl fragment
CH2CHCH2
|<jmol>
<jmolApplet>
<title>Figure C: Allyl Fragment</title><color>white</color>
<size>250</size>
<uploadedFileContents> Yll113CR TS 1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|[[File:Yll11313.jpg|thumb|'''Figure 13. '''Optimised Structure of the allyl fragment]]
|C2v
|<nowiki>-115.82304010</nowiki>
|0.00002945
|}
'''Table 6. '''Summary of the optimised allyl fragment

Then, both chair and boat transition state were drawn and optimised using the '''optimisation to TS (Berny)''' under '''HF/3-21G'''. Figure 14 and Figure C show the optimized structure of the chair transition state while Figure 15 and Figure D show the optimized structure of the boat transition state. Table 7 shows the summary of results.
{| class="wikitable"
! rowspan="2" |
! colspan="2" rowspan="2" |Molecule
! rowspan="2" |Point Group
! rowspan="2" |Energy/ Hartree
! rowspan="2" |RMS Gradient Norm/Hartrees
! colspan="6" |Partial C-C bond length between/ Å
|-
!C14-C6
!C11-C3
!C14-C9
!C6-C1
!C9-C11
!C1-C3
|-
|Optimised Chair Transition State
|<jmol>
<jmolApplet>
<title>Figure D: Optimised Chair transition state</title><color>white</color>
<size>250</size>
<uploadedFileContents>Yll113CHAIR PART1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|[[File:Yll113CHAIR3-21.png|thumb|'''Figure 14. '''Optimised Chair Structure by '''HF/3-21G''' with atoms labelled ]]
|C2h
|<nowiki>-231.61932238</nowiki>
|0.00002645
|2.02016
|2.02016
|1.38929
|1.38930
|1.38930
|1.38929
|-
|Optimised Boat Transition State
|<jmol>
<jmolApplet>
<title>Figure E: Optimised Boat transition state</title><color>white</color>
<size>250</size>
<uploadedFileContents>Yll113BOAT PART1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|[[File:Yll11315.jpg|thumb|'''Figure 15. '''Optimised Boat Structure by '''HF/3-21G''' with atoms labelled]]
|C2v
|<nowiki>-231.60280235</nowiki>
|0.00003872
|2.14060
|2.14060
|1.38138
|1.38138
|1.38138
|1.38138
|}

'''Table 7. '''Summary of the optimised chair and boat transition states by '''optimisation to TS (Berny) '''under '''HF/3-21G''' basis

Furthermore, the transition structures’ '''Frequencies''' were calculated as shown in Table 8.
{| class="wikitable"
!
!Molecule
!IR spectrum
!Imaginary Frequency/ cm-1
|-
|Boat Transition State
|[[File:Yll113Boat Part 1 imag freq.gif]]
|[[File:Yll11317.jpg|thumb|'''Figure 16. '''IR spectrum of the optimised Boat Structure by '''HF/3-21G''']]
|<nowiki>-840</nowiki>
|-
|Chair Transition State
|[[File: Yll113Chair Part 1 imag freq.gif]]
|[[File:Yll11316.jpg|thumb|'''Figure 17. '''IR spectrum of the optimised Chair Structure by '''HF/3-21G''']]
|<nowiki>-818</nowiki>
|}
'''Table 8.''' IR spectra and imaginary frequencies of the boat and chair transition states

As you may notice that, the
imaginary frequency comes up when calculating with the transition states. This
is common, in other words, this should appear to let us know the transition
structure we postulated is correct.

A transition state is the first
order saddle point on the potential energy surface. Therefore, the force
applied to the saddle point against to the displacement. As force and
displacement are vectors, the force constant will be a negative number.Therefore, according to
:<math>\nu = \frac{1}{2 \pi c} \sqrt{\frac{k}{\mu}}</math>

where ''μ'' is the reduced mass, ''k'' is the spring constant for the bond and ''c'' is the light speed,

the square root of a negative
force constant k gives an imaginary wave number/frequency v. In other words,
the appearance of an imaginary frequency tells us that the structure is a
saddle point of the reaction pathway.

The chair transition state
was followed by first 'frozen' then 'relaxed'. The boat transition structure
was followed by '''QST2''' calculation method.

====== Chair Transition State ======
After the above '''optimisation''', the chair transition
state was reoptimised again with another method. This method first freezes the
coordinate of the molecule, in this case, freeze the bond to be made in the
Cope Rearrangement of 1,5-hexadiene as 2.20000 Å. The molecule then optimised with the frozen
coordinate. Details of this optimisation was summarized in Table 9.
{| class="wikitable"
! rowspan="2" |
! colspan="2" rowspan="2" |Molecule
! rowspan="2" |Point Group
! rowspan="2" |
{| class="MsoTableGrid"

|
<nowiki> </nowiki>Energy/ Hartree

|}
! rowspan="2" |RMS Gradient Norm/ Hartree
! colspan="3" |Bond length between/ Å
! rowspan="2" |Imaginary Frequency/ cm-1
! rowspan="2" |IR Spectrum
|-
!C6-C14 and C3-11
!C1-C3 and C9-C14
!C1-C6 and C9-C11
|-
|Optimised Chair Transition Structure with frozen coordinate
|<jmol>
<jmolApplet>
<title>Figure F: Optimised Chair transition state with frozen coordinate</title><color>white</color>
<size>250</size>
<uploadedFileContents>Yll113CHAIR PART2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
[[File: Yll113Chair frozen.gif]]
|[[File:Yll11318.jpg|thumb|'''Figure 18. '''The optimised chair transition structure with frozen coordinate and atoms labelling]]
|
{| class="MsoTableGrid"

|
<nowiki> </nowiki>C2h

|}
|<nowiki>-231.61518510</nowiki>
|0.00325573
|2.20000
|1.38135
|1.38128
|<nowiki>-765</nowiki>
|[[File:Yll11319.jpg|thumb|'''Figure 19. '''IR spectrum of the optimised chair transition structure with frozen coordinate]]
|}

'''Table 9. '''Summary of the optimisation of the chair transition structure with
frozen coordinate(s)

From Table 9, we may notice
that the RMS Gradient Norm value is quite far off from zero. Also, the
imaginary frequency becomes much higher than -818 cm-1 (Shown in
Table 8). With these two pieces of information, we can deduce that the frozen
coordinate(s) affect(s) the force constant of the transition state which does
not give a good optimisation of transition structure. With that, after applying
the frozen coordinate to the molecule, the molecule was reoptimised again with
a degree of '''Derivative '''to the '''Bond'''. Details of the reoptimisation
were presented in Table 10.
{| class="wikitable"
! rowspan="2" |
! colspan="2" rowspan="2" |Molecule
! rowspan="2" |Point Group
! rowspan="2" |Energy/ Hartree
! rowspan="2" |RMS Gradient Norm/ Hartree
! colspan="4" |Bond length between/ Å
! rowspan="2" |Imaginary Frequency/ cm-1
! rowspan="2" |
{| class="MsoTableGrid"

|
<nowiki> </nowiki>IR Spectrum

|}
|-
!C14-C6
!C11-C3
!C14-C9 and C6-C1
!C9-C11 and C1-C3
|-
|Optimised Chair Transition
Structure with a degree of '''Derivative'''
to the '''Bond'''
|<jmol>
<jmolApplet>
<title>Figure G: Optimised Chair transition state with a degree of Derivative to the Bond</title><color>white</color>
<size>250</size>
<uploadedFileContents>Yll113CHAIR PART3.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
[[File: Yll113Chair relax.gif]]
|[[File:Yll11320.jpg|thumb|'''Figure 20. '''The optimised chair transition structure with a degree of '''Derivative''' to the '''Bond''' and atoms labelling]]
|C2h
|<nowiki>-231.61932233</nowiki>
|0.00002127
|2.02075
|2.02071
|1.38929
|1.38930
|<nowiki>-818</nowiki>
|[[File:Yll11321.jpg|thumb|'''Figure 21. '''IR spectrum of the optimised chair transition structure with a degree of '''Derivative''' to the '''Bond''']]
|}
'''Table 10. '''Summary of the reoptimisation of the chair transition
structure with a degree of '''Derivative''' to the '''Bond'''

Now, in Table 10, the RMS
Gradient Norm value is close to zero. Also, the imaginary frequency goes back
to -818 cm-1, indicating that the coordinates no longer be frozen
and the stretching/bending mode of the transition state is able to undergo
freely.

Comparing the bond lengths
in Table 7 and 10, we can see that the difference between bond lengths of the
single bond to be made/ broken calculated in two methods is just less than
0.0006 Å. And also, there is no difference in bond length of the double bond to be make/broken ‘inside’ the system. This tells us that the two optimisation
methods are rather similar under the consideration on the Cope Rearrangement
Reaction.

====== Boat Transition State ======
Instead of using the frozen
coordinate method as for the chair transition state above, another method, '''QST2''', was applied to the boat
transition state under the '''HF/3-21G'''
basis. In order to use this method, without any ‘Link died’, the reactant and
product have to be drawn and labelled carefully. '''QST2''' is a method which interpolates the reactant and product to
give a transition state. Therefore, it will fall if the structure of the
reactant and product are not close to the transition state. And therefore, all
molecules have to be carefully labelled and adjusted.

[[File:Yll11322.png|500px|center]]

'''Figure 22. '''The drawings and adjustments of angles of the reactant (left)
and product (right) for '''QST2''' Method,
i.e. the central C-C-C-C dihedral angle was changed to 0o and inside
C-C-C were reduced to 100o

After the adjustment, the job was run and the optimized molecule converge to the boat transition structure. Summary was shown in Table 11.
{| class="wikitable"
! rowspan="2" |
! colspan="2" rowspan="2" |Molecule
! rowspan="2" |Point Group
! rowspan="2" |Energy/Hartree
! rowspan="2" |RMS Gradient Norm/ Hartree
! colspan="3" |Bond length between/ Å
! rowspan="2" |Imaginary Frequency/ cm-1
! rowspan="2" |IR spectrum
|-
!C1-C6
!C3-C4
!C5-C6, C4-C5, C3-C2 and C1-C2
|-
|Boat transition structure
under '''QST2''' method
|[[File:Yll113Boat qst2.gif]]
|[[File:Yll11323.jpg|thumb|'''Figure 23. '''The optimised Boat transition structure with atom labelling]]
|C2v
|<nowiki>-231.60280241</nowiki>
|0.00002436
|2.13994
|2.14019
|1.38149
|<nowiki>-840</nowiki>
|[[File:Yll11324.jpg|thumb|'''Figure 24. '''IR spectrum of the optimised boat transition structure]]
|}
'''Table 11. '''Summary of the boat transition structure under '''QST2 '''method

====== Intrinsic Reaction Coordinate''' '''('''IRC''') ======
In order to confirm that our transition state is of the
correct one, '''Intrinsic Reaction
Coordinate '''('''IRC''') will be carried
out.

As mentioned above, transition state is the first order
saddle point of the reaction pathway. Therefore, it will start to go to the
product or back to the reactant with it falls off. It resembles that a ball is
at the tip of the mountain, which starts to roll off the mountain on the side
with the steepest slope. Also, when we are doing '''IRC''', we can determine whether the reaction goes forward, backward
or both sides. Also, the number of points, which means the number of little
steps that the geometry of the molecule changes, can be adjusted. A low number
of points will just give us a very rough idea that tell us a little bit about
our transition state. Also, the last point on the '''IRC''' is far from the minimum geometry. A high number of points gives
us more accurate results, however two problems could be raised. First, the time
for calculation will be long and Most importantly, as it goes down the slope
and reaches the minimum geometry, i.e. the plateau of energy, the slope will
become very small or even zero again. However, as the energy difference of the
next or previous geometry compared to the geometry of itself is too small, ''GaussView 5.0'' may not able to know which
direction the molecule should proceed to. And this, therefore, results in ‘Link
died’. Therefore, the most common technique is to have a good number of points,
then take the last point on the IRC and run it with a normal optimisation.

Here, as we know that the
Cope Rearrangement has a symmetric reaction pathway, taking the chair
transition structure, we will run '''IRC'''
on it with 50 points.
{| class="wikitable"
![[File:Yll113hlj29825.jpg|thumb|'''Figure 25. '''Total energy along '''IRC '''of the chair transition structure]]
![[File:Yll11326.jpg|thumb|'''Figure 26. '''RMS Gradient Norm of '''IRC '''of the chair transition structure]]
![[File: Yll113Chair irc.gif]]
|}

[[File:Yll11327.jpg|500px]]

'''Figure 27. '''The product of the Cope Rearrangement after optimisation

The first point on Figure 25 is -231.61932233 Hartree and the last point is -231.69157881 Hartree. Then, we take the last point and optimise it, we get the structure shown in Figure 27.

The structure is of C2
symmetry and the energy calculated is -231.69166702 Hartree. This matches with
Gauche2 C2 on Appendix 1. And therefore, this is how the conformer
of 1,5-hexadiene connects with the chair transition structure.

====== Activation Energy of the Cope Rearrangement ======
Finally, we optimise the chair and boat transition states we got from above, reoptimise it with job Opt + Freq
under a more advanced calculation '''B3LYP/6-31G*'''. And from that, the thermochemistry data were given and we can know the
activation energy of the reaction by comparing to Table 3, which anti2 is used
as a local minimum rather than gauche3 as a global minimum.
{| border="1" cellspacing="1" cellpadding="10"
|-
| rowspan="3" | ''' '''
! colspan="3" |'''HF/3-21G'''
! colspan="3" |'''B3LYP/6-31G*'''
|-
| rowspan="2" width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| rowspan="2" width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619323
| align="center" | -231.466698
| align="center" | -231.461339
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409009
|-
| '''Boat TS'''
| align="center" | -231.602803
| align="center" | -231.450932
| align="center" | -231.445303
| align="center" | -234.543094
| align="center" | -234.402342
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692534
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.611703
| align="center" | -234.469213
| align="center" | -234.461857
|}
'''Table 11'''. Summary of energies of chair, boat and reactant (anti2) structure
{| border="1" cellspacing="1" cellpadding="10"
|-
| rowspan="2" | ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.69
| align="center" | 34.06
| align="center" | 33.16
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.96
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
'''Table 12'''. Summary of activation energies in kcal/mol

==== The Diels-Alder Cycloadditions ====

===== Ethylene and Cis-Butadiene =====
First, the structures of the reactants, ethylene and cis-butadiene, were '''optimised to minimum''' under '''semiempirical AM1''' method. For the butadiene, in order to be in the cis conformer, the dihedral angle was adjusted to be 0o. Details are listed in Table 13.
{| class="wikitable"
!
!Molecule
!Point Group
!HOMO
!LUMO
!Energy/Hartree
!RMS Gradient Norm/ Hartree
!IR spectrum
|-
|Ethylene
|<jmol>
<jmolApplet>
<title>Figure H: Optimised Ethylene</title><color>white</color>
<size>250</size>
<uploadedFileContents>Yll113ETHENE OPTAM1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|D2h
|[[File:Yll11331.jpg|thumb|'''Figure 31. '''HOMO of Ethylene]]
|[[File:Yll11330.jpg|thumb|'''Figure 30.''' LUMO of ethylene]]
|0.02619029
|0.00008755
|[[File:Yll11328.jpg|thumb|'''Figure 28. '''IR spectrum of Ethylene]]
|-
|Cis-Butadiene
|<jmol>
<jmolApplet>
<title>Figure I: Optimised Cis-Butadiene</title><color>white</color>
<size>250</size>
<uploadedFileContents>Yll113CISBUTADIENE OPTAM1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|C2v
|[[File:Yll11332.jpg|thumb|'''Figure 32. '''HOMO of cis-butadiene]]
|[[File:Yll11333.jpg|thumb|'''Figure 33. '''LUMO of cis-butadiene]]
|0.04878534
|0.00000087
|[[File:Yll11329.jpg|thumb|'''Figure 29.''' IR spectrum of cis-butadiene]]
|}
'''Table 13.''' Summary of optimised ethylene and cis-butadiene

Looking into Figure 30-33, as we know that the plane is perpendicular to the molecule, the HOMO of Ethylene is symmetric while that of LUMO is antisymmetric.

Also, the HOMO of cis-butadiene is antisymmetric and that of LUMO is symmetric.

Then, the transition state of the reaction was able to constructed using the optimised structure of the reactants made above. By making the bond lengths of the bonds to be made in Diels-Alder [4+2] cycloaddition to 2.20000 Å in the transition state, which is the 'dash bond', the transition structure was optimized to '''Opt + Freq minimisation to''' '''TS (Berny)''' under the same method of calculation - '''semiempitical AM1'''. Details are listed in Table 14.
{| class="wikitable"
!
! colspan="2" |Molecule
!Point Group
!HOMO
!LUMO
!Energy/Hartree
!RMS Gradient Norm/ Hartree
!Imaginary Frequency/ cm-1
!IR spectrum
|-
|Transition state
|JSMOL
|[[File:Yll11334.jpg|thumb|'''Figure 34. '''Transition state of ethylene + cis-butadiene]]
|C1
|[[File:Yll11336.jpg|thumb|'''Figure 36. '''HOMO of transition state]]
|[[File:Yll11337.jpg|thumb|'''Figure 37. '''LUMO of transition state]]
|0.11165467
|0.00002792
|<nowiki>-956</nowiki>
|[[File:Yll11335.jpg|thumb|'''Figure 35. '''IR spectrum of the transition state]]
|}
'''Table 14.''' Summary of optimised transition state

From Figure 36, we can see that the HOMO of the transition state is antisymmetric whilst the LUMO of the transition state is symmetric. By making very careful comparison between Figure 36, Figure 37 and Figure 30-33, we can see that the HOMO of the transition state in Figure 36 is a combination of Figure 32 and 30; the LUMO of the transition state in Figure 37 is a combination of Figure 31 and 33. We can clearly see that the HOMO and LUMO of the transition state have a complementary combination of HOMO and LUMO of the reactants.

Taking a closer look to HOMO of the transition state. Recalling Woodward Hoffmann’s Rule, (4q+2)s+(4r)a = odd for thermally allowed reaction, we have both π2s and π4s. Therefore, the reaction is thermally allowed by letting q = 0, which gives the value of 1 which is odd.

Furthermore, from Table 14, we notice that there is an imaginary frequency reported at -956 cm-1. As explained above, the transition state should have one imaginary frequency to account for the negative force constant. With that, this imaginary frequency confirms that the transition structure we postulated from the optimised reactants is valid, i.e. it is really a transition state. The animation of where the imaginary frequency originates from, which shows the motion of the transition state - how the two reactants approach to each other and bonds are formed, is shown below.

JSMOL

From the above figure, we can see that the bond formation from the reactant to the product happens at the same time, i.e. synchronous, on both sides of the transition structure. Therefore, we can say that this Diels-Alder cycloaddition is a concerted [4+2] pericyclic cycloaddition, which matches with what we learnt in Pericyclic Reaction course.

On top of that, the geometry of the transition structure was investigated by looking into the optimised bond lengths between carbon atoms Details are shown in Figure 38 and Table 15.[[File:Yll11338.jpg|thumb|'''Figure 38. '''Transition state of ethylene + cis-butadiene with atoms labelled]]
{| class="wikitable"
!
!Bond length/ Å
|-
|C7-C9
|2.11938
|-
|C12-C5
|2.11944
|-
|C12-C9
|1.38284
|-
|C3-C7
|1.38187
|-
|C3-C1
|1.39750
|-
|C5-C1
|1.38175
|}
'''Table 15. '''Geometry analysis of the transition state

According to the literature <ref>M. A. Fox and J. K. Whitesell, ''Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen'', Springer, 1995</ref>, C-C carbon-carbon single bond is 1.54 Å, and C=C carbon-carbon double bond is 1.34 Å. Also, the Van der Waals radius of carbon is 1.70 Å,<ref>A. Bondi,(1964), ''J. Phys. Chem.'', 1964, '''68''' (3), 441</ref>
According to the reaction scheme shown in Figure 3, a single bond is forming between C7 and C9, also another single bond is forming between C12-C5. Comparing the data in Table 15 with the literature, we can see that the bond length of two bonds to be made is longer than C-C, but shorter than the twice of carbon's Van der Waals radius. This tells us some hints that the terminal carbons between two reactants are attracting with each other to a certain degree. Furthermore, taking a look to other carbon-carbon bonds within the system, we notice that they are somewhere between C-C and C=C. This tells us that either C-C is becoming C=C or C=C is becoming C-C in the transition state.
After that, the '''IRC''' of the above optimised transition state was carried out with both direction and force constant calculated always for 50 points to see the reaction profile.
{| class="wikitable"
|[[File:Yll11339.jpg|thumb|'''Figure 39.''' IRC of the transition state of ethylene + cis-butadiene]]
|[[File:Yll11340.jpg|thumb|'''Figure 40. '''RMS Gradient Norm of transition state of ethylene + cis-butadiene]]
|JSMOL
|}
In Figure 39, we can clearly see that the reactants was first passed through the energy barrier to get the transition state and it went down the slope to give the product.
Finally, the activation energy for this reaction was calculated in Table 16.
{| class="wikitable"
!
!Ethylene
!Cis-butadiene
!Transition state
!Activation Energy
|-
|Energy/Hartree
|0.02619029
|0.04878534
|0.11165467
|0.03667904
(23.02 kcal/mol)
|}
'''Table 16. '''Activation energy analysis of Diels-Alder Reaction between ethylene and cis-butadiene

===== Cyclohexa-1,3-diene and Maleic Anhydride =====
First, the structures of
the reactants, ethylene and cis-butadiene, were '''optimised to minimum''' under '''semiempirical AM1''' method. Details are listed in Table 17.
{| class="wikitable"
!Optimised Structure
!Molecule
!Point Group
!Energy/Hartree
|-
|Cyclohexa-1,3-diene
|[[File:Yll113Cycloopt.jpg|thumb|'''Figure 41. '''Optimised structure of cyclohexa-1,3-diene]]
|C2
|0.02771130
|-
|Maleic Anhydride
|[[File:Yll113Maopt.jpg|thumb|'''Figure 42. '''Optimised structure of Maleic Anhydride]]
|Cs
|<nowiki>-0.12182418</nowiki>
|}
'''Table 17. '''Summary
of the optimized reactant

Then, the two transition states, exo and endo, of the reaction were able to construct. By making the bond lengths of the bonds to be made in Diels-Alder [4+2] cycloaddition to 2.20000 Å in the transition state, which is the 'dash bond', the transition structure was optimized to '''Opt + Freq minimisation to''' '''TS (Berny)''' under the same method of calculation - '''semiempitical AM1'''. Details are listed in Table 18.
{| class="wikitable"
!Transition Structure
!Molecule
!Point Group
!HOMO
!LUMO
!Energy/Hartree
!'''RMS Gradient Norm/ Hartree'''
!'''Imaginary Frequency/ cm-1'''
!IR spectrum
|-
|Exo
|[[File:Yll113Exots.jpg|thumb|'''Figure 43.''' Optimised Structure of Exo Transition State]]
|C1
|[[File:Yll113Exo homo.jpg|thumb|'''Figure 45. '''HOMO of Exo Transition State]]
|[[File:Yll113Exo lumo.jpg|thumb|'''Figure 47.''' LUMO of Exo Transition State]]
|<nowiki>-0.05041984</nowiki>
|0.00000883
|<nowiki>-812</nowiki>
|[[File:Yll113Exo ir.jpg|thumb|'''Figure 49. '''Exo Transition State IR spectrum]]
|-
|Endo
|[[File:Yll113Endots.jpg|thumb|'''Figure 44. '''Optimised Structure of Enfo Transition State]]
|C1
|[[File:Yll113Endo homo.jpg|thumb|'''Figure 46'''. HOMO of Endo Transition State]]
|[[File:Yll113Endo lumo.jpg|thumb|'''Figure 48.''' LUMO of Endo Transition States]]
|<nowiki>-0.05150469</nowiki>
|0.00004308
|<nowiki>-807</nowiki>
|[[File:Yll113Endo ir.jpg|thumb|'''Figure 50.''' Endo Transition State IR Spectrum]]
|}
'''Table 18. '''Summary of the optimized transition structures

In Table 18, we can see that both HOMO and LUMO of the exo
transition state are antisymmetric with respect to the plane. Also, both HOMO and LUMO of the endo transition state are antisymmetric as well.

Also, we notice that the energy of exo is higher than that of endo. This can be explained by the poorer overlap between the C=C π and C=O π* compared to that of endo. This is called secondary orbital effect, which will be further discussed below.

There is one imaginary frequency in both transition structure. This confirms that both of them are valid postulated transition structures.
{| class="wikitable"
|JSMOL
'''Imaginary frequency of exo transition state animation'''
|JSMOL
'''Imaginary frequency of exo transition state animation'''
|}
Again, from the above animations, we can see that the bonds are formed in a synchronous fashion, which means the Diels-Alder Cycloaddition reaction is concerted.

Furthermore, the geometries of both transition states were examined carefully in Table 19.
{| class="wikitable"
!Geometry summary of Exo Transition State (Please refer to Figure 43 for atom labelling)
!Geometry summary of Endo Transition State (Please refer to Figure 44 for atom labelling)
|-
|
{| class="wikitable"
!
!Bond length/ Å
!
!Bond length/ Å
|-
|C5-C13
|2.17032
|C6-C14
|2.17035
|-
|C5-C3
|1.39440
|C3-C1
|1.39674
|-
|C1-C6
|1.39440
|C14-C13
|1.4612
|-
|C7-C6
|1.48976
|C7-C10
|1.52208
|-
|C10-C5
|1.48976
|C13-C15
|1.48822
|-
|C14-C16
|1.48822
| colspan="2" |N/A
|-
|C7-C16
(Through space)
|2.94507
|C10-C15
(Through space)
|2.94484
|-
|C1-C16
(Through space)
|3.78172
|C3-C15
(Through Space)
|3.78155
|}
|
{| class="wikitable"
!
!Bond length/ Å
!
!Bond length/ Å
|-
|C15-C7
|2.16230
|C16-C5
|2.16229
|-
|C1-C3
|1.39726
|C3-C7
|1.39296
|-
|C1-C5
|1.39308
|C7-C9
|1.49503
|-
|C9-C12
|1.52300
|C5-C12
|1.49054
|-
|C16-C18
|1.48918
|C15-C17
|1.48903
|-
|C15-C16
|1.40863
| colspan="2" |N/A
|-
|C1-C18
(Through space)
|2.89232
|C3-C17
(Through space)
|2.89203
|}
|}
'''Table 19.''' Geometry analysis of exo and endo transition states

According to the reaction scheme shown in Figure 4, a single bond is forming between C5 and C13, also another single bond is forming between C6-C14 for exo; C15 and C7 plus C16 and C5 for endo, which is what the first row in the two tables in the left and right in Table 19 shows. the single bond to be made Comparing these values with literature, we find that they are longer than C-C but shorter than twice of carbon's Van der Waals' radius. This tells us some hints that these pairs of carbons between two reactants are attracting with each other to a certain degree. Furthermore, taking a look to other carbon-carbon bonds within the system, i.e. except row 1 and those labelled with (through space), we notice that they are somewhere between C-C and C=C. This tells us that either C-C is becoming C=C or C=C is becoming C-C in the transition state.

Now, looking at the through space bond length. In the exo transition structure, the through space distance -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH2-CH2- is smaller than the twice of the Van der Waals radius but larger than C-C which gives us a hint that they might interact. In the endo transition structure, the through space distance -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH=CH- is smaller than the twice of the Van der Waals radius but larger than C-C which gives us a hint that they might interact. However, according to the definition of secondary orbital effect, it is looking for the interaction between the C=C π of the diene and C=O π* of the dienophile. Endo clearly shows that as explained, but exo seems to just demonstrate the sterics clash between the -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH2-CH2- fragment of diene. In order to further confirm that exo has no secondary orbital effect, a measurement of bond length was carried out between -(C=O)-O-(C=O)- fragment of the maleic anhydride and the -CH=CH- in diene in the exo transition state. The result was shown in the last row on the left table in Table 19. This shows that they are too far away which means they are not possible to interact.

Now, looking back to the HOMO of exo and endo transition states in Figure 45 and 46 respectively. We can definitely see that the overlap between the two reactants is relatively smaller in exo. From these two pieces of information, we can conclude that the endo is kinetically controlled, while exo is thermodynamically controlled.

After that, the '''IRC''' of the both optimised transition state was carried out with both direction and force constant calculated always for the reaction profile. 21 points were used for exo transition states and 24 for endo (reasons explained under '''Introduction)''' to see the reaction profiles.
{| class="wikitable"
! colspan="3" |Exo Transition State
! colspan="3" |Endo Transition State
|-
|[[File:Yll113Exo irc.jpg|thumb|'''Figure 51.''' IRC of the exo transtion state]]
|[[File:Yll113Exo rms.jpg|thumb|'''Figure 52. '''RMS of the exo transition structure]]
|JSMOL
|[[File:Yll113Endo irc.jpg|thumb|'''Figure 53. '''IRC of the endo transition state]]
|[[File:Yll113Endo rms.jpg|thumb|'''Figure 54.''' RMS of the endo transition state]]
|JSMOL
|}
And eventually, the activation energies of the reaction via different transition structures were summarised in Table 20.
{| class="wikitable"
!
!Cyclohexa-1,3-diene
!Maleic Anhydride
!Endo Transition State
!ExoTransition State
!Activation Energy via endo
!Activation Energy via exo
|-
|Energy/Hartree
|0.02771130
|<nowiki>-0.12182418</nowiki>
|<nowiki>-0.05150469</nowiki>
|<nowiki>-0.05041984</nowiki>
|0.04260819
(26.74 kcal/mol)
|0.04369304
(27.42 kcal/mol)
|}
'''Table 20.''' Activation energy analysis

=== '''Conclusion''' ===

=== '''Reference''' ===

<gallery>
File:
</gallery>

File:Yll113CISBUTADIENE OPTAM1.LOG

2015-12-17T08:07:58Z

Yll113:

File:Yll113ETHENE OPTAM1.LOG

2015-12-17T08:06:25Z

Yll113:

Rep:Mod:hlj298

2015-12-17T08:03:56Z

Yll113: /* Intrinsic Reaction Coordinate (IRC) */

== '''Study of the reaction profiles of the Cope Rearrangement and the Diels-Alder Cycloadditions''' ==
'''Y. L. J. Lam'''

''Department of Chemistry, South Kensington Campus, Imperial College London, London SW7 2AZ, United Kingdom''

''Received 18 December, 2015''

=== Abstract ===
The reactants, products and transition states of the Cope
Rearrangement of 1,5-hexadiene were investigated by ''GaussView 5.0'' at the '''HF/3-21G''' and '''B3LYP/6-31G*'''levels
of theories respectively. With that, the point groups, vibrational frequencies and different energies at different temperatures of the reactants, products and transition states were calculated. Also, by optimizing the transition structures with different methods, i.e. computing the force constants at the
beginning of the calculations, using the redundant coordinate editor and '''QST2''', at the '''HF/3-21G''' level of theory, closer views of the geometries of the transition states can be observed. Furthermore, by using the '''IRC''' method, the reaction profiles can be
obtained and the activation energies can therefore be calculated. Plus, using '''IRC''' method, all reaction intermediates
can now be observed, which helps us to understand the mechanism of the Cope Rearrangement. Similarly, for Diels-Alder Cycloadditions between ethene and
butadiene and Diels-Alder Cycloadditions between cyclohexa-1,3-diene and maleic anhydride, the reactants, products and transition states were optimized and
their respective energies were calculated at '''AM1 semi-empirical molecular orbital method'''. Furthermore, the symmetries of the
molecular orbitals were visualized and the reaction profiles calculated by '''IRC''' method were obtained.

=== '''Introduction''' ===
Chemical reactions are happening around the world in every second. Some reactions are fast, whilst some are slow. The most common and general reason for that is on the kinetic and thermodynamic aspects. On the kinetic aspect, we might argue that the energy barrier(s) form the reactant(s) to the product(s) is/are huge, and therefore, the reactant(s) cannot overcome the barrier(s) and the reaction is slow or does not proceed. The transformation between crude carbon and diamond is a good example. The energy difference between crude carbon and diamond is just few kcal/mol, however, the energy barrier for the transformation is huge. Hence, the transformation is extremely slow, or even does not proceed. With that, diamond symbolizes eternity. On the other hand, on the thermodynamic aspect, we might argue that the reaction is endothermic, i.e. absorbing/requiring heat from the surroundings in order to proceed. In fact, these two aspects just provide us with a little bit of the story and therefore, chemists, or scientists in general, study the mechanism of the reactions to find out the full story. Unfortunately, some reactions are spontaneous, such as the thiocyanation of the iron complex. Also, some intermediates of the reactions are unstable, which cannot be separated or detected even using very advanced analytical instruments, such as nuclear magnetic resonance (NMR) spectromenter. Therefore, scientists devised some programs and computational methods to find out the mechanism of the reactions. Here we use ''GaussView 5.0'' for our investigation.

==== Computational Theory ====
[[File:Yll113 AM1 and HF.jpg|thumb|463x463px|'''Figure 1.''' HOMO and LUMO (highlighted in yellow) of cis-butadiene under the basis of calculation '''HF/3-21G '''(left) and '''AM1''' (right)]]
In ''GaussView 5.0'', there are numerous methods for calculation, such as '''HF/3-21G''', '''B3LYP/6-31G*''', '''semiempirical AM1''', '''MP4 '''and '''MP2'''. Here, the first two calculation method, namely, '''HF/3-21G''' and '''B3LYP/6-31G* '''were applied for calculation of the Cope Rearrangement Reaction, while '''semiempirical AM1''' was used for the investigation of the two Diels-Alder reactions.

N.B. No matter which method applied, the RMS Gradient Norm in hartress would also be computed. This is a measure of how well does the optimisation go during the calculation of the
structure drawn. The closer to zero, the better the structure is optimised.

===== Hartree-Fock ('''HF''') Method =====
Hartree-Fock theory ('''HF''') is the fundamentals of electronic structure theory. It gives a good starting point for more elaborate theoretical methods which can approximate the electronic Schrödinger equation better. It is the basis of the molecular orbital (MO) theory that assumes the motion of each electron can be described by a single-particle function/orbital and it does not depend on/interact with the instantaneous motions of the other electrons.<ref>C. D. Sherrill, ''An Introduction to Hartree-Fock Molecular Orbital Theory'', 2000</ref>

===== Becke, 3-parameter, Lee-Yeang-Parr ('''B3LYP''') Method =====
Beeke, 3-parameter, Lee-Yang-Parr ('''B3LYP''') is one of the most commonly used hybrid functionals. Hybrid functionals are a class of approximation of the exchange-correlation energy functional in density functional theory.<ref>What is B3LYP?, https://www.quora.com/What-is-B3LYP (accessed December 2015)</ref> '''B3LYP''' contains an '''HF''' exchange with the weight of 0.2, which can be regarded as a uniform screening of
exchange by 80 %.<ref>C. H. Patterson, ''International Journal of Quantum Chemistry'', 2006, '''106 '''(15), 3383</ref> '''B3LYP''' also takes a set of atomization
and ionization energies, proton affinities and total atomic energies into account.<ref>A. D. Becke, ''The Journal of Chemical Physics'', 1993, '''98''', 5648</ref>

===== Semiempirical Austin Model 1 ('''AM1''') =====
Semiempirical Austin Model 1 ('''AM1''') based on the Neglect of Differential Diatomic Overlap (NDDO) integral approximation.<ref>M.
J. S. Dewar, E. G. Zoebisch, E. F. Healy and J. J. P. Stewart, ''J. Am. Chem. Soc.'', 1985, '''107''' (13), 3902</ref> Therefore, when taking the same molecule for '''AM1''' and '''HF/3-21G''', you would find that the numbers of HOMO and LUMO are different, which '''AM1''' gives smaller numbers as shown in Figure 1. This is due to the neglect of the low-lying orbitals overlapping. With that, '''AM1''' proceeds much faster than '''HF/3-21G''' for the sake of time.

==== The Cope Rearrangement ====
The Cope Rearrangement is an organic reaction involving [3,3]-sigmatropic rearrangement of 1,5-dienes, which resembles the Claisen Rearrangement.<ref>A. C. Cope and E. M. Hardy, ''J. Am. Chem. Soc.'', 1940, '''62''' (2), 441</ref> The mechanism of the Rearrangement has sparked a controversy – whether it is concerted, dissociative or stepwise.<ref>O. Wiest, K. A. Black and K. N. Houk, ''J. Am. Chem. Soc.'', 1994, '''116''', 10336</ref> With that, first, each conformer of the reactant, 1,5-hexadiene, was optimised by '''HF/3-21G'''. The lowest energy conformer of 1,5-hexadiene was found. Then, as we know, the Rearrangement undergoes either a chair or boat transition state. So, each transition state was optimised by '''HF/3-21G '''as well. By looking into the energy difference between the transition states and the reactant, the activation energy of the Cope Rearrangement with 1,5-hexadiene was found. In order to find the reaction profile and see how the 1,5-diene rearranges, i.e. the mechanism, the transition state was optimised again with
mainly two methods. The coordinate of the chair transition state was first frozen, with the bond to be made set as 2.20000 Å. 2.20000 Å is a good bond length for partially C-C bond as suggested by the chemists’ observations in the literature. <ref>N. H. Kendall, Y. Li and J. D. Evanseck, ''Angew. Chem. Int. Ed. Engl.'', 1992, '''31''' (6), 682</ref> Then, after the optimization of the frozen coordinate, the partly form 2.20000 Å can be relaxed and the structure was then reoptimised. This methods skips the process of computing the whole force constant matrix i.e. Hessian, which saves time and costs. Furthermore, the boat transition state was optimised by '''QST2'''. '''QST2''' has a higher constrains in which requires a more accurate transition state structure to be put in. In this case, the dihedral angle plays an important role in order to be calculated by ''GaussView'' 5.0. Hence, this method is more expensive and time-consuming. From the optimised transition states, an '''IRC''' can be run for the optimised structure to see the full reaction profile. Also, the intermediates of the reaction can be observed. And finally, the reactant and two transition states
were optimised with '''B3LYP/6-31G*''' similarly. Hence, the two calculation methods can be compared by looking into the numbers obtained. Also, the numbers can be compared against the
experimental values. As explained above, '''B3LYP''' takes a more in-depth consideration, the numbers got from this method should be closer to the reality.

==== The Diels-Alder Cycloaddition ====
The Diels-Alder cycloaddition is a [4+2] cycloaddition between a dienophile and a conjugated alkene to give a cyclohexane system. Here, calculations on two Diels-Alder cycloaddition reactions are reported. They are (1) ethylene and butadiene and (2) cyclohexa-1,3-diene and maleic anhydride.

For Diels-Alder cycloaddition reaction, it is well-known that the reaction gives exo and/or endo product. Exo product implies the reaction pathway is thermodynamically controlled to give more stable product; endo product implies
the reaction pathway is kinetically controlled to give a relatively less stable product. In other words, the activation energy to form the exo product is higher than that of endo, however, the endo product is higher in energy than exo. This can usually be explained by the secondary orbital effects. In our cases, both the exo and endo products were investigated undoubtedly. This time, as you may notice, the molecule is more large in size and there are two reactants instead of just one reactant in the Cope Rearrangement, a simpler method of calculation was implemented, which is '''AM1'''. Also, the electronic distributions and orbitals of the HOMO and LUMO of the transition states were computed and visualised.

=== '''Computational Method''' ===
''All calculations were performed by GaussView 5.0. Relevant JSmol files were uploaded here, however, due to some technique glitches, some bonds, especially double bonds, might not come up properly. Yet, the structures of the molecules are generally correctly shown.''

==== The Cope Rearrangement ====
[[File:Yll113 CR.png|thumb|'''Figure 2.''' The Cope Rearrangement of 1,5-hexadiene]]
An anti and gauche conformation of the 1,5-hexadiene were drawn respectively. The drawn structures were first optimised by a not very accurate technique, i.e. '''Clean'''. Then, the '''clean'''ed structure were optimised by '''HF/3-21G'''. The point group and the energy of each conformer were found and compared to locate the low-energy minima. The optimised structures from '''HF/3-21G''' were then reoptimised by '''B3LYP/6-31G*'''. The point group of each conformer was checked and confirmed. Also, the comparison of the same conformer under different calculation method '''HF/3-21G''' and '''B3LYP/6-31G''' was carried out by looking into energy, bond lengths and bond angles. Furthermore, the '''Thermochemistry''' using job type '''Frequency''' was found in both '''HF/3-21G '''and''' B3LYP/6-31G* '''optimised anti conformers.

The boat and chair transition structures were also drawn and '''clean'''ed. The point group of each transition state was found.

Firstly, the chair transition structure was '''optimised to TS (Berny)''' in '''Opt + Freq '''using the basis of '''HF/3-21G'''. Force constant was calculated '''once'''. The frequency of vibration was checked to make sure there is one imaginary vibrational frequency. Then, '''freeze''' '''coordinate''' of the molecule by freezing the carbon-carbon bond to be made as 2.20000 Å. After that, the frozen coordinate was relaxed so the carbon-carbon bond to be made no longer be restricted to 2.20000 Å. The geometry of the transition state was then compared.

Secondly, at the same time, the boat transition structure was optimised by '''QST2''' method by specifying the reactants and products of the reaction under the basis of '''HF/3-21G'''. Labelling the atoms in
the reactant and product, and adjusting the central '''C-C-C-C '''dihedral angle to 0o plus the two inside '''C-C-C''' angles to 100o, the reactant and product could now be optimised by '''QST2''' method.

Comparing the optimised chair and boat transition structures, the connecting conformer of 1,5-hexadiene was found. The reaction energy profile was then calculated by '''IRC''' under '''HF/3-21G''' with 50 points and force constant as always for every small steps. With that, the mechanism of the reaction, as well as the whole reaction energy profile, could be observed clearly. Take the last point on the '''IRC''' and run a normal '''optimisation''' under '''HF/3-21G''' to obtain a minimized geometry.

Eventually, re'''optimise''' the structures of the two transition states with '''Opt + Freq '''under the basis of '''B3LYP/6-31G*'''. The geometries and energies of the transition structure under two different basis were compared. With that, these computed values were also compared against experimental values.

==== The Diels-Alder Cycloadditions ====
[[File:Yll113DA1.jpg|thumb|'''Figure 3. '''The Diels-Alder Cycloadditions between ethylene and butadiene]]

===== Ethylene and butadiene =====
The structure of cis-butadiene was first optimised under the basis of '''semiempirical AM1'''. The HOMO and LUMO of cis butadiene were visualised and its symmetry was determined.

The transition state of the reaction was drawn and optimised under the basis of '''semiempirical AM1'''. The '''IRC''' was carried out in order to confirm the transition structure postulated is of the correct one. Furthermore, the HOMO of the transition structure was visualised and the nodal
planes and properties of the system were interpreted.

===== Cyclohexa-1,3-diene and maleic anhydride =====
[[File:Yll113DA2.jpg|thumb|'''Figure 4. '''The Diels-Alder Cycloadditions between cyclohexa-1,3-diene and maleic anhydride]]
The transition states of the exo and endo products were drawn and optimised under the basis of '''semiempirical AM1'''. The '''IRC''' was carried out in order to confirm the transition structure postulated is of the correct one. For the number of points, 21 points were used for exo transition states and 24 for endo. This is because the energy was too shallow and the slopes tend to zero after the number of points specified above and ''GaussView 5.0'' cannot predict which direction should it goes on to calculate. Furthermore,
the bond lengths, orientation and the HOMO of the transition structures were investigated.

=== '''Results and Discussion''' ===

==== The Cope Rearrangement ====

===== Optimisation of Reactant =====
1,5-hexadiene has three free rotating carbon-carbon bonds. Each of them has three rotational minima. This gives 27 conformations of the 1,5-hexadiene molecule. Yet, only ten of them were energetically distinct due to symmetry and enantiomeric relationships.<ref>B. G. Rocque, J. M. Gonzales and H. F. Schaefer, ''Molecular Physics'', 2002, '''100''' (4), 441</ref> Two of them, the ''Ci anti ''and ''C1 gauche ''structure in here'' ''were drawn and optimizied as shown in Figure A and B and their energies were calculated as shown in Table 1.
{| class="wikitable"
!Conformation
!Molecule
!Point Group
!Energy/ Hartrees*
HF/3-21G
!RMS Gradient Norm/Hartrees*
HF/3-21G
!Relative Energy#/ kcal/mol
!Newman Projections
|-
|Gauche3
|<jmol>
<jmolApplet>
<title>Figure A: Gauge3 conformer of 1,5-hexadiene</title><color>white</color>
<size>250</size>
<uploadedFileContents> yll113CR_GAUGE_PART1.LOG </uploadedFileContents>
</jmolApplet>
</jmol>
|C1
|<nowiki>-231.69266120</nowiki>
|0.00001556
|0.00
|[[File:Yll113 torsion gauche.jpg|centre|frame|'''Figure 5.''' Newman Projection of C1 gauche3 1,5-hexadiene]]
|-
|Anti2
|<jmol>
<jmolApplet>
<title>Figure B: Anti2 conformer of 1,5-hexadiene</title><color>white</color>
<size>250</size>
<uploadedFileContents>YLL113CR ANTI PART1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|Ci
|<nowiki>-231.69253528</nowiki>
|0.00001891
|0.08
|[[File:Yll113 torsion anti.jpg|centre|frame|'''Figure 6.''' Newman Projection of Ci anti2 1,5-hexadiene]]
|}
<nowiki>*</nowiki>1 hartree = 627.509 kcal/mol

#The difference in energy between the conformer and the lowest energy conformer, in here, which is Gauche3. Then convert Hartree to kcal/mol by *

'''Table 1. '''Conformational analysis of anti2 and gauche3 of 1,5-hexadiene

As shown in Table 1, the energy of Gauche3 is surprisingly lower than the anti2 conformation of 1,5-hexadiene. In most cases, the antiperiplanar conformation of a molecule, such as anti2, is more favourable as it has the least steric clashes. Therefore, usually the antiperiplanar conformation is of the lowest energy. However, here, apart from sterics, the stereoelectroncs concept has also been taken into account. The vinyl proton, in a through space manner, can interact with the π or π* orbital on the sp2 carbon which is separated by four bonds from it.<ref>M. Nishio and M. Hirota, ''Tetrahedron'', 1989, '''45 '''(23), 7201</ref> This is so-called CH-π interaction. The Newman Projection in Figure 5 gives us a closer look on how they are close in space and interact; and the Newman projection in Figure 6 tells us why the vinyl proton cannot interact with the π or π* system through space. Therefore, the gauche3 conformation is more stable than anti2 and of lower energy in 1,5-hexadiene.

Focusing on anti2 conformer of the 1,5-hexadiene, the '''HF/3-21G''' optimised molecule was reoptimised by '''B3LYP/6-31G'''. The energies and the geometries of the anti2 1,5-hexadiene under two basis of calculation method were compared and shown in Table 2.
[[File:Yll113Anti2.png|thumb|'''Figure 7. '''Anti2 conformer of 1,5-hexadiene with labelled atoms]]
{| class="wikitable"
! rowspan="2" |Calculation Method
! rowspan="2" |Point Group
! rowspan="2" |Energy/ Hartrees
! rowspan="2" |RMS Gradient Norm/Hartrees
!Bond length between/ Å
!Bond length between/ Å
!Bond length between/ Å
!Bond angle between
!Bond angle between
|-
|C14-C12 and C4-C1
|C12-C9 and C6-C4
|C6-C9
|C14-C12-C9 and
C1-C4-C6
|C12-C9-C6 and
C4-C6-C9
|-

|'''HF/3-21G'''
|Ci
|<nowiki>-231.69253528</nowiki>
|0.00001891
|1.31613
|1.50891
|1.55275
|124.80579o
|111.34878o
|-
|'''B3LYP/6-31G*'''
|Ci
|<nowiki>-234.61171063</nowiki>
|0.00001249
|1.33350
|1.50419
|1.54816
|125.29968o
|112.67081o
|}
'''Table 2. '''Comparison with the anti2 conformer of 1,5-hexadiene with two calculation method basis ('''HF/3-21G '''and '''B3LYP/6-31G*''')

As seen in Table 2, the point group of the same conformer does not change no matter which method of calculation was carried out. The energy of the conformer with '''B3LYP/6-31G*''' is lower than that of '''HF/3-21G'''. Also, the bond lengths and bond angles changed with a certain amount as shown.

Furthermore, by submitting job type as '''Frequency''' in the optimised structure of anti2 conformer of 1,5-hexadiene under both calculation basis '''HF/3-21G '''and '''B3LYP/6-31G*''', the electronic energy, sum of electronic and zero-point energies at 0 K and sum of electronic and thermal energies at 298.15 K could be determined as shown in Table 3.

{| border="1" cellspacing="1" cellpadding="10"
|-
|rowspan="3"| ''' '''
!colspan="4" |'''HF/3-21G a'''
! colspan="4" |'''B3LYP/6-31G* b'''
|-
! rowspan="2" width="125" align="center" | '''Electronic energy'''
! width="125" align="center" | '''Sum of electronic and zero-point energies'''
! width="125" align="center" | '''Sum of electronic and thermal energies'''
! rowspan="2" width="125" align="center" | '''IR spectrum'''
! rowspan="2" width="125" align="center" | '''Electronic energy'''
! width="125" align="center" | '''Sum of electronic and zero-point energies'''
! width="125" align="center" | '''Sum of electronic and thermal energies'''
! rowspan="2" width="125" align="center" | '''IR spectrum'''
|-
! width="125" align="center" | '''at 0 K'''
! width="125" align="center" | '''at 298.15 K'''
! width="125" align="center" | '''at 0 K'''
! width="125" align="center" | '''at 298.15 K'''
|-
| '''Reactant (anti2)'''
| align="center" | -231.692534
| align="center" |-231.539539
| align="center" | -231.532565
|[[File:Yll113ANTI3-21IR.png|thumb|'''Figure 8. '''IR spectrum of anti2 reactant 1,5-hexadiene calculated by '''HF/3-21G''']]
| align="center" | -234.611703
| align="center" | -234.469213
| align="center" | -234.461857
|[[File:Yll113ANTI6-31IR.png|thumb|'''Figure 9. '''IR spectrum of anti2 reactant 1,5-hexadiene calculated by '''B3LYP/6-31G*''']]
|}

a [https://wiki.ch.ic.ac.uk/wiki/images/5/52/Yll113CR_ANTI_PART4.LOG File]; b [https://wiki.ch.ic.ac.uk/wiki/images/5/54/Yll113_CR_ANTI_PART3.LOG File]

'''Table 3.''' Summary of energies (in hartree) of the reactant (anti2) Comparing Figure 8 and 9, both IR spectra are similar which shows that they have similar stretching/bending mode even under different methods of calculations. However, the strongest peak in Figure 8 is at 1112 cm-1 and that in Figure 9 is 940 cm-1. Yet, they correspond to the same mode of vibration, which is the =C-H bending. Therefore, according to the equation, the wavenumber of absorbance, ν can be calculated:

:<math>\nu = \frac{1}{2 \pi c} \sqrt{\frac{k}{\mu}}</math>

where ''μ'' is the reduced mass, ''k'' is the spring constant for the bond and ''c'' is the light speed,

we can therefore know that the spring constant of the molecule under '''B3LYP/6-31G* '''basis reduced.

Now, focusing on gauche3 conformer of the 1,5-hexadiene, similarly, the '''HF/3-21G''' optimised molecule was reoptimised by '''B3LYP/6-31G'''. The energies and the geometries of the gauche3 1,5-hexadiene under two basis of calculation method were compared and shown in Table 4.
[[File:Yll113Gauche3.png|thumb|'''Figure 10. '''Gauche3 conformer of 1,5-hexadiene with labelled atoms]]
{| class="wikitable"
! rowspan="2" |Calculation Method
! rowspan="2" |Point Group
! rowspan="2" |Energy/ Hartrees
! rowspan="2" |RMS Gradient Norm/Hartrees
! colspan="3" |Bond length between/ Å
! colspan="2" |Bond angle between
|-
|C14-C12 and C4-C1
|C12-C9 and C6-C4
|C6-C9
|C14-C12-C9 and
C1-C4-C6
|C12-C9-C6 and
C4-C6-C9
|-

|'''HF/3-21G'''
|C1
|<nowiki>-231.69266120</nowiki>
|0.00001555
|1.31646
|1.50929
|1.55314
|125.02428o
|111.80728o
|-
|'''B3LYP/6-31G*'''
|C1
|<nowiki>-234.61132605</nowiki>
|0.00000360
|1.33382
|1.50491
|1.55007
|125.49464o
|113.46225o
|}
'''Table 4. '''Comparison with the anti2 conformer of 1,5-hexadiene with two calculation method basis ('''HF/3-21G '''and '''B3LYP/6-31G*''')

As seen in Table 4, the point group of the same conformer, again, does not change no matter which method of calculation was carried out. The energy of the conformer with '''B3LYP/6-31G*''' is lower than that of '''HF/3-21G'''. Also, the bond lengths and bond angles changed with a certain amount as shown.

Furthermore, by submitting job type as '''Frequency''' in the optimised structure of gauche3 conformer of 1,5-hexadiene under both calculation basis '''HF/3-21G '''and '''B3LYP/6-31G*''', the electronic energy, sum of electronic and zero-point energies at 0 K and sum of electronic and thermal energies at 298.15 K could be determined as shown in Table 5.

{| border="1" cellspacing="1" cellpadding="10"
|-
| rowspan="3" | ''' '''
! colspan="4" |'''HF/3-21G c'''
! colspan="4" |'''B3LYP/6-31G* d'''
|-
! rowspan="2" width="125" align="center" | '''Electronic energy'''
! width="125" align="center" | '''Sum of electronic and zero-point energies'''
! width="125" align="center" | '''Sum of electronic and thermal energies'''
! rowspan="2" |IR spectrum
! rowspan="2" width="125" align="center" | '''Electronic energy'''
! width="125" align="center" | '''Sum of electronic and zero-point energies'''
! width="125" align="center" | '''Sum of electronic and thermal energies'''
! rowspan="2" |IR spectrum
|-
! width="125" align="center" | '''at 0 K'''
! width="125" align="center" | '''at 298.15 K'''
! width="125" align="center" | '''at 0 K'''
! width="125" align="center" | '''at 298.15 K'''
|-
| '''Reactant (Gauche 3)'''
| align="center" |-231.692692
| align="center" |-231.539486
| align="center" |-231.532646
|[[File:Yll113GAUCHE3-21IR.png|thumb|'''Figure 11. '''IR spectrum of gauche3 conformer of 1,5-hexadiene calculated by '''HF/3-21G''']]
| align="center" | -234.611326
| align="center" |-234.468719
| align="center" |-234.461477
|[[File:Yll113GAUCHE6-31IR.png|thumb|'''Figure 12.''' IR spectrum of gauche3 conformer of 1,5-hexadiene calculated by '''B3LYP/6-31G*''']]
|}

c [https://wiki.ch.ic.ac.uk/wiki/images/9/98/Yll113CR_GAUGE_PART4.LOG File] ; d [https://wiki.ch.ic.ac.uk/wiki/images/c/ca/Yll113CR_GAUGE_PART3.LOG File]

'''Table 5.''' Summary of energies (in hartree) of the reactant (Gauche3) Comparing Figure 11 and 12, both IR spectra are similar which shows that they have similar stretching/bending mode even under different methods of calculations. However, the strongest peak in Figure 12 is at 939 cm-1 and that in Figure 11 is 1111 cm-1. Yet, they correspond to the same mode of vibration, which is also the =C-H bending. Therefore, similar to the anti2 conformer's case as mentioned above, we can therefore know that the spring constant of the molecule under '''B3LYP/6-31G* '''basis reduced.

===== Optimisation of transition state =====

After optimising the reactants, the chair and boat transition states were optimised accordingly using mainly two different methods. But before that, an allyl fragment (CH2CHCH2) was optimised using '''HF/3-21G''' level of theory for the sake of convenience in constructing the chair and boat transition states. A brief summary was shown in Table 6.
{| class="wikitable"
!
! colspan="2" |Molecule
!Point Group
!Energy/Hartree
!RMS Gradient Norm/ Hartrees
|-
|Allyl fragment
CH2CHCH2
|<jmol>
<jmolApplet>
<title>Figure C: Allyl Fragment</title><color>white</color>
<size>250</size>
<uploadedFileContents> Yll113CR TS 1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|[[File:Yll11313.jpg|thumb|'''Figure 13. '''Optimised Structure of the allyl fragment]]
|C2v
|<nowiki>-115.82304010</nowiki>
|0.00002945
|}
'''Table 6. '''Summary of the optimised allyl fragment

Then, both chair and boat transition state were drawn and optimised using the '''optimisation to TS (Berny)''' under '''HF/3-21G'''. Figure 14 and Figure C show the optimized structure of the chair transition state while Figure 15 and Figure D show the optimized structure of the boat transition state. Table 7 shows the summary of results.
{| class="wikitable"
! rowspan="2" |
! colspan="2" rowspan="2" |Molecule
! rowspan="2" |Point Group
! rowspan="2" |Energy/ Hartree
! rowspan="2" |RMS Gradient Norm/Hartrees
! colspan="6" |Partial C-C bond length between/ Å
|-
!C14-C6
!C11-C3
!C14-C9
!C6-C1
!C9-C11
!C1-C3
|-
|Optimised Chair Transition State
|<jmol>
<jmolApplet>
<title>Figure D: Optimised Chair transition state</title><color>white</color>
<size>250</size>
<uploadedFileContents>Yll113CHAIR PART1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|[[File:Yll113CHAIR3-21.png|thumb|'''Figure 14. '''Optimised Chair Structure by '''HF/3-21G''' with atoms labelled ]]
|C2h
|<nowiki>-231.61932238</nowiki>
|0.00002645
|2.02016
|2.02016
|1.38929
|1.38930
|1.38930
|1.38929
|-
|Optimised Boat Transition State
|<jmol>
<jmolApplet>
<title>Figure E: Optimised Boat transition state</title><color>white</color>
<size>250</size>
<uploadedFileContents>Yll113BOAT PART1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|[[File:Yll11315.jpg|thumb|'''Figure 15. '''Optimised Boat Structure by '''HF/3-21G''' with atoms labelled]]
|C2v
|<nowiki>-231.60280235</nowiki>
|0.00003872
|2.14060
|2.14060
|1.38138
|1.38138
|1.38138
|1.38138
|}

'''Table 7. '''Summary of the optimised chair and boat transition states by '''optimisation to TS (Berny) '''under '''HF/3-21G''' basis

Furthermore, the transition structures’ '''Frequencies''' were calculated as shown in Table 8.
{| class="wikitable"
!
!Molecule
!IR spectrum
!Imaginary Frequency/ cm-1
|-
|Boat Transition State
|[[File:Yll113Boat Part 1 imag freq.gif]]
|[[File:Yll11317.jpg|thumb|'''Figure 16. '''IR spectrum of the optimised Boat Structure by '''HF/3-21G''']]
|<nowiki>-840</nowiki>
|-
|Chair Transition State
|[[File: Yll113Chair Part 1 imag freq.gif]]
|[[File:Yll11316.jpg|thumb|'''Figure 17. '''IR spectrum of the optimised Chair Structure by '''HF/3-21G''']]
|<nowiki>-818</nowiki>
|}
'''Table 8.''' IR spectra and imaginary frequencies of the boat and chair transition states

As you may notice that, the
imaginary frequency comes up when calculating with the transition states. This
is common, in other words, this should appear to let us know the transition
structure we postulated is correct.

A transition state is the first
order saddle point on the potential energy surface. Therefore, the force
applied to the saddle point against to the displacement. As force and
displacement are vectors, the force constant will be a negative number.Therefore, according to
:<math>\nu = \frac{1}{2 \pi c} \sqrt{\frac{k}{\mu}}</math>

where ''μ'' is the reduced mass, ''k'' is the spring constant for the bond and ''c'' is the light speed,

the square root of a negative
force constant k gives an imaginary wave number/frequency v. In other words,
the appearance of an imaginary frequency tells us that the structure is a
saddle point of the reaction pathway.

The chair transition state
was followed by first 'frozen' then 'relaxed'. The boat transition structure
was followed by '''QST2''' calculation method.

====== Chair Transition State ======
After the above '''optimisation''', the chair transition
state was reoptimised again with another method. This method first freezes the
coordinate of the molecule, in this case, freeze the bond to be made in the
Cope Rearrangement of 1,5-hexadiene as 2.20000 Å. The molecule then optimised with the frozen
coordinate. Details of this optimisation was summarized in Table 9.
{| class="wikitable"
! rowspan="2" |
! colspan="2" rowspan="2" |Molecule
! rowspan="2" |Point Group
! rowspan="2" |
{| class="MsoTableGrid"

|
<nowiki> </nowiki>Energy/ Hartree

|}
! rowspan="2" |RMS Gradient Norm/ Hartree
! colspan="3" |Bond length between/ Å
! rowspan="2" |Imaginary Frequency/ cm-1
! rowspan="2" |IR Spectrum
|-
!C6-C14 and C3-11
!C1-C3 and C9-C14
!C1-C6 and C9-C11
|-
|Optimised Chair Transition Structure with frozen coordinate
|<jmol>
<jmolApplet>
<title>Figure F: Optimised Chair transition state with frozen coordinate</title><color>white</color>
<size>250</size>
<uploadedFileContents>Yll113CHAIR PART2.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
[[File: Yll113Chair frozen.gif]]
|[[File:Yll11318.jpg|thumb|'''Figure 18. '''The optimised chair transition structure with frozen coordinate and atoms labelling]]
|
{| class="MsoTableGrid"

|
<nowiki> </nowiki>C2h

|}
|<nowiki>-231.61518510</nowiki>
|0.00325573
|2.20000
|1.38135
|1.38128
|<nowiki>-765</nowiki>
|[[File:Yll11319.jpg|thumb|'''Figure 19. '''IR spectrum of the optimised chair transition structure with frozen coordinate]]
|}

'''Table 9. '''Summary of the optimisation of the chair transition structure with
frozen coordinate(s)

From Table 9, we may notice
that the RMS Gradient Norm value is quite far off from zero. Also, the
imaginary frequency becomes much higher than -818 cm-1 (Shown in
Table 8). With these two pieces of information, we can deduce that the frozen
coordinate(s) affect(s) the force constant of the transition state which does
not give a good optimisation of transition structure. With that, after applying
the frozen coordinate to the molecule, the molecule was reoptimised again with
a degree of '''Derivative '''to the '''Bond'''. Details of the reoptimisation
were presented in Table 10.
{| class="wikitable"
! rowspan="2" |
! colspan="2" rowspan="2" |Molecule
! rowspan="2" |Point Group
! rowspan="2" |Energy/ Hartree
! rowspan="2" |RMS Gradient Norm/ Hartree
! colspan="4" |Bond length between/ Å
! rowspan="2" |Imaginary Frequency/ cm-1
! rowspan="2" |
{| class="MsoTableGrid"

|
<nowiki> </nowiki>IR Spectrum

|}
|-
!C14-C6
!C11-C3
!C14-C9 and C6-C1
!C9-C11 and C1-C3
|-
|Optimised Chair Transition
Structure with a degree of '''Derivative'''
to the '''Bond'''
|<jmol>
<jmolApplet>
<title>Figure G: Optimised Chair transition state with a degree of Derivative to the Bond</title><color>white</color>
<size>250</size>
<uploadedFileContents>Yll113CHAIR PART3.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
[[File: Yll113Chair relax.gif]]
|[[File:Yll11320.jpg|thumb|'''Figure 20. '''The optimised chair transition structure with a degree of '''Derivative''' to the '''Bond''' and atoms labelling]]
|C2h
|<nowiki>-231.61932233</nowiki>
|0.00002127
|2.02075
|2.02071
|1.38929
|1.38930
|<nowiki>-818</nowiki>
|[[File:Yll11321.jpg|thumb|'''Figure 21. '''IR spectrum of the optimised chair transition structure with a degree of '''Derivative''' to the '''Bond''']]
|}
'''Table 10. '''Summary of the reoptimisation of the chair transition
structure with a degree of '''Derivative''' to the '''Bond'''

Now, in Table 10, the RMS
Gradient Norm value is close to zero. Also, the imaginary frequency goes back
to -818 cm-1, indicating that the coordinates no longer be frozen
and the stretching/bending mode of the transition state is able to undergo
freely.

Comparing the bond lengths
in Table 7 and 10, we can see that the difference between bond lengths of the
single bond to be made/ broken calculated in two methods is just less than
0.0006 Å. And also, there is no difference in bond length of the double bond to be make/broken ‘inside’ the system. This tells us that the two optimisation
methods are rather similar under the consideration on the Cope Rearrangement
Reaction.

====== Boat Transition State ======
Instead of using the frozen
coordinate method as for the chair transition state above, another method, '''QST2''', was applied to the boat
transition state under the '''HF/3-21G'''
basis. In order to use this method, without any ‘Link died’, the reactant and
product have to be drawn and labelled carefully. '''QST2''' is a method which interpolates the reactant and product to
give a transition state. Therefore, it will fall if the structure of the
reactant and product are not close to the transition state. And therefore, all
molecules have to be carefully labelled and adjusted.

[[File:Yll11322.png|500px|center]]

'''Figure 22. '''The drawings and adjustments of angles of the reactant (left)
and product (right) for '''QST2''' Method,
i.e. the central C-C-C-C dihedral angle was changed to 0o and inside
C-C-C were reduced to 100o

After the adjustment, the job was run and the optimized molecule converge to the boat transition structure. Summary was shown in Table 11.
{| class="wikitable"
! rowspan="2" |
! colspan="2" rowspan="2" |Molecule
! rowspan="2" |Point Group
! rowspan="2" |Energy/Hartree
! rowspan="2" |RMS Gradient Norm/ Hartree
! colspan="3" |Bond length between/ Å
! rowspan="2" |Imaginary Frequency/ cm-1
! rowspan="2" |IR spectrum
|-
!C1-C6
!C3-C4
!C5-C6, C4-C5, C3-C2 and C1-C2
|-
|Boat transition structure
under '''QST2''' method
|[[File:Yll113Boat qst2.gif]]
|[[File:Yll11323.jpg|thumb|'''Figure 23. '''The optimised Boat transition structure with atom labelling]]
|C2v
|<nowiki>-231.60280241</nowiki>
|0.00002436
|2.13994
|2.14019
|1.38149
|<nowiki>-840</nowiki>
|[[File:Yll11324.jpg|thumb|'''Figure 24. '''IR spectrum of the optimised boat transition structure]]
|}
'''Table 11. '''Summary of the boat transition structure under '''QST2 '''method

====== Intrinsic Reaction Coordinate''' '''('''IRC''') ======
In order to confirm that our transition state is of the
correct one, '''Intrinsic Reaction
Coordinate '''('''IRC''') will be carried
out.

As mentioned above, transition state is the first order
saddle point of the reaction pathway. Therefore, it will start to go to the
product or back to the reactant with it falls off. It resembles that a ball is
at the tip of the mountain, which starts to roll off the mountain on the side
with the steepest slope. Also, when we are doing '''IRC''', we can determine whether the reaction goes forward, backward
or both sides. Also, the number of points, which means the number of little
steps that the geometry of the molecule changes, can be adjusted. A low number
of points will just give us a very rough idea that tell us a little bit about
our transition state. Also, the last point on the '''IRC''' is far from the minimum geometry. A high number of points gives
us more accurate results, however two problems could be raised. First, the time
for calculation will be long and Most importantly, as it goes down the slope
and reaches the minimum geometry, i.e. the plateau of energy, the slope will
become very small or even zero again. However, as the energy difference of the
next or previous geometry compared to the geometry of itself is too small, ''GaussView 5.0'' may not able to know which
direction the molecule should proceed to. And this, therefore, results in ‘Link
died’. Therefore, the most common technique is to have a good number of points,
then take the last point on the IRC and run it with a normal optimisation.

Here, as we know that the
Cope Rearrangement has a symmetric reaction pathway, taking the chair
transition structure, we will run '''IRC'''
on it with 50 points.
{| class="wikitable"
![[File:Yll113hlj29825.jpg|thumb|'''Figure 25. '''Total energy along '''IRC '''of the chair transition structure]]
![[File:Yll11326.jpg|thumb|'''Figure 26. '''RMS Gradient Norm of '''IRC '''of the chair transition structure]]
![[File: Yll113Chair irc.gif]]
|}

[[File:Yll11327.jpg|500px]]

'''Figure 27. '''The product of the Cope Rearrangement after optimisation

The first point on Figure 25 is -231.61932233 Hartree and the last point is -231.69157881 Hartree. Then, we take the last point and optimise it, we get the structure shown in Figure 27.

The structure is of C2
symmetry and the energy calculated is -231.69166702 Hartree. This matches with
Gauche2 C2 on Appendix 1. And therefore, this is how the conformer
of 1,5-hexadiene connects with the chair transition structure.

====== Activation Energy of the Cope Rearrangement ======
Finally, we optimise the chair and boat transition states we got from above, reoptimise it with job Opt + Freq
under a more advanced calculation '''B3LYP/6-31G*'''. And from that, the thermochemistry data were given and we can know the
activation energy of the reaction by comparing to Table 3, which anti2 is used
as a local minimum rather than gauche3 as a global minimum.
{| border="1" cellspacing="1" cellpadding="10"
|-
| rowspan="3" | ''' '''
! colspan="3" |'''HF/3-21G'''
! colspan="3" |'''B3LYP/6-31G*'''
|-
| rowspan="2" width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| rowspan="2" width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619323
| align="center" | -231.466698
| align="center" | -231.461339
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409009
|-
| '''Boat TS'''
| align="center" | -231.602803
| align="center" | -231.450932
| align="center" | -231.445303
| align="center" | -234.543094
| align="center" | -234.402342
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692534
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.611703
| align="center" | -234.469213
| align="center" | -234.461857
|}
'''Table 11'''. Summary of energies of chair, boat and reactant (anti2) structure
{| border="1" cellspacing="1" cellpadding="10"
|-
| rowspan="2" | ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.69
| align="center" | 34.06
| align="center" | 33.16
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.96
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
'''Table 12'''. Summary of activation energies in kcal/mol

==== The Diels-Alder Cycloadditions ====

===== Ethylene and Cis-Butadiene =====
First, the structures of the reactants, ethylene and cis-butadiene, were '''optimised to minimum''' under '''semiempirical AM1''' method. For the butadiene, in order to be in the cis conformer, the dihedral angle was adjusted to be 0o. Details are listed in Table 13.
{| class="wikitable"
!
!Molecule
!Point Group
!HOMO
!LUMO
!Energy/Hartree
!RMS Gradient Norm/ Hartree
!IR spectrum
|-
|Ethylene
|JSMOL
|D2h
|[[File:Yll11331.jpg|thumb|'''Figure 31. '''HOMO of Ethylene]]
|[[File:Yll11330.jpg|thumb|'''Figure 30.''' LUMO of ethylene]]
|0.02619029
|0.00008755
|[[File:Yll11328.jpg|thumb|'''Figure 28. '''IR spectrum of Ethylene]]
|-
|Cis-Butadiene
|JSMOL
|C2v
|[[File:Yll11332.jpg|thumb|'''Figure 32. '''HOMO of cis-butadiene]]
|[[File:Yll11333.jpg|thumb|'''Figure 33. '''LUMO of cis-butadiene]]
|0.04878534
|0.00000087
|[[File:Yll11329.jpg|thumb|'''Figure 29.''' IR spectrum of cis-butadiene]]
|}
'''Table 13.''' Summary of optimised ethylene and cis-butadiene

Looking into Figure 30-33, as we know that the plane is perpendicular to the molecule, the HOMO of Ethylene is symmetric while that of LUMO is antisymmetric.

Also, the HOMO of cis-butadiene is antisymmetric and that of LUMO is symmetric.

Then, the transition state of the reaction was able to constructed using the optimised structure of the reactants made above. By making the bond lengths of the bonds to be made in Diels-Alder [4+2] cycloaddition to 2.20000 Å in the transition state, which is the 'dash bond', the transition structure was optimized to '''Opt + Freq minimisation to''' '''TS (Berny)''' under the same method of calculation - '''semiempitical AM1'''. Details are listed in Table 14.
{| class="wikitable"
!
! colspan="2" |Molecule
!Point Group
!HOMO
!LUMO
!Energy/Hartree
!RMS Gradient Norm/ Hartree
!Imaginary Frequency/ cm-1
!IR spectrum
|-
|Transition state
|JSMOL
|[[File:Yll11334.jpg|thumb|'''Figure 34. '''Transition state of ethylene + cis-butadiene]]
|C1
|[[File:Yll11336.jpg|thumb|'''Figure 36. '''HOMO of transition state]]
|[[File:Yll11337.jpg|thumb|'''Figure 37. '''LUMO of transition state]]
|0.11165467
|0.00002792
|<nowiki>-956</nowiki>
|[[File:Yll11335.jpg|thumb|'''Figure 35. '''IR spectrum of the transition state]]
|}
'''Table 14.''' Summary of optimised transition state

From Figure 36, we can see that the HOMO of the transition state is antisymmetric whilst the LUMO of the transition state is symmetric. By making very careful comparison between Figure 36, Figure 37 and Figure 30-33, we can see that the HOMO of the transition state in Figure 36 is a combination of Figure 32 and 30; the LUMO of the transition state in Figure 37 is a combination of Figure 31 and 33. We can clearly see that the HOMO and LUMO of the transition state have a complementary combination of HOMO and LUMO of the reactants.

Taking a closer look to HOMO of the transition state. Recalling Woodward Hoffmann’s Rule, (4q+2)s+(4r)a = odd for thermally allowed reaction, we have both π2s and π4s. Therefore, the reaction is thermally allowed by letting q = 0, which gives the value of 1 which is odd.

Furthermore, from Table 14, we notice that there is an imaginary frequency reported at -956 cm-1. As explained above, the transition state should have one imaginary frequency to account for the negative force constant. With that, this imaginary frequency confirms that the transition structure we postulated from the optimised reactants is valid, i.e. it is really a transition state. The animation of where the imaginary frequency originates from, which shows the motion of the transition state - how the two reactants approach to each other and bonds are formed, is shown below.

JSMOL

From the above figure, we can see that the bond formation from the reactant to the product happens at the same time, i.e. synchronous, on both sides of the transition structure. Therefore, we can say that this Diels-Alder cycloaddition is a concerted [4+2] pericyclic cycloaddition, which matches with what we learnt in Pericyclic Reaction course.

On top of that, the geometry of the transition structure was investigated by looking into the optimised bond lengths between carbon atoms Details are shown in Figure 38 and Table 15.[[File:Yll11338.jpg|thumb|'''Figure 38. '''Transition state of ethylene + cis-butadiene with atoms labelled]]
{| class="wikitable"
!
!Bond length/ Å
|-
|C7-C9
|2.11938
|-
|C12-C5
|2.11944
|-
|C12-C9
|1.38284
|-
|C3-C7
|1.38187
|-
|C3-C1
|1.39750
|-
|C5-C1
|1.38175
|}
'''Table 15. '''Geometry analysis of the transition state

According to the literature <ref>M. A. Fox and J. K. Whitesell, ''Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen'', Springer, 1995</ref>, C-C carbon-carbon single bond is 1.54 Å, and C=C carbon-carbon double bond is 1.34 Å. Also, the Van der Waals radius of carbon is 1.70 Å,<ref>A. Bondi,(1964), ''J. Phys. Chem.'', 1964, '''68''' (3), 441</ref>
According to the reaction scheme shown in Figure 3, a single bond is forming between C7 and C9, also another single bond is forming between C12-C5. Comparing the data in Table 15 with the literature, we can see that the bond length of two bonds to be made is longer than C-C, but shorter than the twice of carbon's Van der Waals radius. This tells us some hints that the terminal carbons between two reactants are attracting with each other to a certain degree. Furthermore, taking a look to other carbon-carbon bonds within the system, we notice that they are somewhere between C-C and C=C. This tells us that either C-C is becoming C=C or C=C is becoming C-C in the transition state.
After that, the '''IRC''' of the above optimised transition state was carried out with both direction and force constant calculated always for 50 points to see the reaction profile.
{| class="wikitable"
|[[File:Yll11339.jpg|thumb|'''Figure 39.''' IRC of the transition state of ethylene + cis-butadiene]]
|[[File:Yll11340.jpg|thumb|'''Figure 40. '''RMS Gradient Norm of transition state of ethylene + cis-butadiene]]
|JSMOL
|}
In Figure 39, we can clearly see that the reactants was first passed through the energy barrier to get the transition state and it went down the slope to give the product.
Finally, the activation energy for this reaction was calculated in Table 16.
{| class="wikitable"
!
!Ethylene
!Cis-butadiene
!Transition state
!Activation Energy
|-
|Energy/Hartree
|0.02619029
|0.04878534
|0.11165467
|0.03667904
(23.02 kcal/mol)
|}
'''Table 16. '''Activation energy analysis of Diels-Alder Reaction between ethylene and cis-butadiene
===== Cyclohexa-1,3-diene and Maleic Anhydride =====
First, the structures of
the reactants, ethylene and cis-butadiene, were '''optimised to minimum''' under '''semiempirical AM1''' method. Details are listed in Table 17.
{| class="wikitable"
!Optimised Structure
!Molecule
!Point Group
!Energy/Hartree
|-
|Cyclohexa-1,3-diene
|[[File:Yll113Cycloopt.jpg|thumb|'''Figure 41. '''Optimised structure of cyclohexa-1,3-diene]]
|C2
|0.02771130
|-
|Maleic Anhydride
|[[File:Yll113Maopt.jpg|thumb|'''Figure 42. '''Optimised structure of Maleic Anhydride]]
|Cs
|<nowiki>-0.12182418</nowiki>
|}
'''Table 17. '''Summary
of the optimized reactant

Then, the two transition states, exo and endo, of the reaction were able to construct. By making the bond lengths of the bonds to be made in Diels-Alder [4+2] cycloaddition to 2.20000 Å in the transition state, which is the 'dash bond', the transition structure was optimized to '''Opt + Freq minimisation to''' '''TS (Berny)''' under the same method of calculation - '''semiempitical AM1'''. Details are listed in Table 18.
{| class="wikitable"
!Transition Structure
!Molecule
!Point Group
!HOMO
!LUMO
!Energy/Hartree
!'''RMS Gradient Norm/ Hartree'''
!'''Imaginary Frequency/ cm-1'''
!IR spectrum
|-
|Exo
|[[File:Yll113Exots.jpg|thumb|'''Figure 43.''' Optimised Structure of Exo Transition State]]
|C1
|[[File:Yll113Exo homo.jpg|thumb|'''Figure 45. '''HOMO of Exo Transition State]]
|[[File:Yll113Exo lumo.jpg|thumb|'''Figure 47.''' LUMO of Exo Transition State]]
|<nowiki>-0.05041984</nowiki>
|0.00000883
|<nowiki>-812</nowiki>
|[[File:Yll113Exo ir.jpg|thumb|'''Figure 49. '''Exo Transition State IR spectrum]]
|-
|Endo
|[[File:Yll113Endots.jpg|thumb|'''Figure 44. '''Optimised Structure of Enfo Transition State]]
|C1
|[[File:Yll113Endo homo.jpg|thumb|'''Figure 46'''. HOMO of Endo Transition State]]
|[[File:Yll113Endo lumo.jpg|thumb|'''Figure 48.''' LUMO of Endo Transition States]]
|<nowiki>-0.05150469</nowiki>
|0.00004308
|<nowiki>-807</nowiki>
|[[File:Yll113Endo ir.jpg|thumb|'''Figure 50.''' Endo Transition State IR Spectrum]]
|}
'''Table 18. '''Summary of the optimized transition structures

In Table 18, we can see that both HOMO and LUMO of the exo
transition state are antisymmetric with respect to the plane. Also, both HOMO and LUMO of the endo transition state are antisymmetric as well.

Also, we notice that the energy of exo is higher than that of endo. This can be explained by the poorer overlap between the C=C π and C=O π* compared to that of endo. This is called secondary orbital effect, which will be further discussed below.

There is one imaginary frequency in both transition structure. This confirms that both of them are valid postulated transition structures.
{| class="wikitable"
|JSMOL
'''Imaginary frequency of exo transition state animation'''
|JSMOL
'''Imaginary frequency of exo transition state animation'''
|}
Again, from the above animations, we can see that the bonds are formed in a synchronous fashion, which means the Diels-Alder Cycloaddition reaction is concerted.

Furthermore, the geometries of both transition states were examined carefully in Table 19.
{| class="wikitable"
!Geometry summary of Exo Transition State (Please refer to Figure 43 for atom labelling)
!Geometry summary of Endo Transition State (Please refer to Figure 44 for atom labelling)
|-
|
{| class="wikitable"
!
!Bond length/ Å
!
!Bond length/ Å
|-
|C5-C13
|2.17032
|C6-C14
|2.17035
|-
|C5-C3
|1.39440
|C3-C1
|1.39674
|-
|C1-C6
|1.39440
|C14-C13
|1.4612
|-
|C7-C6
|1.48976
|C7-C10
|1.52208
|-
|C10-C5
|1.48976
|C13-C15
|1.48822
|-
|C14-C16
|1.48822
| colspan="2" |N/A
|-
|C7-C16
(Through space)
|2.94507
|C10-C15
(Through space)
|2.94484
|-
|C1-C16
(Through space)
|3.78172
|C3-C15
(Through Space)
|3.78155
|}
|
{| class="wikitable"
!
!Bond length/ Å
!
!Bond length/ Å
|-
|C15-C7
|2.16230
|C16-C5
|2.16229
|-
|C1-C3
|1.39726
|C3-C7
|1.39296
|-
|C1-C5
|1.39308
|C7-C9
|1.49503
|-
|C9-C12
|1.52300
|C5-C12
|1.49054
|-
|C16-C18
|1.48918
|C15-C17
|1.48903
|-
|C15-C16
|1.40863
| colspan="2" |N/A
|-
|C1-C18
(Through space)
|2.89232
|C3-C17
(Through space)
|2.89203
|}
|}
'''Table 19.''' Geometry analysis of exo and endo transition states

According to the reaction scheme shown in Figure 4, a single bond is forming between C5 and C13, also another single bond is forming between C6-C14 for exo; C15 and C7 plus C16 and C5 for endo, which is what the first row in the two tables in the left and right in Table 19 shows. the single bond to be made Comparing these values with literature, we find that they are longer than C-C but shorter than twice of carbon's Van der Waals' radius. This tells us some hints that these pairs of carbons between two reactants are attracting with each other to a certain degree. Furthermore, taking a look to other carbon-carbon bonds within the system, i.e. except row 1 and those labelled with (through space), we notice that they are somewhere between C-C and C=C. This tells us that either C-C is becoming C=C or C=C is becoming C-C in the transition state.

Now, looking at the through space bond length. In the exo transition structure, the through space distance -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH2-CH2- is smaller than the twice of the Van der Waals radius but larger than C-C which gives us a hint that they might interact. In the endo transition structure, the through space distance -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH=CH- is smaller than the twice of the Van der Waals radius but larger than C-C which gives us a hint that they might interact. However, according to the definition of secondary orbital effect, it is looking for the interaction between the C=C π of the diene and C=O π* of the dienophile. Endo clearly shows that as explained, but exo seems to just demonstrate the sterics clash between the -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH2-CH2- fragment of diene. In order to further confirm that exo has no secondary orbital effect, a measurement of bond length was carried out between -(C=O)-O-(C=O)- fragment of the maleic anhydride and the -CH=CH- in diene in the exo transition state. The result was shown in the last row on the left table in Table 19. This shows that they are too far away which means they are not possible to interact.

Now, looking back to the HOMO of exo and endo transition states in Figure 45 and 46 respectively. We can definitely see that the overlap between the two reactants is relatively smaller in exo. From these two pieces of information, we can conclude that the endo is kinetically controlled, while exo is thermodynamically controlled.

After that, the '''IRC''' of the both optimised transition state was carried out with both direction and force constant calculated always for the reaction profile. 21 points were used for exo transition states and 24 for endo (reasons explained under '''Introduction)''' to see the reaction profiles.
{| class="wikitable"
! colspan="3" |Exo Transition State
! colspan="3" |Endo Transition State
|-
|[[File:Yll113Exo irc.jpg|thumb|'''Figure 51.''' IRC of the exo transtion state]]
|[[File:Yll113Exo rms.jpg|thumb|'''Figure 52. '''RMS of the exo transition structure]]
|JSMOL
|[[File:Yll113Endo irc.jpg|thumb|'''Figure 53. '''IRC of the endo transition state]]
|[[File:Yll113Endo rms.jpg|thumb|'''Figure 54.''' RMS of the endo transition state]]
|JSMOL
|}
And eventually, the activation energies of the reaction via different transition structures were summarised in Table 20.
{| class="wikitable"
!
!Cyclohexa-1,3-diene
!Maleic Anhydride
!Endo Transition State
!ExoTransition State
!Activation Energy via endo
!Activation Energy via exo
|-
|Energy/Hartree
|0.02771130
|<nowiki>-0.12182418</nowiki>
|<nowiki>-0.05150469</nowiki>
|<nowiki>-0.05041984</nowiki>
|0.04260819
(26.74 kcal/mol)
|0.04369304
(27.42 kcal/mol)
|}
'''Table 20.''' Activation energy analysis

=== '''Conclusion''' ===

=== '''Reference''' ===

<gallery>
File:
</gallery>

File:Yll113Chair irc.gif

2015-12-17T08:02:41Z

Yll113:

Rep:Mod:hlj298

2015-12-17T07:59:21Z

2015-12-17T07:39:58Z

Yll113: /* Optimisation of transition state */

== '''Study of the reaction profiles of the Cope Rearrangement and the Diels-Alder Cycloadditions''' ==
'''Y. L. J. Lam'''

''Department of Chemistry, South Kensington Campus, Imperial College London, London SW7 2AZ, United Kingdom''

''Received 18 December, 2015''

=== Abstract ===
The reactants, products and transition states of the Cope
Rearrangement of 1,5-hexadiene were investigated by ''GaussView 5.0'' at the '''HF/3-21G''' and '''B3LYP/6-31G*'''levels
of theories respectively. With that, the point groups, vibrational frequencies and different energies at different temperatures of the reactants, products and transition states were calculated. Also, by optimizing the transition structures with different methods, i.e. computing the force constants at the
beginning of the calculations, using the redundant coordinate editor and '''QST2''', at the '''HF/3-21G''' level of theory, closer views of the geometries of the transition states can be observed. Furthermore, by using the '''IRC''' method, the reaction profiles can be
obtained and the activation energies can therefore be calculated. Plus, using '''IRC''' method, all reaction intermediates
can now be observed, which helps us to understand the mechanism of the Cope Rearrangement. Similarly, for Diels-Alder Cycloadditions between ethene and
butadiene and Diels-Alder Cycloadditions between cyclohexa-1,3-diene and maleic anhydride, the reactants, products and transition states were optimized and
their respective energies were calculated at '''AM1 semi-empirical molecular orbital method'''. Furthermore, the symmetries of the
molecular orbitals were visualized and the reaction profiles calculated by '''IRC''' method were obtained.

=== '''Introduction''' ===
Chemical reactions are happening around the world in every second. Some reactions are fast, whilst some are slow. The most common and general reason for that is on the kinetic and thermodynamic aspects. On the kinetic aspect, we might argue that the energy barrier(s) form the reactant(s) to the product(s) is/are huge, and therefore, the reactant(s) cannot overcome the barrier(s) and the reaction is slow or does not proceed. The transformation between crude carbon and diamond is a good example. The energy difference between crude carbon and diamond is just few kcal/mol, however, the energy barrier for the transformation is huge. Hence, the transformation is extremely slow, or even does not proceed. With that, diamond symbolizes eternity. On the other hand, on the thermodynamic aspect, we might argue that the reaction is endothermic, i.e. absorbing/requiring heat from the surroundings in order to proceed. In fact, these two aspects just provide us with a little bit of the story and therefore, chemists, or scientists in general, study the mechanism of the reactions to find out the full story. Unfortunately, some reactions are spontaneous, such as the thiocyanation of the iron complex. Also, some intermediates of the reactions are unstable, which cannot be separated or detected even using very advanced analytical instruments, such as nuclear magnetic resonance (NMR) spectromenter. Therefore, scientists devised some programs and computational methods to find out the mechanism of the reactions. Here we use ''GaussView 5.0'' for our investigation.

==== Computational Theory ====
[[File:Yll113 AM1 and HF.jpg|thumb|463x463px|'''Figure 1.''' HOMO and LUMO (highlighted in yellow) of cis-butadiene under the basis of calculation '''HF/3-21G '''(left) and '''AM1''' (right)]]
In ''GaussView 5.0'', there are numerous methods for calculation, such as '''HF/3-21G''', '''B3LYP/6-31G*''', '''semiempirical AM1''', '''MP4 '''and '''MP2'''. Here, the first two calculation method, namely, '''HF/3-21G''' and '''B3LYP/6-31G* '''were applied for calculation of the Cope Rearrangement Reaction, while '''semiempirical AM1''' was used for the investigation of the two Diels-Alder reactions.

N.B. No matter which method applied, the RMS Gradient Norm in hartress would also be computed. This is a measure of how well does the optimisation go during the calculation of the
structure drawn. The closer to zero, the better the structure is optimised.

===== Hartree-Fock ('''HF''') Method =====
Hartree-Fock theory ('''HF''') is the fundamentals of electronic structure theory. It gives a good starting point for more elaborate theoretical methods which can approximate the electronic Schrödinger equation better. It is the basis of the molecular orbital (MO) theory that assumes the motion of each electron can be described by a single-particle function/orbital and it does not depend on/interact with the instantaneous motions of the other electrons.<ref>C. D. Sherrill, ''An Introduction to Hartree-Fock Molecular Orbital Theory'', 2000</ref>

===== Becke, 3-parameter, Lee-Yeang-Parr ('''B3LYP''') Method =====
Beeke, 3-parameter, Lee-Yang-Parr ('''B3LYP''') is one of the most commonly used hybrid functionals. Hybrid functionals are a class of approximation of the exchange-correlation energy functional in density functional theory.<ref>What is B3LYP?, https://www.quora.com/What-is-B3LYP (accessed December 2015)</ref> '''B3LYP''' contains an '''HF''' exchange with the weight of 0.2, which can be regarded as a uniform screening of
exchange by 80 %.<ref>C. H. Patterson, ''International Journal of Quantum Chemistry'', 2006, '''106 '''(15), 3383</ref> '''B3LYP''' also takes a set of atomization
and ionization energies, proton affinities and total atomic energies into account.<ref>A. D. Becke, ''The Journal of Chemical Physics'', 1993, '''98''', 5648</ref>

===== Semiempirical Austin Model 1 ('''AM1''') =====
Semiempirical Austin Model 1 ('''AM1''') based on the Neglect of Differential Diatomic Overlap (NDDO) integral approximation.<ref>M.
J. S. Dewar, E. G. Zoebisch, E. F. Healy and J. J. P. Stewart, ''J. Am. Chem. Soc.'', 1985, '''107''' (13), 3902</ref> Therefore, when taking the same molecule for '''AM1''' and '''HF/3-21G''', you would find that the numbers of HOMO and LUMO are different, which '''AM1''' gives smaller numbers as shown in Figure 1. This is due to the neglect of the low-lying orbitals overlapping. With that, '''AM1''' proceeds much faster than '''HF/3-21G''' for the sake of time.

==== The Cope Rearrangement ====
The Cope Rearrangement is an organic reaction involving [3,3]-sigmatropic rearrangement of 1,5-dienes, which resembles the Claisen Rearrangement.<ref>A. C. Cope and E. M. Hardy, ''J. Am. Chem. Soc.'', 1940, '''62''' (2), 441</ref> The mechanism of the Rearrangement has sparked a controversy – whether it is concerted, dissociative or stepwise.<ref>O. Wiest, K. A. Black and K. N. Houk, ''J. Am. Chem. Soc.'', 1994, '''116''', 10336</ref> With that, first, each conformer of the reactant, 1,5-hexadiene, was optimised by '''HF/3-21G'''. The lowest energy conformer of 1,5-hexadiene was found. Then, as we know, the Rearrangement undergoes either a chair or boat transition state. So, each transition state was optimised by '''HF/3-21G '''as well. By looking into the energy difference between the transition states and the reactant, the activation energy of the Cope Rearrangement with 1,5-hexadiene was found. In order to find the reaction profile and see how the 1,5-diene rearranges, i.e. the mechanism, the transition state was optimised again with
mainly two methods. The coordinate of the chair transition state was first frozen, with the bond to be made set as 2.20000 Å. 2.20000 Å is a good bond length for partially C-C bond as suggested by the chemists’ observations in the literature. <ref>N. H. Kendall, Y. Li and J. D. Evanseck, ''Angew. Chem. Int. Ed. Engl.'', 1992, '''31''' (6), 682</ref> Then, after the optimization of the frozen coordinate, the partly form 2.20000 Å can be relaxed and the structure was then reoptimised. This methods skips the process of computing the whole force constant matrix i.e. Hessian, which saves time and costs. Furthermore, the boat transition state was optimised by '''QST2'''. '''QST2''' has a higher constrains in which requires a more accurate transition state structure to be put in. In this case, the dihedral angle plays an important role in order to be calculated by ''GaussView'' 5.0. Hence, this method is more expensive and time-consuming. From the optimised transition states, an '''IRC''' can be run for the optimised structure to see the full reaction profile. Also, the intermediates of the reaction can be observed. And finally, the reactant and two transition states
were optimised with '''B3LYP/6-31G*''' similarly. Hence, the two calculation methods can be compared by looking into the numbers obtained. Also, the numbers can be compared against the
experimental values. As explained above, '''B3LYP''' takes a more in-depth consideration, the numbers got from this method should be closer to the reality.

==== The Diels-Alder Cycloaddition ====
The Diels-Alder cycloaddition is a [4+2] cycloaddition between a dienophile and a conjugated alkene to give a cyclohexane system. Here, calculations on two Diels-Alder cycloaddition reactions are reported. They are (1) ethylene and butadiene and (2) cyclohexa-1,3-diene and maleic anhydride.

For Diels-Alder cycloaddition reaction, it is well-known that the reaction gives exo and/or endo product. Exo product implies the reaction pathway is thermodynamically controlled to give more stable product; endo product implies
the reaction pathway is kinetically controlled to give a relatively less stable product. In other words, the activation energy to form the exo product is higher than that of endo, however, the endo product is higher in energy than exo. This can usually be explained by the secondary orbital effects. In our cases, both the exo and endo products were investigated undoubtedly. This time, as you may notice, the molecule is more large in size and there are two reactants instead of just one reactant in the Cope Rearrangement, a simpler method of calculation was implemented, which is '''AM1'''. Also, the electronic distributions and orbitals of the HOMO and LUMO of the transition states were computed and visualised.

=== '''Computational Method''' ===
''All calculations were performed by GaussView 5.0. Relevant JSmol files were uploaded here, however, due to some technique glitches, some bonds, especially double bonds, might not come up properly. Yet, the structures of the molecules are generally correctly shown.''

==== The Cope Rearrangement ====
[[File:Yll113 CR.png|thumb|'''Figure 2.''' The Cope Rearrangement of 1,5-hexadiene]]
An anti and gauche conformation of the 1,5-hexadiene were drawn respectively. The drawn structures were first optimised by a not very accurate technique, i.e. '''Clean'''. Then, the '''clean'''ed structure were optimised by '''HF/3-21G'''. The point group and the energy of each conformer were found and compared to locate the low-energy minima. The optimised structures from '''HF/3-21G''' were then reoptimised by '''B3LYP/6-31G*'''. The point group of each conformer was checked and confirmed. Also, the comparison of the same conformer under different calculation method '''HF/3-21G''' and '''B3LYP/6-31G''' was carried out by looking into energy, bond lengths and bond angles. Furthermore, the '''Thermochemistry''' using job type '''Frequency''' was found in both '''HF/3-21G '''and''' B3LYP/6-31G* '''optimised anti conformers.

The boat and chair transition structures were also drawn and '''clean'''ed. The point group of each transition state was found.

Firstly, the chair transition structure was '''optimised to TS (Berny)''' in '''Opt + Freq '''using the basis of '''HF/3-21G'''. Force constant was calculated '''once'''. The frequency of vibration was checked to make sure there is one imaginary vibrational frequency. Then, '''freeze''' '''coordinate''' of the molecule by freezing the carbon-carbon bond to be made as 2.20000 Å. After that, the frozen coordinate was relaxed so the carbon-carbon bond to be made no longer be restricted to 2.20000 Å. The geometry of the transition state was then compared.

Secondly, at the same time, the boat transition structure was optimised by '''QST2''' method by specifying the reactants and products of the reaction under the basis of '''HF/3-21G'''. Labelling the atoms in
the reactant and product, and adjusting the central '''C-C-C-C '''dihedral angle to 0o plus the two inside '''C-C-C''' angles to 100o, the reactant and product could now be optimised by '''QST2''' method.

Comparing the optimised chair and boat transition structures, the connecting conformer of 1,5-hexadiene was found. The reaction energy profile was then calculated by '''IRC''' under '''HF/3-21G''' with 50 points and force constant as always for every small steps. With that, the mechanism of the reaction, as well as the whole reaction energy profile, could be observed clearly. Take the last point on the '''IRC''' and run a normal '''optimisation''' under '''HF/3-21G''' to obtain a minimized geometry.

Eventually, re'''optimise''' the structures of the two transition states with '''Opt + Freq '''under the basis of '''B3LYP/6-31G*'''. The geometries and energies of the transition structure under two different basis were compared. With that, these computed values were also compared against experimental values.

==== The Diels-Alder Cycloadditions ====
[[File:Yll113DA1.jpg|thumb|'''Figure 3. '''The Diels-Alder Cycloadditions between ethylene and butadiene]]

===== Ethylene and butadiene =====
The structure of cis-butadiene was first optimised under the basis of '''semiempirical AM1'''. The HOMO and LUMO of cis butadiene were visualised and its symmetry was determined.

The transition state of the reaction was drawn and optimised under the basis of '''semiempirical AM1'''. The '''IRC''' was carried out in order to confirm the transition structure postulated is of the correct one. Furthermore, the HOMO of the transition structure was visualised and the nodal
planes and properties of the system were interpreted.

===== Cyclohexa-1,3-diene and maleic anhydride =====
[[File:Yll113DA2.jpg|thumb|'''Figure 4. '''The Diels-Alder Cycloadditions between cyclohexa-1,3-diene and maleic anhydride]]
The transition states of the exo and endo products were drawn and optimised under the basis of '''semiempirical AM1'''. The '''IRC''' was carried out in order to confirm the transition structure postulated is of the correct one. For the number of points, 21 points were used for exo transition states and 24 for endo. This is because the energy was too shallow and the slopes tend to zero after the number of points specified above and ''GaussView 5.0'' cannot predict which direction should it goes on to calculate. Furthermore,
the bond lengths, orientation and the HOMO of the transition structures were investigated.

=== '''Results and Discussion''' ===

==== The Cope Rearrangement ====

===== Optimisation of Reactant =====
1,5-hexadiene has three free rotating carbon-carbon bonds. Each of them has three rotational minima. This gives 27 conformations of the 1,5-hexadiene molecule. Yet, only ten of them were energetically distinct due to symmetry and enantiomeric relationships.<ref>B. G. Rocque, J. M. Gonzales and H. F. Schaefer, ''Molecular Physics'', 2002, '''100''' (4), 441</ref> Two of them, the ''Ci anti ''and ''C1 gauche ''structure in here'' ''were drawn and optimizied as shown in Figure A and B and their energies were calculated as shown in Table 1.
{| class="wikitable"
!Conformation
!Molecule
!Point Group
!Energy/ Hartrees*
HF/3-21G
!RMS Gradient Norm/Hartrees*
HF/3-21G
!Relative Energy#/ kcal/mol
!Newman Projections
|-
|Gauche3
|<jmol>
<jmolApplet>
<title>Figure A: Gauge3 conformer of 1,5-hexadiene</title><color>white</color>
<size>250</size>
<uploadedFileContents> yll113CR_GAUGE_PART1.LOG </uploadedFileContents>
</jmolApplet>
</jmol>
|C1
|<nowiki>-231.69266120</nowiki>
|0.00001556
|0.00
|[[File:Yll113 torsion gauche.jpg|centre|frame|'''Figure 5.''' Newman Projection of C1 gauche3 1,5-hexadiene]]
|-
|Anti2
|<jmol>
<jmolApplet>
<title>Figure B: Anti2 conformer of 1,5-hexadiene</title><color>white</color>
<size>250</size>
<uploadedFileContents>YLL113CR ANTI PART1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|Ci
|<nowiki>-231.69253528</nowiki>
|0.00001891
|0.08
|[[File:Yll113 torsion anti.jpg|centre|frame|'''Figure 6.''' Newman Projection of Ci anti2 1,5-hexadiene]]
|}
<nowiki>*</nowiki>1 hartree = 627.509 kcal/mol

#The difference in energy between the conformer and the lowest energy conformer, in here, which is Gauche3. Then convert Hartree to kcal/mol by *

'''Table 1. '''Conformational analysis of anti2 and gauche3 of 1,5-hexadiene

As shown in Table 1, the energy of Gauche3 is surprisingly lower than the anti2 conformation of 1,5-hexadiene. In most cases, the antiperiplanar conformation of a molecule, such as anti2, is more favourable as it has the least steric clashes. Therefore, usually the antiperiplanar conformation is of the lowest energy. However, here, apart from sterics, the stereoelectroncs concept has also been taken into account. The vinyl proton, in a through space manner, can interact with the π or π* orbital on the sp2 carbon which is separated by four bonds from it.<ref>M. Nishio and M. Hirota, ''Tetrahedron'', 1989, '''45 '''(23), 7201</ref> This is so-called CH-π interaction. The Newman Projection in Figure 5 gives us a closer look on how they are close in space and interact; and the Newman projection in Figure 6 tells us why the vinyl proton cannot interact with the π or π* system through space. Therefore, the gauche3 conformation is more stable than anti2 and of lower energy in 1,5-hexadiene.

Focusing on anti2 conformer of the 1,5-hexadiene, the '''HF/3-21G''' optimised molecule was reoptimised by '''B3LYP/6-31G'''. The energies and the geometries of the anti2 1,5-hexadiene under two basis of calculation method were compared and shown in Table 2.
[[File:Yll113Anti2.png|thumb|'''Figure 7. '''Anti2 conformer of 1,5-hexadiene with labelled atoms]]
{| class="wikitable"
! rowspan="2" |Calculation Method
! rowspan="2" |Point Group
! rowspan="2" |Energy/ Hartrees
! rowspan="2" |RMS Gradient Norm/Hartrees
!Bond length between/ Å
!Bond length between/ Å
!Bond length between/ Å
!Bond angle between
!Bond angle between
|-
|C14-C12 and C4-C1
|C12-C9 and C6-C4
|C6-C9
|C14-C12-C9 and
C1-C4-C6
|C12-C9-C6 and
C4-C6-C9
|-

|'''HF/3-21G'''
|Ci
|<nowiki>-231.69253528</nowiki>
|0.00001891
|1.31613
|1.50891
|1.55275
|124.80579o
|111.34878o
|-
|'''B3LYP/6-31G*'''
|Ci
|<nowiki>-234.61171063</nowiki>
|0.00001249
|1.33350
|1.50419
|1.54816
|125.29968o
|112.67081o
|}
'''Table 2. '''Comparison with the anti2 conformer of 1,5-hexadiene with two calculation method basis ('''HF/3-21G '''and '''B3LYP/6-31G*''')

As seen in Table 2, the point group of the same conformer does not change no matter which method of calculation was carried out. The energy of the conformer with '''B3LYP/6-31G*''' is lower than that of '''HF/3-21G'''. Also, the bond lengths and bond angles changed with a certain amount as shown.

Furthermore, by submitting job type as '''Frequency''' in the optimised structure of anti2 conformer of 1,5-hexadiene under both calculation basis '''HF/3-21G '''and '''B3LYP/6-31G*''', the electronic energy, sum of electronic and zero-point energies at 0 K and sum of electronic and thermal energies at 298.15 K could be determined as shown in Table 3.

{| border="1" cellspacing="1" cellpadding="10"
|-
|rowspan="3"| ''' '''
!colspan="4" |'''HF/3-21G a'''
! colspan="4" |'''B3LYP/6-31G* b'''
|-
! rowspan="2" width="125" align="center" | '''Electronic energy'''
! width="125" align="center" | '''Sum of electronic and zero-point energies'''
! width="125" align="center" | '''Sum of electronic and thermal energies'''
! rowspan="2" width="125" align="center" | '''IR spectrum'''
! rowspan="2" width="125" align="center" | '''Electronic energy'''
! width="125" align="center" | '''Sum of electronic and zero-point energies'''
! width="125" align="center" | '''Sum of electronic and thermal energies'''
! rowspan="2" width="125" align="center" | '''IR spectrum'''
|-
! width="125" align="center" | '''at 0 K'''
! width="125" align="center" | '''at 298.15 K'''
! width="125" align="center" | '''at 0 K'''
! width="125" align="center" | '''at 298.15 K'''
|-
| '''Reactant (anti2)'''
| align="center" | -231.692534
| align="center" |-231.539539
| align="center" | -231.532565
|[[File:Yll113ANTI3-21IR.png|thumb|'''Figure 8. '''IR spectrum of anti2 reactant 1,5-hexadiene calculated by '''HF/3-21G''']]
| align="center" | -234.611703
| align="center" | -234.469213
| align="center" | -234.461857
|[[File:Yll113ANTI6-31IR.png|thumb|'''Figure 9. '''IR spectrum of anti2 reactant 1,5-hexadiene calculated by '''B3LYP/6-31G*''']]
|}

a [https://wiki.ch.ic.ac.uk/wiki/images/5/52/Yll113CR_ANTI_PART4.LOG File]; b [https://wiki.ch.ic.ac.uk/wiki/images/5/54/Yll113_CR_ANTI_PART3.LOG File]

'''Table 3.''' Summary of energies (in hartree) of the reactant (anti2) Comparing Figure 8 and 9, both IR spectra are similar which shows that they have similar stretching/bending mode even under different methods of calculations. However, the strongest peak in Figure 8 is at 1112 cm-1 and that in Figure 9 is 940 cm-1. Yet, they correspond to the same mode of vibration, which is the =C-H bending. Therefore, according to the equation, the wavenumber of absorbance, ν can be calculated:

:<math>\nu = \frac{1}{2 \pi c} \sqrt{\frac{k}{\mu}}</math>

where ''μ'' is the reduced mass, ''k'' is the spring constant for the bond and ''c'' is the light speed,

we can therefore know that the spring constant of the molecule under '''B3LYP/6-31G* '''basis reduced.

Now, focusing on gauche3 conformer of the 1,5-hexadiene, similarly, the '''HF/3-21G''' optimised molecule was reoptimised by '''B3LYP/6-31G'''. The energies and the geometries of the gauche3 1,5-hexadiene under two basis of calculation method were compared and shown in Table 4.
[[File:Yll113Gauche3.png|thumb|'''Figure 10. '''Gauche3 conformer of 1,5-hexadiene with labelled atoms]]
{| class="wikitable"
! rowspan="2" |Calculation Method
! rowspan="2" |Point Group
! rowspan="2" |Energy/ Hartrees
! rowspan="2" |RMS Gradient Norm/Hartrees
! colspan="3" |Bond length between/ Å
! colspan="2" |Bond angle between
|-
|C14-C12 and C4-C1
|C12-C9 and C6-C4
|C6-C9
|C14-C12-C9 and
C1-C4-C6
|C12-C9-C6 and
C4-C6-C9
|-

|'''HF/3-21G'''
|C1
|<nowiki>-231.69266120</nowiki>
|0.00001555
|1.31646
|1.50929
|1.55314
|125.02428o
|111.80728o
|-
|'''B3LYP/6-31G*'''
|C1
|<nowiki>-234.61132605</nowiki>
|0.00000360
|1.33382
|1.50491
|1.55007
|125.49464o
|113.46225o
|}
'''Table 4. '''Comparison with the anti2 conformer of 1,5-hexadiene with two calculation method basis ('''HF/3-21G '''and '''B3LYP/6-31G*''')

As seen in Table 4, the point group of the same conformer, again, does not change no matter which method of calculation was carried out. The energy of the conformer with '''B3LYP/6-31G*''' is lower than that of '''HF/3-21G'''. Also, the bond lengths and bond angles changed with a certain amount as shown.

Furthermore, by submitting job type as '''Frequency''' in the optimised structure of gauche3 conformer of 1,5-hexadiene under both calculation basis '''HF/3-21G '''and '''B3LYP/6-31G*''', the electronic energy, sum of electronic and zero-point energies at 0 K and sum of electronic and thermal energies at 298.15 K could be determined as shown in Table 5.

{| border="1" cellspacing="1" cellpadding="10"
|-
| rowspan="3" | ''' '''
! colspan="4" |'''HF/3-21G c'''
! colspan="4" |'''B3LYP/6-31G* d'''
|-
! rowspan="2" width="125" align="center" | '''Electronic energy'''
! width="125" align="center" | '''Sum of electronic and zero-point energies'''
! width="125" align="center" | '''Sum of electronic and thermal energies'''
! rowspan="2" |IR spectrum
! rowspan="2" width="125" align="center" | '''Electronic energy'''
! width="125" align="center" | '''Sum of electronic and zero-point energies'''
! width="125" align="center" | '''Sum of electronic and thermal energies'''
! rowspan="2" |IR spectrum
|-
! width="125" align="center" | '''at 0 K'''
! width="125" align="center" | '''at 298.15 K'''
! width="125" align="center" | '''at 0 K'''
! width="125" align="center" | '''at 298.15 K'''
|-
| '''Reactant (Gauche 3)'''
| align="center" |-231.692692
| align="center" |-231.539486
| align="center" |-231.532646
|[[File:Yll113GAUCHE3-21IR.png|thumb|'''Figure 11. '''IR spectrum of gauche3 conformer of 1,5-hexadiene calculated by '''HF/3-21G''']]
| align="center" | -234.611326
| align="center" |-234.468719
| align="center" |-234.461477
|[[File:Yll113GAUCHE6-31IR.png|thumb|'''Figure 12.''' IR spectrum of gauche3 conformer of 1,5-hexadiene calculated by '''B3LYP/6-31G*''']]
|}

c [https://wiki.ch.ic.ac.uk/wiki/images/9/98/Yll113CR_GAUGE_PART4.LOG File] ; d [https://wiki.ch.ic.ac.uk/wiki/images/c/ca/Yll113CR_GAUGE_PART3.LOG File]

'''Table 5.''' Summary of energies (in hartree) of the reactant (Gauche3) Comparing Figure 11 and 12, both IR spectra are similar which shows that they have similar stretching/bending mode even under different methods of calculations. However, the strongest peak in Figure 12 is at 939 cm-1 and that in Figure 11 is 1111 cm-1. Yet, they correspond to the same mode of vibration, which is also the =C-H bending. Therefore, similar to the anti2 conformer's case as mentioned above, we can therefore know that the spring constant of the molecule under '''B3LYP/6-31G* '''basis reduced.

===== Optimisation of transition state =====

After optimising the reactants, the chair and boat transition states were optimised accordingly using mainly two different methods. But before that, an allyl fragment (CH2CHCH2) was optimised using '''HF/3-21G''' level of theory for the sake of convenience in constructing the chair and boat transition states. A brief summary was shown in Table 6.
{| class="wikitable"
!
! colspan="2" |Molecule
!Point Group
!Energy/Hartree
!RMS Gradient Norm/ Hartrees
|-
|Allyl fragment
CH2CHCH2
|<jmol>
<jmolApplet>
<title>Figure C: Allyl Fragment</title><color>white</color>
<size>250</size>
<uploadedFileContents> Yll113CR TS 1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|[[File:Yll11313.jpg|thumb|'''Figure 13. '''Optimised Structure of the allyl fragment]]
|C2v
|<nowiki>-115.82304010</nowiki>
|0.00002945
|}
'''Table 6. '''Summary of the optimised allyl fragment

Then, both chair and boat transition state were drawn and optimised using the '''optimisation to TS (Berny)''' under '''HF/3-21G'''. Figure 14 and Figure C show the optimized structure of the chair transition state while Figure 15 and Figure D show the optimized structure of the boat transition state. Table 7 shows the summary of results.
{| class="wikitable"
! rowspan="2" |
! colspan="2" rowspan="2" |Molecule
! rowspan="2" |Point Group
! rowspan="2" |Energy/ Hartree
! rowspan="2" |RMS Gradient Norm/Hartrees
! colspan="6" |Partial C-C bond length between/ Å
|-
!C14-C6
!C11-C3
!C14-C9
!C6-C1
!C9-C11
!C1-C3
|-
|Optimised Chair Transition State
|<jmol>
<jmolApplet>
<title>Figure D: Optimised Chair transition state</title><color>white</color>
<size>250</size>
<uploadedFileContents>Yll113CHAIR PART1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|[[File:Yll113CHAIR3-21.png|thumb|'''Figure 14. '''Optimised Chair Structure by '''HF/3-21G''' with atoms labelled ]]
|C2h
|<nowiki>-231.61932238</nowiki>
|0.00002645
|2.02016
|2.02016
|1.38929
|1.38930
|1.38930
|1.38929
|-
|Optimised Boat Transition State
|<jmol>
<jmolApplet>
<title>Figure E: Optimised Boat transition state</title><color>white</color>
<size>250</size>
<uploadedFileContents>Yll113BOAT PART1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|[[File:Yll11315.jpg|thumb|'''Figure 15. '''Optimised Boat Structure by '''HF/3-21G''' with atoms labelled]]
|C2v
|<nowiki>-231.60280235</nowiki>
|0.00003872
|2.14060
|2.14060
|1.38138
|1.38138
|1.38138
|1.38138
|}

'''Table 7. '''Summary of the optimised chair and boat transition states by '''optimisation to TS (Berny) '''under '''HF/3-21G''' basis

Furthermore, the transition structures’ '''Frequencies''' were calculated as shown in Table 8.
{| class="wikitable"
!
!Molecule
!IR spectrum
!Imaginary Frequency/ cm-1
|-
|Boat Transition State
|
|[[File:Yll11317.jpg|thumb|'''Figure 16. '''IR spectrum of the optimised Boat Structure by '''HF/3-21G''']]
|<nowiki>-840</nowiki>
|-
|Chair Transition State
|[[File: Yll113Chair Part 1 imag freq.gif]]
|[[File:Yll11316.jpg|thumb|'''Figure 17. '''IR spectrum of the optimised Chair Structure by '''HF/3-21G''']]
|<nowiki>-818</nowiki>
|}
'''Table 8.''' IR spectra and imaginary frequencies of the boat and chair transition states

As you may notice that, the
imaginary frequency comes up when calculating with the transition states. This
is common, in other words, this should appear to let us know the transition
structure we postulated is correct.

A transition state is the first
order saddle point on the potential energy surface. Therefore, the force
applied to the saddle point against to the displacement. As force and
displacement are vectors, the force constant will be a negative number.Therefore, according to
:<math>\nu = \frac{1}{2 \pi c} \sqrt{\frac{k}{\mu}}</math>

where ''μ'' is the reduced mass, ''k'' is the spring constant for the bond and ''c'' is the light speed,

the square root of a negative
force constant k gives an imaginary wave number/frequency v. In other words,
the appearance of an imaginary frequency tells us that the structure is a
saddle point of the reaction pathway.

The chair transition state
was followed by first 'frozen' then 'relaxed'. The boat transition structure
was followed by '''QST2''' calculation method.

====== Chair Transition State ======
After the above '''optimisation''', the chair transition
state was reoptimised again with another method. This method first freezes the
coordinate of the molecule, in this case, freeze the bond to be made in the
Cope Rearrangement of 1,5-hexadiene as 2.20000 Å. The molecule then optimised with the frozen
coordinate. Details of this optimisation was summarized in Table 9.
{| class="wikitable"
! rowspan="2" |
! colspan="2" rowspan="2" |Molecule
! rowspan="2" |Point Group
! rowspan="2" |
{| class="MsoTableGrid"

|
<nowiki> </nowiki>Energy/ Hartree

|}
! rowspan="2" |RMS Gradient Norm/ Hartree
! colspan="3" |Bond length between/ Å
! rowspan="2" |Imaginary Frequency/ cm-1
! rowspan="2" |IR Spectrum
|-
!C6-C14 and C3-11
!C1-C3 and C9-C14
!C1-C6 and C9-C11
|-
|Optimised Chair Transition Structure with frozen coordinate
|JSMOL
|[[File:Yll11318.jpg|thumb|'''Figure 18. '''The optimised chair transition structure with frozen coordinate and atoms labelling]]
|
{| class="MsoTableGrid"

|
<nowiki> </nowiki>C2h

|}
|<nowiki>-231.61518510</nowiki>
|0.00325573
|2.20000
|1.38135
|1.38128
|<nowiki>-765</nowiki>
|[[File:Yll11319.jpg|thumb|'''Figure 19. '''IR spectrum of the optimised chair transition structure with frozen coordinate]]
|}

'''Table 9. '''Summary of the optimisation of the chair transition structure with
frozen coordinate(s)

From Table 9, we may notice
that the RMS Gradient Norm value is quite far off from zero. Also, the
imaginary frequency becomes much higher than -818 cm-1 (Shown in
Table 8). With these two pieces of information, we can deduce that the frozen
coordinate(s) affect(s) the force constant of the transition state which does
not give a good optimisation of transition structure. With that, after applying
the frozen coordinate to the molecule, the molecule was reoptimised again with
a degree of '''Derivative '''to the '''Bond'''. Details of the reoptimisation
were presented in Table 10.
{| class="wikitable"
! rowspan="2" |
! colspan="2" rowspan="2" |Molecule
! rowspan="2" |Point Group
! rowspan="2" |Energy/ Hartree
! rowspan="2" |RMS Gradient Norm/ Hartree
! colspan="4" |Bond length between/ Å
! rowspan="2" |Imaginary Frequency/ cm-1
! rowspan="2" |
{| class="MsoTableGrid"

|
<nowiki> </nowiki>IR Spectrum

|}
|-
!C14-C6
!C11-C3
!C14-C9 and C6-C1
!C9-C11 and C1-C3
|-
|Optimised Chair Transition
Structure with a degree of '''Derivative'''
to the '''Bond'''
|JSMOL
|[[File:Yll11320.jpg|thumb|'''Figure 20. '''The optimised chair transition structure with a degree of '''Derivative''' to the '''Bond''' and atoms labelling]]
|C2h
|<nowiki>-231.61932233</nowiki>
|0.00002127
|2.02075
|2.02071
|1.38929
|1.38930
|<nowiki>-818</nowiki>
|[[File:Yll11321.jpg|thumb|'''Figure 21. '''IR spectrum of the optimised chair transition structure with a degree of '''Derivative''' to the '''Bond''']]
|}
'''Table 10. '''Summary of the reoptimisation of the chair transition
structure with a degree of '''Derivative''' to the '''Bond'''

Now, in Table 10, the RMS
Gradient Norm value is close to zero. Also, the imaginary frequency goes back
to -818 cm-1, indicating that the coordinates no longer be frozen
and the stretching/bending mode of the transition state is able to undergo
freely.

Comparing the bond lengths
in Table 7 and 10, we can see that the difference between bond lengths of the
single bond to be made/ broken calculated in two methods is just less than
0.0006 Å. And also, there is no difference in bond length of the double bond to be make/broken ‘inside’ the system. This tells us that the two optimisation
methods are rather similar under the consideration on the Cope Rearrangement
Reaction.

====== Boat Transition State ======
Instead of using the frozen
coordinate method as for the chair transition state above, another method, '''QST2''', was applied to the boat
transition state under the '''HF/3-21G'''
basis. In order to use this method, without any ‘Link died’, the reactant and
product have to be drawn and labelled carefully. '''QST2''' is a method which interpolates the reactant and product to
give a transition state. Therefore, it will fall if the structure of the
reactant and product are not close to the transition state. And therefore, all
molecules have to be carefully labelled and adjusted.

[[File:Yll11322.png|500px|center]]

'''Figure 22. '''The drawings and adjustments of angles of the reactant (left)
and product (right) for '''QST2''' Method,
i.e. the central C-C-C-C dihedral angle was changed to 0o and inside
C-C-C were reduced to 100o

After the adjustment, the job was run and the optimized molecule converge to the boat transition structure. Summary was shown in Table 11.
{| class="wikitable"
! rowspan="2" |
! colspan="2" rowspan="2" |Molecule
! rowspan="2" |Point Group
! rowspan="2" |Energy/Hartree
! rowspan="2" |RMS Gradient Norm/ Hartree
! colspan="3" |Bond length between/ Å
! rowspan="2" |Imaginary Frequency/ cm-1
! rowspan="2" |IR spectrum
|-
!C1-C6
!C3-C4
!C5-C6, C4-C5, C3-C2 and C1-C2
|-
|Boat transition structure
under '''QST2''' method
|JSMOL
|[[File:Yll11323.jpg|thumb|'''Figure 23. '''The optimised Boat transition structure with atom labelling]]
|C2v
|<nowiki>-231.60280241</nowiki>
|0.00002436
|2.13994
|2.14019
|1.38149
|<nowiki>-840</nowiki>
|[[File:Yll11324.jpg|thumb|'''Figure 24. '''IR spectrum of the optimised boat transition structure]]
|}
'''Table 11. '''Summary of the boat transition structure under '''QST2 '''method

====== Intrinsic Reaction Coordinate''' '''('''IRC''') ======
In order to confirm that our transition state is of the
correct one, '''Intrinsic Reaction
Coordinate '''('''IRC''') will be carried
out.

As mentioned above, transition state is the first order
saddle point of the reaction pathway. Therefore, it will start to go to the
product or back to the reactant with it falls off. It resembles that a ball is
at the tip of the mountain, which starts to roll off the mountain on the side
with the steepest slope. Also, when we are doing '''IRC''', we can determine whether the reaction goes forward, backward
or both sides. Also, the number of points, which means the number of little
steps that the geometry of the molecule changes, can be adjusted. A low number
of points will just give us a very rough idea that tell us a little bit about
our transition state. Also, the last point on the '''IRC''' is far from the minimum geometry. A high number of points gives
us more accurate results, however two problems could be raised. First, the time
for calculation will be long and Most importantly, as it goes down the slope
and reaches the minimum geometry, i.e. the plateau of energy, the slope will
become very small or even zero again. However, as the energy difference of the
next or previous geometry compared to the geometry of itself is too small, ''GaussView 5.0'' may not able to know which
direction the molecule should proceed to. And this, therefore, results in ‘Link
died’. Therefore, the most common technique is to have a good number of points,
then take the last point on the IRC and run it with a normal optimisation.

Here, as we know that the
Cope Rearrangement has a symmetric reaction pathway, taking the chair
transition structure, we will run '''IRC'''
on it with 50 points.
{| class="wikitable"
![[File:Yll113hlj29825.jpg|thumb|'''Figure 25. '''Total energy along '''IRC '''of the chair transition structure]]
![[File:Yll11326.jpg|thumb|'''Figure 26. '''RMS Gradient Norm of '''IRC '''of the chair transition structure]]
!JSMOL
|}

[[File:Yll11327.jpg|500px]]

'''Figure 27. '''The product of the Cope Rearrangement after optimisation

The first point on Figure 25 is -231.61932233 Hartree and the last point is -231.69157881 Hartree. Then, we take the last point and optimise it, we get the structure shown in Figure 27.

The structure is of C2
symmetry and the energy calculated is -231.69166702 Hartree. This matches with
Gauche2 C2 on Appendix 1. And therefore, this is how the conformer
of 1,5-hexadiene connects with the chair transition structure.

====== Activation Energy of the Cope Rearrangement ======
Finally, we optimise the chair and boat transition states we got from above, reoptimise it with job Opt + Freq
under a more advanced calculation '''B3LYP/6-31G*'''. And from that, the thermochemistry data were given and we can know the
activation energy of the reaction by comparing to Table 3, which anti2 is used
as a local minimum rather than gauche3 as a global minimum.
{| border="1" cellspacing="1" cellpadding="10"
|-
| rowspan="3" | ''' '''
! colspan="3" |'''HF/3-21G'''
! colspan="3" |'''B3LYP/6-31G*'''
|-
| rowspan="2" width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| rowspan="2" width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619323
| align="center" | -231.466698
| align="center" | -231.461339
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409009
|-
| '''Boat TS'''
| align="center" | -231.602803
| align="center" | -231.450932
| align="center" | -231.445303
| align="center" | -234.543094
| align="center" | -234.402342
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692534
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.611703
| align="center" | -234.469213
| align="center" | -234.461857
|}
'''Table 11'''. Summary of energies of chair, boat and reactant (anti2) structure
{| border="1" cellspacing="1" cellpadding="10"
|-
| rowspan="2" | ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.69
| align="center" | 34.06
| align="center" | 33.16
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.96
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
'''Table 12'''. Summary of activation energies in kcal/mol

==== The Diels-Alder Cycloadditions ====

===== Ethylene and Cis-Butadiene =====
First, the structures of the reactants, ethylene and cis-butadiene, were '''optimised to minimum''' under '''semiempirical AM1''' method. For the butadiene, in order to be in the cis conformer, the dihedral angle was adjusted to be 0o. Details are listed in Table 13.
{| class="wikitable"
!
!Molecule
!Point Group
!HOMO
!LUMO
!Energy/Hartree
!RMS Gradient Norm/ Hartree
!IR spectrum
|-
|Ethylene
|JSMOL
|D2h
|[[File:Yll11331.jpg|thumb|'''Figure 31. '''HOMO of Ethylene]]
|[[File:Yll11330.jpg|thumb|'''Figure 30.''' LUMO of ethylene]]
|0.02619029
|0.00008755
|[[File:Yll11328.jpg|thumb|'''Figure 28. '''IR spectrum of Ethylene]]
|-
|Cis-Butadiene
|JSMOL
|C2v
|[[File:Yll11332.jpg|thumb|'''Figure 32. '''HOMO of cis-butadiene]]
|[[File:Yll11333.jpg|thumb|'''Figure 33. '''LUMO of cis-butadiene]]
|0.04878534
|0.00000087
|[[File:Yll11329.jpg|thumb|'''Figure 29.''' IR spectrum of cis-butadiene]]
|}
'''Table 13.''' Summary of optimised ethylene and cis-butadiene

Looking into Figure 30-33, as we know that the plane is perpendicular to the molecule, the HOMO of Ethylene is symmetric while that of LUMO is antisymmetric.

Also, the HOMO of cis-butadiene is antisymmetric and that of LUMO is symmetric.

Then, the transition state of the reaction was able to constructed using the optimised structure of the reactants made above. By making the bond lengths of the bonds to be made in Diels-Alder [4+2] cycloaddition to 2.20000 Å in the transition state, which is the 'dash bond', the transition structure was optimized to '''Opt + Freq minimisation to''' '''TS (Berny)''' under the same method of calculation - '''semiempitical AM1'''. Details are listed in Table 14.
{| class="wikitable"
!
! colspan="2" |Molecule
!Point Group
!HOMO
!LUMO
!Energy/Hartree
!RMS Gradient Norm/ Hartree
!Imaginary Frequency/ cm-1
!IR spectrum
|-
|Transition state
|JSMOL
|[[File:Yll11334.jpg|thumb|'''Figure 34. '''Transition state of ethylene + cis-butadiene]]
|C1
|[[File:Yll11336.jpg|thumb|'''Figure 36. '''HOMO of transition state]]
|[[File:Yll11337.jpg|thumb|'''Figure 37. '''LUMO of transition state]]
|0.11165467
|0.00002792
|<nowiki>-956</nowiki>
|[[File:Yll11335.jpg|thumb|'''Figure 35. '''IR spectrum of the transition state]]
|}
'''Table 14.''' Summary of optimised transition state

From Figure 36, we can see that the HOMO of the transition state is antisymmetric whilst the LUMO of the transition state is symmetric. By making very careful comparison between Figure 36, Figure 37 and Figure 30-33, we can see that the HOMO of the transition state in Figure 36 is a combination of Figure 32 and 30; the LUMO of the transition state in Figure 37 is a combination of Figure 31 and 33. We can clearly see that the HOMO and LUMO of the transition state have a complementary combination of HOMO and LUMO of the reactants.

Taking a closer look to HOMO of the transition state. Recalling Woodward Hoffmann’s Rule, (4q+2)s+(4r)a = odd for thermally allowed reaction, we have both π2s and π4s. Therefore, the reaction is thermally allowed by letting q = 0, which gives the value of 1 which is odd.

Furthermore, from Table 14, we notice that there is an imaginary frequency reported at -956 cm-1. As explained above, the transition state should have one imaginary frequency to account for the negative force constant. With that, this imaginary frequency confirms that the transition structure we postulated from the optimised reactants is valid, i.e. it is really a transition state. The animation of where the imaginary frequency originates from, which shows the motion of the transition state - how the two reactants approach to each other and bonds are formed, is shown below.

JSMOL

From the above figure, we can see that the bond formation from the reactant to the product happens at the same time, i.e. synchronous, on both sides of the transition structure. Therefore, we can say that this Diels-Alder cycloaddition is a concerted [4+2] pericyclic cycloaddition, which matches with what we learnt in Pericyclic Reaction course.

On top of that, the geometry of the transition structure was investigated by looking into the optimised bond lengths between carbon atoms Details are shown in Figure 38 and Table 15.[[File:Yll11338.jpg|thumb|'''Figure 38. '''Transition state of ethylene + cis-butadiene with atoms labelled]]
{| class="wikitable"
!
!Bond length/ Å
|-
|C7-C9
|2.11938
|-
|C12-C5
|2.11944
|-
|C12-C9
|1.38284
|-
|C3-C7
|1.38187
|-
|C3-C1
|1.39750
|-
|C5-C1
|1.38175
|}
'''Table 15. '''Geometry analysis of the transition state

According to the literature <ref>M. A. Fox and J. K. Whitesell, ''Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen'', Springer, 1995</ref>, C-C carbon-carbon single bond is 1.54 Å, and C=C carbon-carbon double bond is 1.34 Å. Also, the Van der Waals radius of carbon is 1.70 Å,<ref>A. Bondi,(1964), ''J. Phys. Chem.'', 1964, '''68''' (3), 441</ref>
According to the reaction scheme shown in Figure 3, a single bond is forming between C7 and C9, also another single bond is forming between C12-C5. Comparing the data in Table 15 with the literature, we can see that the bond length of two bonds to be made is longer than C-C, but shorter than the twice of carbon's Van der Waals radius. This tells us some hints that the terminal carbons between two reactants are attracting with each other to a certain degree. Furthermore, taking a look to other carbon-carbon bonds within the system, we notice that they are somewhere between C-C and C=C. This tells us that either C-C is becoming C=C or C=C is becoming C-C in the transition state.
After that, the '''IRC''' of the above optimised transition state was carried out with both direction and force constant calculated always for 50 points to see the reaction profile.
{| class="wikitable"
|[[File:Yll11339.jpg|thumb|'''Figure 39.''' IRC of the transition state of ethylene + cis-butadiene]]
|[[File:Yll11340.jpg|thumb|'''Figure 40. '''RMS Gradient Norm of transition state of ethylene + cis-butadiene]]
|JSMOL
|}
In Figure 39, we can clearly see that the reactants was first passed through the energy barrier to get the transition state and it went down the slope to give the product.
Finally, the activation energy for this reaction was calculated in Table 16.
{| class="wikitable"
!
!Ethylene
!Cis-butadiene
!Transition state
!Activation Energy
|-
|Energy/Hartree
|0.02619029
|0.04878534
|0.11165467
|0.03667904
(23.02 kcal/mol)
|}
'''Table 16. '''Activation energy analysis of Diels-Alder Reaction between ethylene and cis-butadiene
===== Cyclohexa-1,3-diene and Maleic Anhydride =====
First, the structures of
the reactants, ethylene and cis-butadiene, were '''optimised to minimum''' under '''semiempirical AM1''' method. Details are listed in Table 17.
{| class="wikitable"
!Optimised Structure
!Molecule
!Point Group
!Energy/Hartree
|-
|Cyclohexa-1,3-diene
|[[File:Yll113Cycloopt.jpg|thumb|'''Figure 41. '''Optimised structure of cyclohexa-1,3-diene]]
|C2
|0.02771130
|-
|Maleic Anhydride
|[[File:Yll113Maopt.jpg|thumb|'''Figure 42. '''Optimised structure of Maleic Anhydride]]
|Cs
|<nowiki>-0.12182418</nowiki>
|}
'''Table 17. '''Summary
of the optimized reactant

Then, the two transition states, exo and endo, of the reaction were able to construct. By making the bond lengths of the bonds to be made in Diels-Alder [4+2] cycloaddition to 2.20000 Å in the transition state, which is the 'dash bond', the transition structure was optimized to '''Opt + Freq minimisation to''' '''TS (Berny)''' under the same method of calculation - '''semiempitical AM1'''. Details are listed in Table 18.
{| class="wikitable"
!Transition Structure
!Molecule
!Point Group
!HOMO
!LUMO
!Energy/Hartree
!'''RMS Gradient Norm/ Hartree'''
!'''Imaginary Frequency/ cm-1'''
!IR spectrum
|-
|Exo
|[[File:Yll113Exots.jpg|thumb|'''Figure 43.''' Optimised Structure of Exo Transition State]]
|C1
|[[File:Yll113Exo homo.jpg|thumb|'''Figure 45. '''HOMO of Exo Transition State]]
|[[File:Yll113Exo lumo.jpg|thumb|'''Figure 47.''' LUMO of Exo Transition State]]
|<nowiki>-0.05041984</nowiki>
|0.00000883
|<nowiki>-812</nowiki>
|[[File:Yll113Exo ir.jpg|thumb|'''Figure 49. '''Exo Transition State IR spectrum]]
|-
|Endo
|[[File:Yll113Endots.jpg|thumb|'''Figure 44. '''Optimised Structure of Enfo Transition State]]
|C1
|[[File:Yll113Endo homo.jpg|thumb|'''Figure 46'''. HOMO of Endo Transition State]]
|[[File:Yll113Endo lumo.jpg|thumb|'''Figure 48.''' LUMO of Endo Transition States]]
|<nowiki>-0.05150469</nowiki>
|0.00004308
|<nowiki>-807</nowiki>
|[[File:Yll113Endo ir.jpg|thumb|'''Figure 50.''' Endo Transition State IR Spectrum]]
|}
'''Table 18. '''Summary of the optimized transition structures

In Table 18, we can see that both HOMO and LUMO of the exo
transition state are antisymmetric with respect to the plane. Also, both HOMO and LUMO of the endo transition state are antisymmetric as well.

Also, we notice that the energy of exo is higher than that of endo. This can be explained by the poorer overlap between the C=C π and C=O π* compared to that of endo. This is called secondary orbital effect, which will be further discussed below.

There is one imaginary frequency in both transition structure. This confirms that both of them are valid postulated transition structures.
{| class="wikitable"
|JSMOL
'''Imaginary frequency of exo transition state animation'''
|JSMOL
'''Imaginary frequency of exo transition state animation'''
|}
Again, from the above animations, we can see that the bonds are formed in a synchronous fashion, which means the Diels-Alder Cycloaddition reaction is concerted.

Furthermore, the geometries of both transition states were examined carefully in Table 19.
{| class="wikitable"
!Geometry summary of Exo Transition State (Please refer to Figure 43 for atom labelling)
!Geometry summary of Endo Transition State (Please refer to Figure 44 for atom labelling)
|-
|
{| class="wikitable"
!
!Bond length/ Å
!
!Bond length/ Å
|-
|C5-C13
|2.17032
|C6-C14
|2.17035
|-
|C5-C3
|1.39440
|C3-C1
|1.39674
|-
|C1-C6
|1.39440
|C14-C13
|1.4612
|-
|C7-C6
|1.48976
|C7-C10
|1.52208
|-
|C10-C5
|1.48976
|C13-C15
|1.48822
|-
|C14-C16
|1.48822
| colspan="2" |N/A
|-
|C7-C16
(Through space)
|2.94507
|C10-C15
(Through space)
|2.94484
|-
|C1-C16
(Through space)
|3.78172
|C3-C15
(Through Space)
|3.78155
|}
|
{| class="wikitable"
!
!Bond length/ Å
!
!Bond length/ Å
|-
|C15-C7
|2.16230
|C16-C5
|2.16229
|-
|C1-C3
|1.39726
|C3-C7
|1.39296
|-
|C1-C5
|1.39308
|C7-C9
|1.49503
|-
|C9-C12
|1.52300
|C5-C12
|1.49054
|-
|C16-C18
|1.48918
|C15-C17
|1.48903
|-
|C15-C16
|1.40863
| colspan="2" |N/A
|-
|C1-C18
(Through space)
|2.89232
|C3-C17
(Through space)
|2.89203
|}
|}
'''Table 19.''' Geometry analysis of exo and endo transition states

According to the reaction scheme shown in Figure 4, a single bond is forming between C5 and C13, also another single bond is forming between C6-C14 for exo; C15 and C7 plus C16 and C5 for endo, which is what the first row in the two tables in the left and right in Table 19 shows. the single bond to be made Comparing these values with literature, we find that they are longer than C-C but shorter than twice of carbon's Van der Waals' radius. This tells us some hints that these pairs of carbons between two reactants are attracting with each other to a certain degree. Furthermore, taking a look to other carbon-carbon bonds within the system, i.e. except row 1 and those labelled with (through space), we notice that they are somewhere between C-C and C=C. This tells us that either C-C is becoming C=C or C=C is becoming C-C in the transition state.

Now, looking at the through space bond length. In the exo transition structure, the through space distance -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH2-CH2- is smaller than the twice of the Van der Waals radius but larger than C-C which gives us a hint that they might interact. In the endo transition structure, the through space distance -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH=CH- is smaller than the twice of the Van der Waals radius but larger than C-C which gives us a hint that they might interact. However, according to the definition of secondary orbital effect, it is looking for the interaction between the C=C π of the diene and C=O π* of the dienophile. Endo clearly shows that as explained, but exo seems to just demonstrate the sterics clash between the -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH2-CH2- fragment of diene. In order to further confirm that exo has no secondary orbital effect, a measurement of bond length was carried out between -(C=O)-O-(C=O)- fragment of the maleic anhydride and the -CH=CH- in diene in the exo transition state. The result was shown in the last row on the left table in Table 19. This shows that they are too far away which means they are not possible to interact.

Now, looking back to the HOMO of exo and endo transition states in Figure 45 and 46 respectively. We can definitely see that the overlap between the two reactants is relatively smaller in exo. From these two pieces of information, we can conclude that the endo is kinetically controlled, while exo is thermodynamically controlled.

After that, the '''IRC''' of the both optimised transition state was carried out with both direction and force constant calculated always for the reaction profile. 21 points were used for exo transition states and 24 for endo (reasons explained under '''Introduction)''' to see the reaction profiles.
{| class="wikitable"
! colspan="3" |Exo Transition State
! colspan="3" |Endo Transition State
|-
|[[File:Yll113Exo irc.jpg|thumb|'''Figure 51.''' IRC of the exo transtion state]]
|[[File:Yll113Exo rms.jpg|thumb|'''Figure 52. '''RMS of the exo transition structure]]
|JSMOL
|[[File:Yll113Endo irc.jpg|thumb|'''Figure 53. '''IRC of the endo transition state]]
|[[File:Yll113Endo rms.jpg|thumb|'''Figure 54.''' RMS of the endo transition state]]
|JSMOL
|}
And eventually, the activation energies of the reaction via different transition structures were summarised in Table 20.
{| class="wikitable"
!
!Cyclohexa-1,3-diene
!Maleic Anhydride
!Endo Transition State
!ExoTransition State
!Activation Energy via endo
!Activation Energy via exo
|-
|Energy/Hartree
|0.02771130
|<nowiki>-0.12182418</nowiki>
|<nowiki>-0.05150469</nowiki>
|<nowiki>-0.05041984</nowiki>
|0.04260819
(26.74 kcal/mol)
|0.04369304
(27.42 kcal/mol)
|}
'''Table 20.''' Activation energy analysis

=== '''Conclusion''' ===

=== '''Reference''' ===

<gallery>
File:
</gallery>

File:Yll113BOAT PART1.LOG

2015-12-17T07:38:44Z

Yll113:

Rep:Mod:hlj298

2015-12-17T07:36:37Z

Yll113: /* Optimisation of transition state */

== '''Study of the reaction profiles of the Cope Rearrangement and the Diels-Alder Cycloadditions''' ==
'''Y. L. J. Lam'''

''Department of Chemistry, South Kensington Campus, Imperial College London, London SW7 2AZ, United Kingdom''

''Received 18 December, 2015''

=== Abstract ===
The reactants, products and transition states of the Cope
Rearrangement of 1,5-hexadiene were investigated by ''GaussView 5.0'' at the '''HF/3-21G''' and '''B3LYP/6-31G*'''levels
of theories respectively. With that, the point groups, vibrational frequencies and different energies at different temperatures of the reactants, products and transition states were calculated. Also, by optimizing the transition structures with different methods, i.e. computing the force constants at the
beginning of the calculations, using the redundant coordinate editor and '''QST2''', at the '''HF/3-21G''' level of theory, closer views of the geometries of the transition states can be observed. Furthermore, by using the '''IRC''' method, the reaction profiles can be
obtained and the activation energies can therefore be calculated. Plus, using '''IRC''' method, all reaction intermediates
can now be observed, which helps us to understand the mechanism of the Cope Rearrangement. Similarly, for Diels-Alder Cycloadditions between ethene and
butadiene and Diels-Alder Cycloadditions between cyclohexa-1,3-diene and maleic anhydride, the reactants, products and transition states were optimized and
their respective energies were calculated at '''AM1 semi-empirical molecular orbital method'''. Furthermore, the symmetries of the
molecular orbitals were visualized and the reaction profiles calculated by '''IRC''' method were obtained.

=== '''Introduction''' ===
Chemical reactions are happening around the world in every second. Some reactions are fast, whilst some are slow. The most common and general reason for that is on the kinetic and thermodynamic aspects. On the kinetic aspect, we might argue that the energy barrier(s) form the reactant(s) to the product(s) is/are huge, and therefore, the reactant(s) cannot overcome the barrier(s) and the reaction is slow or does not proceed. The transformation between crude carbon and diamond is a good example. The energy difference between crude carbon and diamond is just few kcal/mol, however, the energy barrier for the transformation is huge. Hence, the transformation is extremely slow, or even does not proceed. With that, diamond symbolizes eternity. On the other hand, on the thermodynamic aspect, we might argue that the reaction is endothermic, i.e. absorbing/requiring heat from the surroundings in order to proceed. In fact, these two aspects just provide us with a little bit of the story and therefore, chemists, or scientists in general, study the mechanism of the reactions to find out the full story. Unfortunately, some reactions are spontaneous, such as the thiocyanation of the iron complex. Also, some intermediates of the reactions are unstable, which cannot be separated or detected even using very advanced analytical instruments, such as nuclear magnetic resonance (NMR) spectromenter. Therefore, scientists devised some programs and computational methods to find out the mechanism of the reactions. Here we use ''GaussView 5.0'' for our investigation.

==== Computational Theory ====
[[File:Yll113 AM1 and HF.jpg|thumb|463x463px|'''Figure 1.''' HOMO and LUMO (highlighted in yellow) of cis-butadiene under the basis of calculation '''HF/3-21G '''(left) and '''AM1''' (right)]]
In ''GaussView 5.0'', there are numerous methods for calculation, such as '''HF/3-21G''', '''B3LYP/6-31G*''', '''semiempirical AM1''', '''MP4 '''and '''MP2'''. Here, the first two calculation method, namely, '''HF/3-21G''' and '''B3LYP/6-31G* '''were applied for calculation of the Cope Rearrangement Reaction, while '''semiempirical AM1''' was used for the investigation of the two Diels-Alder reactions.

N.B. No matter which method applied, the RMS Gradient Norm in hartress would also be computed. This is a measure of how well does the optimisation go during the calculation of the
structure drawn. The closer to zero, the better the structure is optimised.

===== Hartree-Fock ('''HF''') Method =====
Hartree-Fock theory ('''HF''') is the fundamentals of electronic structure theory. It gives a good starting point for more elaborate theoretical methods which can approximate the electronic Schrödinger equation better. It is the basis of the molecular orbital (MO) theory that assumes the motion of each electron can be described by a single-particle function/orbital and it does not depend on/interact with the instantaneous motions of the other electrons.<ref>C. D. Sherrill, ''An Introduction to Hartree-Fock Molecular Orbital Theory'', 2000</ref>

===== Becke, 3-parameter, Lee-Yeang-Parr ('''B3LYP''') Method =====
Beeke, 3-parameter, Lee-Yang-Parr ('''B3LYP''') is one of the most commonly used hybrid functionals. Hybrid functionals are a class of approximation of the exchange-correlation energy functional in density functional theory.<ref>What is B3LYP?, https://www.quora.com/What-is-B3LYP (accessed December 2015)</ref> '''B3LYP''' contains an '''HF''' exchange with the weight of 0.2, which can be regarded as a uniform screening of
exchange by 80 %.<ref>C. H. Patterson, ''International Journal of Quantum Chemistry'', 2006, '''106 '''(15), 3383</ref> '''B3LYP''' also takes a set of atomization
and ionization energies, proton affinities and total atomic energies into account.<ref>A. D. Becke, ''The Journal of Chemical Physics'', 1993, '''98''', 5648</ref>

===== Semiempirical Austin Model 1 ('''AM1''') =====
Semiempirical Austin Model 1 ('''AM1''') based on the Neglect of Differential Diatomic Overlap (NDDO) integral approximation.<ref>M.
J. S. Dewar, E. G. Zoebisch, E. F. Healy and J. J. P. Stewart, ''J. Am. Chem. Soc.'', 1985, '''107''' (13), 3902</ref> Therefore, when taking the same molecule for '''AM1''' and '''HF/3-21G''', you would find that the numbers of HOMO and LUMO are different, which '''AM1''' gives smaller numbers as shown in Figure 1. This is due to the neglect of the low-lying orbitals overlapping. With that, '''AM1''' proceeds much faster than '''HF/3-21G''' for the sake of time.

==== The Cope Rearrangement ====
The Cope Rearrangement is an organic reaction involving [3,3]-sigmatropic rearrangement of 1,5-dienes, which resembles the Claisen Rearrangement.<ref>A. C. Cope and E. M. Hardy, ''J. Am. Chem. Soc.'', 1940, '''62''' (2), 441</ref> The mechanism of the Rearrangement has sparked a controversy – whether it is concerted, dissociative or stepwise.<ref>O. Wiest, K. A. Black and K. N. Houk, ''J. Am. Chem. Soc.'', 1994, '''116''', 10336</ref> With that, first, each conformer of the reactant, 1,5-hexadiene, was optimised by '''HF/3-21G'''. The lowest energy conformer of 1,5-hexadiene was found. Then, as we know, the Rearrangement undergoes either a chair or boat transition state. So, each transition state was optimised by '''HF/3-21G '''as well. By looking into the energy difference between the transition states and the reactant, the activation energy of the Cope Rearrangement with 1,5-hexadiene was found. In order to find the reaction profile and see how the 1,5-diene rearranges, i.e. the mechanism, the transition state was optimised again with
mainly two methods. The coordinate of the chair transition state was first frozen, with the bond to be made set as 2.20000 Å. 2.20000 Å is a good bond length for partially C-C bond as suggested by the chemists’ observations in the literature. <ref>N. H. Kendall, Y. Li and J. D. Evanseck, ''Angew. Chem. Int. Ed. Engl.'', 1992, '''31''' (6), 682</ref> Then, after the optimization of the frozen coordinate, the partly form 2.20000 Å can be relaxed and the structure was then reoptimised. This methods skips the process of computing the whole force constant matrix i.e. Hessian, which saves time and costs. Furthermore, the boat transition state was optimised by '''QST2'''. '''QST2''' has a higher constrains in which requires a more accurate transition state structure to be put in. In this case, the dihedral angle plays an important role in order to be calculated by ''GaussView'' 5.0. Hence, this method is more expensive and time-consuming. From the optimised transition states, an '''IRC''' can be run for the optimised structure to see the full reaction profile. Also, the intermediates of the reaction can be observed. And finally, the reactant and two transition states
were optimised with '''B3LYP/6-31G*''' similarly. Hence, the two calculation methods can be compared by looking into the numbers obtained. Also, the numbers can be compared against the
experimental values. As explained above, '''B3LYP''' takes a more in-depth consideration, the numbers got from this method should be closer to the reality.

==== The Diels-Alder Cycloaddition ====
The Diels-Alder cycloaddition is a [4+2] cycloaddition between a dienophile and a conjugated alkene to give a cyclohexane system. Here, calculations on two Diels-Alder cycloaddition reactions are reported. They are (1) ethylene and butadiene and (2) cyclohexa-1,3-diene and maleic anhydride.

For Diels-Alder cycloaddition reaction, it is well-known that the reaction gives exo and/or endo product. Exo product implies the reaction pathway is thermodynamically controlled to give more stable product; endo product implies
the reaction pathway is kinetically controlled to give a relatively less stable product. In other words, the activation energy to form the exo product is higher than that of endo, however, the endo product is higher in energy than exo. This can usually be explained by the secondary orbital effects. In our cases, both the exo and endo products were investigated undoubtedly. This time, as you may notice, the molecule is more large in size and there are two reactants instead of just one reactant in the Cope Rearrangement, a simpler method of calculation was implemented, which is '''AM1'''. Also, the electronic distributions and orbitals of the HOMO and LUMO of the transition states were computed and visualised.

=== '''Computational Method''' ===
''All calculations were performed by GaussView 5.0. Relevant JSmol files were uploaded here, however, due to some technique glitches, some bonds, especially double bonds, might not come up properly. Yet, the structures of the molecules are generally correctly shown.''

==== The Cope Rearrangement ====
[[File:Yll113 CR.png|thumb|'''Figure 2.''' The Cope Rearrangement of 1,5-hexadiene]]
An anti and gauche conformation of the 1,5-hexadiene were drawn respectively. The drawn structures were first optimised by a not very accurate technique, i.e. '''Clean'''. Then, the '''clean'''ed structure were optimised by '''HF/3-21G'''. The point group and the energy of each conformer were found and compared to locate the low-energy minima. The optimised structures from '''HF/3-21G''' were then reoptimised by '''B3LYP/6-31G*'''. The point group of each conformer was checked and confirmed. Also, the comparison of the same conformer under different calculation method '''HF/3-21G''' and '''B3LYP/6-31G''' was carried out by looking into energy, bond lengths and bond angles. Furthermore, the '''Thermochemistry''' using job type '''Frequency''' was found in both '''HF/3-21G '''and''' B3LYP/6-31G* '''optimised anti conformers.

The boat and chair transition structures were also drawn and '''clean'''ed. The point group of each transition state was found.

Firstly, the chair transition structure was '''optimised to TS (Berny)''' in '''Opt + Freq '''using the basis of '''HF/3-21G'''. Force constant was calculated '''once'''. The frequency of vibration was checked to make sure there is one imaginary vibrational frequency. Then, '''freeze''' '''coordinate''' of the molecule by freezing the carbon-carbon bond to be made as 2.20000 Å. After that, the frozen coordinate was relaxed so the carbon-carbon bond to be made no longer be restricted to 2.20000 Å. The geometry of the transition state was then compared.

Secondly, at the same time, the boat transition structure was optimised by '''QST2''' method by specifying the reactants and products of the reaction under the basis of '''HF/3-21G'''. Labelling the atoms in
the reactant and product, and adjusting the central '''C-C-C-C '''dihedral angle to 0o plus the two inside '''C-C-C''' angles to 100o, the reactant and product could now be optimised by '''QST2''' method.

Comparing the optimised chair and boat transition structures, the connecting conformer of 1,5-hexadiene was found. The reaction energy profile was then calculated by '''IRC''' under '''HF/3-21G''' with 50 points and force constant as always for every small steps. With that, the mechanism of the reaction, as well as the whole reaction energy profile, could be observed clearly. Take the last point on the '''IRC''' and run a normal '''optimisation''' under '''HF/3-21G''' to obtain a minimized geometry.

Eventually, re'''optimise''' the structures of the two transition states with '''Opt + Freq '''under the basis of '''B3LYP/6-31G*'''. The geometries and energies of the transition structure under two different basis were compared. With that, these computed values were also compared against experimental values.

==== The Diels-Alder Cycloadditions ====
[[File:Yll113DA1.jpg|thumb|'''Figure 3. '''The Diels-Alder Cycloadditions between ethylene and butadiene]]

===== Ethylene and butadiene =====
The structure of cis-butadiene was first optimised under the basis of '''semiempirical AM1'''. The HOMO and LUMO of cis butadiene were visualised and its symmetry was determined.

The transition state of the reaction was drawn and optimised under the basis of '''semiempirical AM1'''. The '''IRC''' was carried out in order to confirm the transition structure postulated is of the correct one. Furthermore, the HOMO of the transition structure was visualised and the nodal
planes and properties of the system were interpreted.

===== Cyclohexa-1,3-diene and maleic anhydride =====
[[File:Yll113DA2.jpg|thumb|'''Figure 4. '''The Diels-Alder Cycloadditions between cyclohexa-1,3-diene and maleic anhydride]]
The transition states of the exo and endo products were drawn and optimised under the basis of '''semiempirical AM1'''. The '''IRC''' was carried out in order to confirm the transition structure postulated is of the correct one. For the number of points, 21 points were used for exo transition states and 24 for endo. This is because the energy was too shallow and the slopes tend to zero after the number of points specified above and ''GaussView 5.0'' cannot predict which direction should it goes on to calculate. Furthermore,
the bond lengths, orientation and the HOMO of the transition structures were investigated.

=== '''Results and Discussion''' ===

==== The Cope Rearrangement ====

===== Optimisation of Reactant =====
1,5-hexadiene has three free rotating carbon-carbon bonds. Each of them has three rotational minima. This gives 27 conformations of the 1,5-hexadiene molecule. Yet, only ten of them were energetically distinct due to symmetry and enantiomeric relationships.<ref>B. G. Rocque, J. M. Gonzales and H. F. Schaefer, ''Molecular Physics'', 2002, '''100''' (4), 441</ref> Two of them, the ''Ci anti ''and ''C1 gauche ''structure in here'' ''were drawn and optimizied as shown in Figure A and B and their energies were calculated as shown in Table 1.
{| class="wikitable"
!Conformation
!Molecule
!Point Group
!Energy/ Hartrees*
HF/3-21G
!RMS Gradient Norm/Hartrees*
HF/3-21G
!Relative Energy#/ kcal/mol
!Newman Projections
|-
|Gauche3
|<jmol>
<jmolApplet>
<title>Figure A: Gauge3 conformer of 1,5-hexadiene</title><color>white</color>
<size>250</size>
<uploadedFileContents> yll113CR_GAUGE_PART1.LOG </uploadedFileContents>
</jmolApplet>
</jmol>
|C1
|<nowiki>-231.69266120</nowiki>
|0.00001556
|0.00
|[[File:Yll113 torsion gauche.jpg|centre|frame|'''Figure 5.''' Newman Projection of C1 gauche3 1,5-hexadiene]]
|-
|Anti2
|<jmol>
<jmolApplet>
<title>Figure B: Anti2 conformer of 1,5-hexadiene</title><color>white</color>
<size>250</size>
<uploadedFileContents>YLL113CR ANTI PART1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|Ci
|<nowiki>-231.69253528</nowiki>
|0.00001891
|0.08
|[[File:Yll113 torsion anti.jpg|centre|frame|'''Figure 6.''' Newman Projection of Ci anti2 1,5-hexadiene]]
|}
<nowiki>*</nowiki>1 hartree = 627.509 kcal/mol

#The difference in energy between the conformer and the lowest energy conformer, in here, which is Gauche3. Then convert Hartree to kcal/mol by *

'''Table 1. '''Conformational analysis of anti2 and gauche3 of 1,5-hexadiene

As shown in Table 1, the energy of Gauche3 is surprisingly lower than the anti2 conformation of 1,5-hexadiene. In most cases, the antiperiplanar conformation of a molecule, such as anti2, is more favourable as it has the least steric clashes. Therefore, usually the antiperiplanar conformation is of the lowest energy. However, here, apart from sterics, the stereoelectroncs concept has also been taken into account. The vinyl proton, in a through space manner, can interact with the π or π* orbital on the sp2 carbon which is separated by four bonds from it.<ref>M. Nishio and M. Hirota, ''Tetrahedron'', 1989, '''45 '''(23), 7201</ref> This is so-called CH-π interaction. The Newman Projection in Figure 5 gives us a closer look on how they are close in space and interact; and the Newman projection in Figure 6 tells us why the vinyl proton cannot interact with the π or π* system through space. Therefore, the gauche3 conformation is more stable than anti2 and of lower energy in 1,5-hexadiene.

Focusing on anti2 conformer of the 1,5-hexadiene, the '''HF/3-21G''' optimised molecule was reoptimised by '''B3LYP/6-31G'''. The energies and the geometries of the anti2 1,5-hexadiene under two basis of calculation method were compared and shown in Table 2.
[[File:Yll113Anti2.png|thumb|'''Figure 7. '''Anti2 conformer of 1,5-hexadiene with labelled atoms]]
{| class="wikitable"
! rowspan="2" |Calculation Method
! rowspan="2" |Point Group
! rowspan="2" |Energy/ Hartrees
! rowspan="2" |RMS Gradient Norm/Hartrees
!Bond length between/ Å
!Bond length between/ Å
!Bond length between/ Å
!Bond angle between
!Bond angle between
|-
|C14-C12 and C4-C1
|C12-C9 and C6-C4
|C6-C9
|C14-C12-C9 and
C1-C4-C6
|C12-C9-C6 and
C4-C6-C9
|-

|'''HF/3-21G'''
|Ci
|<nowiki>-231.69253528</nowiki>
|0.00001891
|1.31613
|1.50891
|1.55275
|124.80579o
|111.34878o
|-
|'''B3LYP/6-31G*'''
|Ci
|<nowiki>-234.61171063</nowiki>
|0.00001249
|1.33350
|1.50419
|1.54816
|125.29968o
|112.67081o
|}
'''Table 2. '''Comparison with the anti2 conformer of 1,5-hexadiene with two calculation method basis ('''HF/3-21G '''and '''B3LYP/6-31G*''')

As seen in Table 2, the point group of the same conformer does not change no matter which method of calculation was carried out. The energy of the conformer with '''B3LYP/6-31G*''' is lower than that of '''HF/3-21G'''. Also, the bond lengths and bond angles changed with a certain amount as shown.

Furthermore, by submitting job type as '''Frequency''' in the optimised structure of anti2 conformer of 1,5-hexadiene under both calculation basis '''HF/3-21G '''and '''B3LYP/6-31G*''', the electronic energy, sum of electronic and zero-point energies at 0 K and sum of electronic and thermal energies at 298.15 K could be determined as shown in Table 3.

{| border="1" cellspacing="1" cellpadding="10"
|-
|rowspan="3"| ''' '''
!colspan="4" |'''HF/3-21G a'''
! colspan="4" |'''B3LYP/6-31G* b'''
|-
! rowspan="2" width="125" align="center" | '''Electronic energy'''
! width="125" align="center" | '''Sum of electronic and zero-point energies'''
! width="125" align="center" | '''Sum of electronic and thermal energies'''
! rowspan="2" width="125" align="center" | '''IR spectrum'''
! rowspan="2" width="125" align="center" | '''Electronic energy'''
! width="125" align="center" | '''Sum of electronic and zero-point energies'''
! width="125" align="center" | '''Sum of electronic and thermal energies'''
! rowspan="2" width="125" align="center" | '''IR spectrum'''
|-
! width="125" align="center" | '''at 0 K'''
! width="125" align="center" | '''at 298.15 K'''
! width="125" align="center" | '''at 0 K'''
! width="125" align="center" | '''at 298.15 K'''
|-
| '''Reactant (anti2)'''
| align="center" | -231.692534
| align="center" |-231.539539
| align="center" | -231.532565
|[[File:Yll113ANTI3-21IR.png|thumb|'''Figure 8. '''IR spectrum of anti2 reactant 1,5-hexadiene calculated by '''HF/3-21G''']]
| align="center" | -234.611703
| align="center" | -234.469213
| align="center" | -234.461857
|[[File:Yll113ANTI6-31IR.png|thumb|'''Figure 9. '''IR spectrum of anti2 reactant 1,5-hexadiene calculated by '''B3LYP/6-31G*''']]
|}

a [https://wiki.ch.ic.ac.uk/wiki/images/5/52/Yll113CR_ANTI_PART4.LOG File]; b [https://wiki.ch.ic.ac.uk/wiki/images/5/54/Yll113_CR_ANTI_PART3.LOG File]

'''Table 3.''' Summary of energies (in hartree) of the reactant (anti2) Comparing Figure 8 and 9, both IR spectra are similar which shows that they have similar stretching/bending mode even under different methods of calculations. However, the strongest peak in Figure 8 is at 1112 cm-1 and that in Figure 9 is 940 cm-1. Yet, they correspond to the same mode of vibration, which is the =C-H bending. Therefore, according to the equation, the wavenumber of absorbance, ν can be calculated:

:<math>\nu = \frac{1}{2 \pi c} \sqrt{\frac{k}{\mu}}</math>

where ''μ'' is the reduced mass, ''k'' is the spring constant for the bond and ''c'' is the light speed,

we can therefore know that the spring constant of the molecule under '''B3LYP/6-31G* '''basis reduced.

Now, focusing on gauche3 conformer of the 1,5-hexadiene, similarly, the '''HF/3-21G''' optimised molecule was reoptimised by '''B3LYP/6-31G'''. The energies and the geometries of the gauche3 1,5-hexadiene under two basis of calculation method were compared and shown in Table 4.
[[File:Yll113Gauche3.png|thumb|'''Figure 10. '''Gauche3 conformer of 1,5-hexadiene with labelled atoms]]
{| class="wikitable"
! rowspan="2" |Calculation Method
! rowspan="2" |Point Group
! rowspan="2" |Energy/ Hartrees
! rowspan="2" |RMS Gradient Norm/Hartrees
! colspan="3" |Bond length between/ Å
! colspan="2" |Bond angle between
|-
|C14-C12 and C4-C1
|C12-C9 and C6-C4
|C6-C9
|C14-C12-C9 and
C1-C4-C6
|C12-C9-C6 and
C4-C6-C9
|-

|'''HF/3-21G'''
|C1
|<nowiki>-231.69266120</nowiki>
|0.00001555
|1.31646
|1.50929
|1.55314
|125.02428o
|111.80728o
|-
|'''B3LYP/6-31G*'''
|C1
|<nowiki>-234.61132605</nowiki>
|0.00000360
|1.33382
|1.50491
|1.55007
|125.49464o
|113.46225o
|}
'''Table 4. '''Comparison with the anti2 conformer of 1,5-hexadiene with two calculation method basis ('''HF/3-21G '''and '''B3LYP/6-31G*''')

As seen in Table 4, the point group of the same conformer, again, does not change no matter which method of calculation was carried out. The energy of the conformer with '''B3LYP/6-31G*''' is lower than that of '''HF/3-21G'''. Also, the bond lengths and bond angles changed with a certain amount as shown.

Furthermore, by submitting job type as '''Frequency''' in the optimised structure of gauche3 conformer of 1,5-hexadiene under both calculation basis '''HF/3-21G '''and '''B3LYP/6-31G*''', the electronic energy, sum of electronic and zero-point energies at 0 K and sum of electronic and thermal energies at 298.15 K could be determined as shown in Table 5.

{| border="1" cellspacing="1" cellpadding="10"
|-
| rowspan="3" | ''' '''
! colspan="4" |'''HF/3-21G c'''
! colspan="4" |'''B3LYP/6-31G* d'''
|-
! rowspan="2" width="125" align="center" | '''Electronic energy'''
! width="125" align="center" | '''Sum of electronic and zero-point energies'''
! width="125" align="center" | '''Sum of electronic and thermal energies'''
! rowspan="2" |IR spectrum
! rowspan="2" width="125" align="center" | '''Electronic energy'''
! width="125" align="center" | '''Sum of electronic and zero-point energies'''
! width="125" align="center" | '''Sum of electronic and thermal energies'''
! rowspan="2" |IR spectrum
|-
! width="125" align="center" | '''at 0 K'''
! width="125" align="center" | '''at 298.15 K'''
! width="125" align="center" | '''at 0 K'''
! width="125" align="center" | '''at 298.15 K'''
|-
| '''Reactant (Gauche 3)'''
| align="center" |-231.692692
| align="center" |-231.539486
| align="center" |-231.532646
|[[File:Yll113GAUCHE3-21IR.png|thumb|'''Figure 11. '''IR spectrum of gauche3 conformer of 1,5-hexadiene calculated by '''HF/3-21G''']]
| align="center" | -234.611326
| align="center" |-234.468719
| align="center" |-234.461477
|[[File:Yll113GAUCHE6-31IR.png|thumb|'''Figure 12.''' IR spectrum of gauche3 conformer of 1,5-hexadiene calculated by '''B3LYP/6-31G*''']]
|}

c [https://wiki.ch.ic.ac.uk/wiki/images/9/98/Yll113CR_GAUGE_PART4.LOG File] ; d [https://wiki.ch.ic.ac.uk/wiki/images/c/ca/Yll113CR_GAUGE_PART3.LOG File]

'''Table 5.''' Summary of energies (in hartree) of the reactant (Gauche3) Comparing Figure 11 and 12, both IR spectra are similar which shows that they have similar stretching/bending mode even under different methods of calculations. However, the strongest peak in Figure 12 is at 939 cm-1 and that in Figure 11 is 1111 cm-1. Yet, they correspond to the same mode of vibration, which is also the =C-H bending. Therefore, similar to the anti2 conformer's case as mentioned above, we can therefore know that the spring constant of the molecule under '''B3LYP/6-31G* '''basis reduced.

===== Optimisation of transition state =====

After optimising the reactants, the chair and boat transition states were optimised accordingly using mainly two different methods. But before that, an allyl fragment (CH2CHCH2) was optimised using '''HF/3-21G''' level of theory for the sake of convenience in constructing the chair and boat transition states. A brief summary was shown in Table 6.
{| class="wikitable"
!
! colspan="2" |Molecule
!Point Group
!Energy/Hartree
!RMS Gradient Norm/ Hartrees
|-
|Allyl fragment
CH2CHCH2
|<jmol>
<jmolApplet>
<title>Figure C: Allyl Fragment</title><color>white</color>
<size>250</size>
<uploadedFileContents> Yll113CR TS 1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|[[File:Yll11313.jpg|thumb|'''Figure 13. '''Optimised Structure of the allyl fragment]]
|C2v
|<nowiki>-115.82304010</nowiki>
|0.00002945
|}
'''Table 6. '''Summary of the optimised allyl fragment

Then, both chair and boat transition state were drawn and optimised using the '''optimisation to TS (Berny)''' under '''HF/3-21G'''. Figure 14 and Figure C show the optimized structure of the chair transition state while Figure 15 and Figure D show the optimized structure of the boat transition state. Table 7 shows the summary of results.
{| class="wikitable"
! rowspan="2" |
! colspan="2" rowspan="2" |Molecule
! rowspan="2" |Point Group
! rowspan="2" |Energy/ Hartree
! rowspan="2" |RMS Gradient Norm/Hartrees
! colspan="6" |Partial C-C bond length between/ Å
|-
!C14-C6
!C11-C3
!C14-C9
!C6-C1
!C9-C11
!C1-C3
|-
|Optimised Chair Transition State
|<jmol>
<jmolApplet>
<title>Figure D: Optimised Chair transition state</title><color>white</color>
<size>250</size>
<uploadedFileContents>Yll113CHAIR PART1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|[[File:Yll113CHAIR3-21.png|thumb|'''Figure 14. '''Optimised Chair Structure by '''HF/3-21G''' with atoms labelled ]]
|C2h
|<nowiki>-231.61932238</nowiki>
|0.00002645
|2.02016
|2.02016
|1.38929
|1.38930
|1.38930
|1.38929
|-
|Optimised Boat Transition State
|JSMOL
|[[File:Yll11315.jpg|thumb|'''Figure 15. '''Optimised Boat Structure by '''HF/3-21G''' with atoms labelled]]
|C2v
|<nowiki>-231.60280235</nowiki>
|0.00003872
|2.14060
|2.14060
|1.38138
|1.38138
|1.38138
|1.38138
|}

'''Table 7. '''Summary of the optimised chair and boat transition states by '''optimisation to TS (Berny) '''under '''HF/3-21G''' basis

Furthermore, the transition structures’ '''Frequencies''' were calculated as shown in Table 8.
{| class="wikitable"
!
!Molecule
!IR spectrum
!Imaginary Frequency/ cm-1
|-
|Boat Transition State
|
|[[File:Yll11317.jpg|thumb|'''Figure 16. '''IR spectrum of the optimised Boat Structure by '''HF/3-21G''']]
|<nowiki>-840</nowiki>
|-
|Chair Transition State
|[[File: Yll113Chair Part 1 imag freq.gif]]
|[[File:Yll11316.jpg|thumb|'''Figure 17. '''IR spectrum of the optimised Chair Structure by '''HF/3-21G''']]
|<nowiki>-818</nowiki>
|}
'''Table 8.''' IR spectra and imaginary frequencies of the boat and chair transition states

As you may notice that, the
imaginary frequency comes up when calculating with the transition states. This
is common, in other words, this should appear to let us know the transition
structure we postulated is correct.

A transition state is the first
order saddle point on the potential energy surface. Therefore, the force
applied to the saddle point against to the displacement. As force and
displacement are vectors, the force constant will be a negative number.Therefore, according to
:<math>\nu = \frac{1}{2 \pi c} \sqrt{\frac{k}{\mu}}</math>

where ''μ'' is the reduced mass, ''k'' is the spring constant for the bond and ''c'' is the light speed,

the square root of a negative
force constant k gives an imaginary wave number/frequency v. In other words,
the appearance of an imaginary frequency tells us that the structure is a
saddle point of the reaction pathway.

The chair transition state
was followed by first 'frozen' then 'relaxed'. The boat transition structure
was followed by '''QST2''' calculation method.

====== Chair Transition State ======
After the above '''optimisation''', the chair transition
state was reoptimised again with another method. This method first freezes the
coordinate of the molecule, in this case, freeze the bond to be made in the
Cope Rearrangement of 1,5-hexadiene as 2.20000 Å. The molecule then optimised with the frozen
coordinate. Details of this optimisation was summarized in Table 9.
{| class="wikitable"
! rowspan="2" |
! colspan="2" rowspan="2" |Molecule
! rowspan="2" |Point Group
! rowspan="2" |
{| class="MsoTableGrid"

|
<nowiki> </nowiki>Energy/ Hartree

|}
! rowspan="2" |RMS Gradient Norm/ Hartree
! colspan="3" |Bond length between/ Å
! rowspan="2" |Imaginary Frequency/ cm-1
! rowspan="2" |IR Spectrum
|-
!C6-C14 and C3-11
!C1-C3 and C9-C14
!C1-C6 and C9-C11
|-
|Optimised Chair Transition Structure with frozen coordinate
|JSMOL
|[[File:Yll11318.jpg|thumb|'''Figure 18. '''The optimised chair transition structure with frozen coordinate and atoms labelling]]
|
{| class="MsoTableGrid"

|
<nowiki> </nowiki>C2h

|}
|<nowiki>-231.61518510</nowiki>
|0.00325573
|2.20000
|1.38135
|1.38128
|<nowiki>-765</nowiki>
|[[File:Yll11319.jpg|thumb|'''Figure 19. '''IR spectrum of the optimised chair transition structure with frozen coordinate]]
|}

'''Table 9. '''Summary of the optimisation of the chair transition structure with
frozen coordinate(s)

From Table 9, we may notice
that the RMS Gradient Norm value is quite far off from zero. Also, the
imaginary frequency becomes much higher than -818 cm-1 (Shown in
Table 8). With these two pieces of information, we can deduce that the frozen
coordinate(s) affect(s) the force constant of the transition state which does
not give a good optimisation of transition structure. With that, after applying
the frozen coordinate to the molecule, the molecule was reoptimised again with
a degree of '''Derivative '''to the '''Bond'''. Details of the reoptimisation
were presented in Table 10.
{| class="wikitable"
! rowspan="2" |
! colspan="2" rowspan="2" |Molecule
! rowspan="2" |Point Group
! rowspan="2" |Energy/ Hartree
! rowspan="2" |RMS Gradient Norm/ Hartree
! colspan="4" |Bond length between/ Å
! rowspan="2" |Imaginary Frequency/ cm-1
! rowspan="2" |
{| class="MsoTableGrid"

|
<nowiki> </nowiki>IR Spectrum

|}
|-
!C14-C6
!C11-C3
!C14-C9 and C6-C1
!C9-C11 and C1-C3
|-
|Optimised Chair Transition
Structure with a degree of '''Derivative'''
to the '''Bond'''
|JSMOL
|[[File:Yll11320.jpg|thumb|'''Figure 20. '''The optimised chair transition structure with a degree of '''Derivative''' to the '''Bond''' and atoms labelling]]
|C2h
|<nowiki>-231.61932233</nowiki>
|0.00002127
|2.02075
|2.02071
|1.38929
|1.38930
|<nowiki>-818</nowiki>
|[[File:Yll11321.jpg|thumb|'''Figure 21. '''IR spectrum of the optimised chair transition structure with a degree of '''Derivative''' to the '''Bond''']]
|}
'''Table 10. '''Summary of the reoptimisation of the chair transition
structure with a degree of '''Derivative''' to the '''Bond'''

Now, in Table 10, the RMS
Gradient Norm value is close to zero. Also, the imaginary frequency goes back
to -818 cm-1, indicating that the coordinates no longer be frozen
and the stretching/bending mode of the transition state is able to undergo
freely.

Comparing the bond lengths
in Table 7 and 10, we can see that the difference between bond lengths of the
single bond to be made/ broken calculated in two methods is just less than
0.0006 Å. And also, there is no difference in bond length of the double bond to be make/broken ‘inside’ the system. This tells us that the two optimisation
methods are rather similar under the consideration on the Cope Rearrangement
Reaction.

====== Boat Transition State ======
Instead of using the frozen
coordinate method as for the chair transition state above, another method, '''QST2''', was applied to the boat
transition state under the '''HF/3-21G'''
basis. In order to use this method, without any ‘Link died’, the reactant and
product have to be drawn and labelled carefully. '''QST2''' is a method which interpolates the reactant and product to
give a transition state. Therefore, it will fall if the structure of the
reactant and product are not close to the transition state. And therefore, all
molecules have to be carefully labelled and adjusted.

[[File:Yll11322.png|500px|center]]

'''Figure 22. '''The drawings and adjustments of angles of the reactant (left)
and product (right) for '''QST2''' Method,
i.e. the central C-C-C-C dihedral angle was changed to 0o and inside
C-C-C were reduced to 100o

After the adjustment, the job was run and the optimized molecule converge to the boat transition structure. Summary was shown in Table 11.
{| class="wikitable"
! rowspan="2" |
! colspan="2" rowspan="2" |Molecule
! rowspan="2" |Point Group
! rowspan="2" |Energy/Hartree
! rowspan="2" |RMS Gradient Norm/ Hartree
! colspan="3" |Bond length between/ Å
! rowspan="2" |Imaginary Frequency/ cm-1
! rowspan="2" |IR spectrum
|-
!C1-C6
!C3-C4
!C5-C6, C4-C5, C3-C2 and C1-C2
|-
|Boat transition structure
under '''QST2''' method
|JSMOL
|[[File:Yll11323.jpg|thumb|'''Figure 23. '''The optimised Boat transition structure with atom labelling]]
|C2v
|<nowiki>-231.60280241</nowiki>
|0.00002436
|2.13994
|2.14019
|1.38149
|<nowiki>-840</nowiki>
|[[File:Yll11324.jpg|thumb|'''Figure 24. '''IR spectrum of the optimised boat transition structure]]
|}
'''Table 11. '''Summary of the boat transition structure under '''QST2 '''method

====== Intrinsic Reaction Coordinate''' '''('''IRC''') ======
In order to confirm that our transition state is of the
correct one, '''Intrinsic Reaction
Coordinate '''('''IRC''') will be carried
out.

As mentioned above, transition state is the first order
saddle point of the reaction pathway. Therefore, it will start to go to the
product or back to the reactant with it falls off. It resembles that a ball is
at the tip of the mountain, which starts to roll off the mountain on the side
with the steepest slope. Also, when we are doing '''IRC''', we can determine whether the reaction goes forward, backward
or both sides. Also, the number of points, which means the number of little
steps that the geometry of the molecule changes, can be adjusted. A low number
of points will just give us a very rough idea that tell us a little bit about
our transition state. Also, the last point on the '''IRC''' is far from the minimum geometry. A high number of points gives
us more accurate results, however two problems could be raised. First, the time
for calculation will be long and Most importantly, as it goes down the slope
and reaches the minimum geometry, i.e. the plateau of energy, the slope will
become very small or even zero again. However, as the energy difference of the
next or previous geometry compared to the geometry of itself is too small, ''GaussView 5.0'' may not able to know which
direction the molecule should proceed to. And this, therefore, results in ‘Link
died’. Therefore, the most common technique is to have a good number of points,
then take the last point on the IRC and run it with a normal optimisation.

Here, as we know that the
Cope Rearrangement has a symmetric reaction pathway, taking the chair
transition structure, we will run '''IRC'''
on it with 50 points.
{| class="wikitable"
![[File:Yll113hlj29825.jpg|thumb|'''Figure 25. '''Total energy along '''IRC '''of the chair transition structure]]
![[File:Yll11326.jpg|thumb|'''Figure 26. '''RMS Gradient Norm of '''IRC '''of the chair transition structure]]
!JSMOL
|}

[[File:Yll11327.jpg|500px]]

'''Figure 27. '''The product of the Cope Rearrangement after optimisation

The first point on Figure 25 is -231.61932233 Hartree and the last point is -231.69157881 Hartree. Then, we take the last point and optimise it, we get the structure shown in Figure 27.

The structure is of C2
symmetry and the energy calculated is -231.69166702 Hartree. This matches with
Gauche2 C2 on Appendix 1. And therefore, this is how the conformer
of 1,5-hexadiene connects with the chair transition structure.

====== Activation Energy of the Cope Rearrangement ======
Finally, we optimise the chair and boat transition states we got from above, reoptimise it with job Opt + Freq
under a more advanced calculation '''B3LYP/6-31G*'''. And from that, the thermochemistry data were given and we can know the
activation energy of the reaction by comparing to Table 3, which anti2 is used
as a local minimum rather than gauche3 as a global minimum.
{| border="1" cellspacing="1" cellpadding="10"
|-
| rowspan="3" | ''' '''
! colspan="3" |'''HF/3-21G'''
! colspan="3" |'''B3LYP/6-31G*'''
|-
| rowspan="2" width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| rowspan="2" width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619323
| align="center" | -231.466698
| align="center" | -231.461339
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409009
|-
| '''Boat TS'''
| align="center" | -231.602803
| align="center" | -231.450932
| align="center" | -231.445303
| align="center" | -234.543094
| align="center" | -234.402342
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692534
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.611703
| align="center" | -234.469213
| align="center" | -234.461857
|}
'''Table 11'''. Summary of energies of chair, boat and reactant (anti2) structure
{| border="1" cellspacing="1" cellpadding="10"
|-
| rowspan="2" | ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.69
| align="center" | 34.06
| align="center" | 33.16
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.96
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
'''Table 12'''. Summary of activation energies in kcal/mol

==== The Diels-Alder Cycloadditions ====

===== Ethylene and Cis-Butadiene =====
First, the structures of the reactants, ethylene and cis-butadiene, were '''optimised to minimum''' under '''semiempirical AM1''' method. For the butadiene, in order to be in the cis conformer, the dihedral angle was adjusted to be 0o. Details are listed in Table 13.
{| class="wikitable"
!
!Molecule
!Point Group
!HOMO
!LUMO
!Energy/Hartree
!RMS Gradient Norm/ Hartree
!IR spectrum
|-
|Ethylene
|JSMOL
|D2h
|[[File:Yll11331.jpg|thumb|'''Figure 31. '''HOMO of Ethylene]]
|[[File:Yll11330.jpg|thumb|'''Figure 30.''' LUMO of ethylene]]
|0.02619029
|0.00008755
|[[File:Yll11328.jpg|thumb|'''Figure 28. '''IR spectrum of Ethylene]]
|-
|Cis-Butadiene
|JSMOL
|C2v
|[[File:Yll11332.jpg|thumb|'''Figure 32. '''HOMO of cis-butadiene]]
|[[File:Yll11333.jpg|thumb|'''Figure 33. '''LUMO of cis-butadiene]]
|0.04878534
|0.00000087
|[[File:Yll11329.jpg|thumb|'''Figure 29.''' IR spectrum of cis-butadiene]]
|}
'''Table 13.''' Summary of optimised ethylene and cis-butadiene

Looking into Figure 30-33, as we know that the plane is perpendicular to the molecule, the HOMO of Ethylene is symmetric while that of LUMO is antisymmetric.

Also, the HOMO of cis-butadiene is antisymmetric and that of LUMO is symmetric.

Then, the transition state of the reaction was able to constructed using the optimised structure of the reactants made above. By making the bond lengths of the bonds to be made in Diels-Alder [4+2] cycloaddition to 2.20000 Å in the transition state, which is the 'dash bond', the transition structure was optimized to '''Opt + Freq minimisation to''' '''TS (Berny)''' under the same method of calculation - '''semiempitical AM1'''. Details are listed in Table 14.
{| class="wikitable"
!
! colspan="2" |Molecule
!Point Group
!HOMO
!LUMO
!Energy/Hartree
!RMS Gradient Norm/ Hartree
!Imaginary Frequency/ cm-1
!IR spectrum
|-
|Transition state
|JSMOL
|[[File:Yll11334.jpg|thumb|'''Figure 34. '''Transition state of ethylene + cis-butadiene]]
|C1
|[[File:Yll11336.jpg|thumb|'''Figure 36. '''HOMO of transition state]]
|[[File:Yll11337.jpg|thumb|'''Figure 37. '''LUMO of transition state]]
|0.11165467
|0.00002792
|<nowiki>-956</nowiki>
|[[File:Yll11335.jpg|thumb|'''Figure 35. '''IR spectrum of the transition state]]
|}
'''Table 14.''' Summary of optimised transition state

From Figure 36, we can see that the HOMO of the transition state is antisymmetric whilst the LUMO of the transition state is symmetric. By making very careful comparison between Figure 36, Figure 37 and Figure 30-33, we can see that the HOMO of the transition state in Figure 36 is a combination of Figure 32 and 30; the LUMO of the transition state in Figure 37 is a combination of Figure 31 and 33. We can clearly see that the HOMO and LUMO of the transition state have a complementary combination of HOMO and LUMO of the reactants.

Taking a closer look to HOMO of the transition state. Recalling Woodward Hoffmann’s Rule, (4q+2)s+(4r)a = odd for thermally allowed reaction, we have both π2s and π4s. Therefore, the reaction is thermally allowed by letting q = 0, which gives the value of 1 which is odd.

Furthermore, from Table 14, we notice that there is an imaginary frequency reported at -956 cm-1. As explained above, the transition state should have one imaginary frequency to account for the negative force constant. With that, this imaginary frequency confirms that the transition structure we postulated from the optimised reactants is valid, i.e. it is really a transition state. The animation of where the imaginary frequency originates from, which shows the motion of the transition state - how the two reactants approach to each other and bonds are formed, is shown below.

JSMOL

From the above figure, we can see that the bond formation from the reactant to the product happens at the same time, i.e. synchronous, on both sides of the transition structure. Therefore, we can say that this Diels-Alder cycloaddition is a concerted [4+2] pericyclic cycloaddition, which matches with what we learnt in Pericyclic Reaction course.

On top of that, the geometry of the transition structure was investigated by looking into the optimised bond lengths between carbon atoms Details are shown in Figure 38 and Table 15.[[File:Yll11338.jpg|thumb|'''Figure 38. '''Transition state of ethylene + cis-butadiene with atoms labelled]]
{| class="wikitable"
!
!Bond length/ Å
|-
|C7-C9
|2.11938
|-
|C12-C5
|2.11944
|-
|C12-C9
|1.38284
|-
|C3-C7
|1.38187
|-
|C3-C1
|1.39750
|-
|C5-C1
|1.38175
|}
'''Table 15. '''Geometry analysis of the transition state

According to the literature <ref>M. A. Fox and J. K. Whitesell, ''Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen'', Springer, 1995</ref>, C-C carbon-carbon single bond is 1.54 Å, and C=C carbon-carbon double bond is 1.34 Å. Also, the Van der Waals radius of carbon is 1.70 Å,<ref>A. Bondi,(1964), ''J. Phys. Chem.'', 1964, '''68''' (3), 441</ref>
According to the reaction scheme shown in Figure 3, a single bond is forming between C7 and C9, also another single bond is forming between C12-C5. Comparing the data in Table 15 with the literature, we can see that the bond length of two bonds to be made is longer than C-C, but shorter than the twice of carbon's Van der Waals radius. This tells us some hints that the terminal carbons between two reactants are attracting with each other to a certain degree. Furthermore, taking a look to other carbon-carbon bonds within the system, we notice that they are somewhere between C-C and C=C. This tells us that either C-C is becoming C=C or C=C is becoming C-C in the transition state.
After that, the '''IRC''' of the above optimised transition state was carried out with both direction and force constant calculated always for 50 points to see the reaction profile.
{| class="wikitable"
|[[File:Yll11339.jpg|thumb|'''Figure 39.''' IRC of the transition state of ethylene + cis-butadiene]]
|[[File:Yll11340.jpg|thumb|'''Figure 40. '''RMS Gradient Norm of transition state of ethylene + cis-butadiene]]
|JSMOL
|}
In Figure 39, we can clearly see that the reactants was first passed through the energy barrier to get the transition state and it went down the slope to give the product.
Finally, the activation energy for this reaction was calculated in Table 16.
{| class="wikitable"
!
!Ethylene
!Cis-butadiene
!Transition state
!Activation Energy
|-
|Energy/Hartree
|0.02619029
|0.04878534
|0.11165467
|0.03667904
(23.02 kcal/mol)
|}
'''Table 16. '''Activation energy analysis of Diels-Alder Reaction between ethylene and cis-butadiene
===== Cyclohexa-1,3-diene and Maleic Anhydride =====
First, the structures of
the reactants, ethylene and cis-butadiene, were '''optimised to minimum''' under '''semiempirical AM1''' method. Details are listed in Table 17.
{| class="wikitable"
!Optimised Structure
!Molecule
!Point Group
!Energy/Hartree
|-
|Cyclohexa-1,3-diene
|[[File:Yll113Cycloopt.jpg|thumb|'''Figure 41. '''Optimised structure of cyclohexa-1,3-diene]]
|C2
|0.02771130
|-
|Maleic Anhydride
|[[File:Yll113Maopt.jpg|thumb|'''Figure 42. '''Optimised structure of Maleic Anhydride]]
|Cs
|<nowiki>-0.12182418</nowiki>
|}
'''Table 17. '''Summary
of the optimized reactant

Then, the two transition states, exo and endo, of the reaction were able to construct. By making the bond lengths of the bonds to be made in Diels-Alder [4+2] cycloaddition to 2.20000 Å in the transition state, which is the 'dash bond', the transition structure was optimized to '''Opt + Freq minimisation to''' '''TS (Berny)''' under the same method of calculation - '''semiempitical AM1'''. Details are listed in Table 18.
{| class="wikitable"
!Transition Structure
!Molecule
!Point Group
!HOMO
!LUMO
!Energy/Hartree
!'''RMS Gradient Norm/ Hartree'''
!'''Imaginary Frequency/ cm-1'''
!IR spectrum
|-
|Exo
|[[File:Yll113Exots.jpg|thumb|'''Figure 43.''' Optimised Structure of Exo Transition State]]
|C1
|[[File:Yll113Exo homo.jpg|thumb|'''Figure 45. '''HOMO of Exo Transition State]]
|[[File:Yll113Exo lumo.jpg|thumb|'''Figure 47.''' LUMO of Exo Transition State]]
|<nowiki>-0.05041984</nowiki>
|0.00000883
|<nowiki>-812</nowiki>
|[[File:Yll113Exo ir.jpg|thumb|'''Figure 49. '''Exo Transition State IR spectrum]]
|-
|Endo
|[[File:Yll113Endots.jpg|thumb|'''Figure 44. '''Optimised Structure of Enfo Transition State]]
|C1
|[[File:Yll113Endo homo.jpg|thumb|'''Figure 46'''. HOMO of Endo Transition State]]
|[[File:Yll113Endo lumo.jpg|thumb|'''Figure 48.''' LUMO of Endo Transition States]]
|<nowiki>-0.05150469</nowiki>
|0.00004308
|<nowiki>-807</nowiki>
|[[File:Yll113Endo ir.jpg|thumb|'''Figure 50.''' Endo Transition State IR Spectrum]]
|}
'''Table 18. '''Summary of the optimized transition structures

In Table 18, we can see that both HOMO and LUMO of the exo
transition state are antisymmetric with respect to the plane. Also, both HOMO and LUMO of the endo transition state are antisymmetric as well.

Also, we notice that the energy of exo is higher than that of endo. This can be explained by the poorer overlap between the C=C π and C=O π* compared to that of endo. This is called secondary orbital effect, which will be further discussed below.

There is one imaginary frequency in both transition structure. This confirms that both of them are valid postulated transition structures.
{| class="wikitable"
|JSMOL
'''Imaginary frequency of exo transition state animation'''
|JSMOL
'''Imaginary frequency of exo transition state animation'''
|}
Again, from the above animations, we can see that the bonds are formed in a synchronous fashion, which means the Diels-Alder Cycloaddition reaction is concerted.

Furthermore, the geometries of both transition states were examined carefully in Table 19.
{| class="wikitable"
!Geometry summary of Exo Transition State (Please refer to Figure 43 for atom labelling)
!Geometry summary of Endo Transition State (Please refer to Figure 44 for atom labelling)
|-
|
{| class="wikitable"
!
!Bond length/ Å
!
!Bond length/ Å
|-
|C5-C13
|2.17032
|C6-C14
|2.17035
|-
|C5-C3
|1.39440
|C3-C1
|1.39674
|-
|C1-C6
|1.39440
|C14-C13
|1.4612
|-
|C7-C6
|1.48976
|C7-C10
|1.52208
|-
|C10-C5
|1.48976
|C13-C15
|1.48822
|-
|C14-C16
|1.48822
| colspan="2" |N/A
|-
|C7-C16
(Through space)
|2.94507
|C10-C15
(Through space)
|2.94484
|-
|C1-C16
(Through space)
|3.78172
|C3-C15
(Through Space)
|3.78155
|}
|
{| class="wikitable"
!
!Bond length/ Å
!
!Bond length/ Å
|-
|C15-C7
|2.16230
|C16-C5
|2.16229
|-
|C1-C3
|1.39726
|C3-C7
|1.39296
|-
|C1-C5
|1.39308
|C7-C9
|1.49503
|-
|C9-C12
|1.52300
|C5-C12
|1.49054
|-
|C16-C18
|1.48918
|C15-C17
|1.48903
|-
|C15-C16
|1.40863
| colspan="2" |N/A
|-
|C1-C18
(Through space)
|2.89232
|C3-C17
(Through space)
|2.89203
|}
|}
'''Table 19.''' Geometry analysis of exo and endo transition states

According to the reaction scheme shown in Figure 4, a single bond is forming between C5 and C13, also another single bond is forming between C6-C14 for exo; C15 and C7 plus C16 and C5 for endo, which is what the first row in the two tables in the left and right in Table 19 shows. the single bond to be made Comparing these values with literature, we find that they are longer than C-C but shorter than twice of carbon's Van der Waals' radius. This tells us some hints that these pairs of carbons between two reactants are attracting with each other to a certain degree. Furthermore, taking a look to other carbon-carbon bonds within the system, i.e. except row 1 and those labelled with (through space), we notice that they are somewhere between C-C and C=C. This tells us that either C-C is becoming C=C or C=C is becoming C-C in the transition state.

Now, looking at the through space bond length. In the exo transition structure, the through space distance -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH2-CH2- is smaller than the twice of the Van der Waals radius but larger than C-C which gives us a hint that they might interact. In the endo transition structure, the through space distance -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH=CH- is smaller than the twice of the Van der Waals radius but larger than C-C which gives us a hint that they might interact. However, according to the definition of secondary orbital effect, it is looking for the interaction between the C=C π of the diene and C=O π* of the dienophile. Endo clearly shows that as explained, but exo seems to just demonstrate the sterics clash between the -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH2-CH2- fragment of diene. In order to further confirm that exo has no secondary orbital effect, a measurement of bond length was carried out between -(C=O)-O-(C=O)- fragment of the maleic anhydride and the -CH=CH- in diene in the exo transition state. The result was shown in the last row on the left table in Table 19. This shows that they are too far away which means they are not possible to interact.

Now, looking back to the HOMO of exo and endo transition states in Figure 45 and 46 respectively. We can definitely see that the overlap between the two reactants is relatively smaller in exo. From these two pieces of information, we can conclude that the endo is kinetically controlled, while exo is thermodynamically controlled.

After that, the '''IRC''' of the both optimised transition state was carried out with both direction and force constant calculated always for the reaction profile. 21 points were used for exo transition states and 24 for endo (reasons explained under '''Introduction)''' to see the reaction profiles.
{| class="wikitable"
! colspan="3" |Exo Transition State
! colspan="3" |Endo Transition State
|-
|[[File:Yll113Exo irc.jpg|thumb|'''Figure 51.''' IRC of the exo transtion state]]
|[[File:Yll113Exo rms.jpg|thumb|'''Figure 52. '''RMS of the exo transition structure]]
|JSMOL
|[[File:Yll113Endo irc.jpg|thumb|'''Figure 53. '''IRC of the endo transition state]]
|[[File:Yll113Endo rms.jpg|thumb|'''Figure 54.''' RMS of the endo transition state]]
|JSMOL
|}
And eventually, the activation energies of the reaction via different transition structures were summarised in Table 20.
{| class="wikitable"
!
!Cyclohexa-1,3-diene
!Maleic Anhydride
!Endo Transition State
!ExoTransition State
!Activation Energy via endo
!Activation Energy via exo
|-
|Energy/Hartree
|0.02771130
|<nowiki>-0.12182418</nowiki>
|<nowiki>-0.05150469</nowiki>
|<nowiki>-0.05041984</nowiki>
|0.04260819
(26.74 kcal/mol)
|0.04369304
(27.42 kcal/mol)
|}
'''Table 20.''' Activation energy analysis

=== '''Conclusion''' ===

=== '''Reference''' ===

<gallery>
File:
</gallery>

Rep:Mod:hlj298

2015-12-17T07:35:30Z

Yll113: /* Optimisation of transition state */

== '''Study of the reaction profiles of the Cope Rearrangement and the Diels-Alder Cycloadditions''' ==
'''Y. L. J. Lam'''

''Department of Chemistry, South Kensington Campus, Imperial College London, London SW7 2AZ, United Kingdom''

''Received 18 December, 2015''

=== Abstract ===
The reactants, products and transition states of the Cope
Rearrangement of 1,5-hexadiene were investigated by ''GaussView 5.0'' at the '''HF/3-21G''' and '''B3LYP/6-31G*'''levels
of theories respectively. With that, the point groups, vibrational frequencies and different energies at different temperatures of the reactants, products and transition states were calculated. Also, by optimizing the transition structures with different methods, i.e. computing the force constants at the
beginning of the calculations, using the redundant coordinate editor and '''QST2''', at the '''HF/3-21G''' level of theory, closer views of the geometries of the transition states can be observed. Furthermore, by using the '''IRC''' method, the reaction profiles can be
obtained and the activation energies can therefore be calculated. Plus, using '''IRC''' method, all reaction intermediates
can now be observed, which helps us to understand the mechanism of the Cope Rearrangement. Similarly, for Diels-Alder Cycloadditions between ethene and
butadiene and Diels-Alder Cycloadditions between cyclohexa-1,3-diene and maleic anhydride, the reactants, products and transition states were optimized and
their respective energies were calculated at '''AM1 semi-empirical molecular orbital method'''. Furthermore, the symmetries of the
molecular orbitals were visualized and the reaction profiles calculated by '''IRC''' method were obtained.

=== '''Introduction''' ===
Chemical reactions are happening around the world in every second. Some reactions are fast, whilst some are slow. The most common and general reason for that is on the kinetic and thermodynamic aspects. On the kinetic aspect, we might argue that the energy barrier(s) form the reactant(s) to the product(s) is/are huge, and therefore, the reactant(s) cannot overcome the barrier(s) and the reaction is slow or does not proceed. The transformation between crude carbon and diamond is a good example. The energy difference between crude carbon and diamond is just few kcal/mol, however, the energy barrier for the transformation is huge. Hence, the transformation is extremely slow, or even does not proceed. With that, diamond symbolizes eternity. On the other hand, on the thermodynamic aspect, we might argue that the reaction is endothermic, i.e. absorbing/requiring heat from the surroundings in order to proceed. In fact, these two aspects just provide us with a little bit of the story and therefore, chemists, or scientists in general, study the mechanism of the reactions to find out the full story. Unfortunately, some reactions are spontaneous, such as the thiocyanation of the iron complex. Also, some intermediates of the reactions are unstable, which cannot be separated or detected even using very advanced analytical instruments, such as nuclear magnetic resonance (NMR) spectromenter. Therefore, scientists devised some programs and computational methods to find out the mechanism of the reactions. Here we use ''GaussView 5.0'' for our investigation.

==== Computational Theory ====
[[File:Yll113 AM1 and HF.jpg|thumb|463x463px|'''Figure 1.''' HOMO and LUMO (highlighted in yellow) of cis-butadiene under the basis of calculation '''HF/3-21G '''(left) and '''AM1''' (right)]]
In ''GaussView 5.0'', there are numerous methods for calculation, such as '''HF/3-21G''', '''B3LYP/6-31G*''', '''semiempirical AM1''', '''MP4 '''and '''MP2'''. Here, the first two calculation method, namely, '''HF/3-21G''' and '''B3LYP/6-31G* '''were applied for calculation of the Cope Rearrangement Reaction, while '''semiempirical AM1''' was used for the investigation of the two Diels-Alder reactions.

N.B. No matter which method applied, the RMS Gradient Norm in hartress would also be computed. This is a measure of how well does the optimisation go during the calculation of the
structure drawn. The closer to zero, the better the structure is optimised.

===== Hartree-Fock ('''HF''') Method =====
Hartree-Fock theory ('''HF''') is the fundamentals of electronic structure theory. It gives a good starting point for more elaborate theoretical methods which can approximate the electronic Schrödinger equation better. It is the basis of the molecular orbital (MO) theory that assumes the motion of each electron can be described by a single-particle function/orbital and it does not depend on/interact with the instantaneous motions of the other electrons.<ref>C. D. Sherrill, ''An Introduction to Hartree-Fock Molecular Orbital Theory'', 2000</ref>

===== Becke, 3-parameter, Lee-Yeang-Parr ('''B3LYP''') Method =====
Beeke, 3-parameter, Lee-Yang-Parr ('''B3LYP''') is one of the most commonly used hybrid functionals. Hybrid functionals are a class of approximation of the exchange-correlation energy functional in density functional theory.<ref>What is B3LYP?, https://www.quora.com/What-is-B3LYP (accessed December 2015)</ref> '''B3LYP''' contains an '''HF''' exchange with the weight of 0.2, which can be regarded as a uniform screening of
exchange by 80 %.<ref>C. H. Patterson, ''International Journal of Quantum Chemistry'', 2006, '''106 '''(15), 3383</ref> '''B3LYP''' also takes a set of atomization
and ionization energies, proton affinities and total atomic energies into account.<ref>A. D. Becke, ''The Journal of Chemical Physics'', 1993, '''98''', 5648</ref>

===== Semiempirical Austin Model 1 ('''AM1''') =====
Semiempirical Austin Model 1 ('''AM1''') based on the Neglect of Differential Diatomic Overlap (NDDO) integral approximation.<ref>M.
J. S. Dewar, E. G. Zoebisch, E. F. Healy and J. J. P. Stewart, ''J. Am. Chem. Soc.'', 1985, '''107''' (13), 3902</ref> Therefore, when taking the same molecule for '''AM1''' and '''HF/3-21G''', you would find that the numbers of HOMO and LUMO are different, which '''AM1''' gives smaller numbers as shown in Figure 1. This is due to the neglect of the low-lying orbitals overlapping. With that, '''AM1''' proceeds much faster than '''HF/3-21G''' for the sake of time.

==== The Cope Rearrangement ====
The Cope Rearrangement is an organic reaction involving [3,3]-sigmatropic rearrangement of 1,5-dienes, which resembles the Claisen Rearrangement.<ref>A. C. Cope and E. M. Hardy, ''J. Am. Chem. Soc.'', 1940, '''62''' (2), 441</ref> The mechanism of the Rearrangement has sparked a controversy – whether it is concerted, dissociative or stepwise.<ref>O. Wiest, K. A. Black and K. N. Houk, ''J. Am. Chem. Soc.'', 1994, '''116''', 10336</ref> With that, first, each conformer of the reactant, 1,5-hexadiene, was optimised by '''HF/3-21G'''. The lowest energy conformer of 1,5-hexadiene was found. Then, as we know, the Rearrangement undergoes either a chair or boat transition state. So, each transition state was optimised by '''HF/3-21G '''as well. By looking into the energy difference between the transition states and the reactant, the activation energy of the Cope Rearrangement with 1,5-hexadiene was found. In order to find the reaction profile and see how the 1,5-diene rearranges, i.e. the mechanism, the transition state was optimised again with
mainly two methods. The coordinate of the chair transition state was first frozen, with the bond to be made set as 2.20000 Å. 2.20000 Å is a good bond length for partially C-C bond as suggested by the chemists’ observations in the literature. <ref>N. H. Kendall, Y. Li and J. D. Evanseck, ''Angew. Chem. Int. Ed. Engl.'', 1992, '''31''' (6), 682</ref> Then, after the optimization of the frozen coordinate, the partly form 2.20000 Å can be relaxed and the structure was then reoptimised. This methods skips the process of computing the whole force constant matrix i.e. Hessian, which saves time and costs. Furthermore, the boat transition state was optimised by '''QST2'''. '''QST2''' has a higher constrains in which requires a more accurate transition state structure to be put in. In this case, the dihedral angle plays an important role in order to be calculated by ''GaussView'' 5.0. Hence, this method is more expensive and time-consuming. From the optimised transition states, an '''IRC''' can be run for the optimised structure to see the full reaction profile. Also, the intermediates of the reaction can be observed. And finally, the reactant and two transition states
were optimised with '''B3LYP/6-31G*''' similarly. Hence, the two calculation methods can be compared by looking into the numbers obtained. Also, the numbers can be compared against the
experimental values. As explained above, '''B3LYP''' takes a more in-depth consideration, the numbers got from this method should be closer to the reality.

==== The Diels-Alder Cycloaddition ====
The Diels-Alder cycloaddition is a [4+2] cycloaddition between a dienophile and a conjugated alkene to give a cyclohexane system. Here, calculations on two Diels-Alder cycloaddition reactions are reported. They are (1) ethylene and butadiene and (2) cyclohexa-1,3-diene and maleic anhydride.

For Diels-Alder cycloaddition reaction, it is well-known that the reaction gives exo and/or endo product. Exo product implies the reaction pathway is thermodynamically controlled to give more stable product; endo product implies
the reaction pathway is kinetically controlled to give a relatively less stable product. In other words, the activation energy to form the exo product is higher than that of endo, however, the endo product is higher in energy than exo. This can usually be explained by the secondary orbital effects. In our cases, both the exo and endo products were investigated undoubtedly. This time, as you may notice, the molecule is more large in size and there are two reactants instead of just one reactant in the Cope Rearrangement, a simpler method of calculation was implemented, which is '''AM1'''. Also, the electronic distributions and orbitals of the HOMO and LUMO of the transition states were computed and visualised.

=== '''Computational Method''' ===
''All calculations were performed by GaussView 5.0. Relevant JSmol files were uploaded here, however, due to some technique glitches, some bonds, especially double bonds, might not come up properly. Yet, the structures of the molecules are generally correctly shown.''

==== The Cope Rearrangement ====
[[File:Yll113 CR.png|thumb|'''Figure 2.''' The Cope Rearrangement of 1,5-hexadiene]]
An anti and gauche conformation of the 1,5-hexadiene were drawn respectively. The drawn structures were first optimised by a not very accurate technique, i.e. '''Clean'''. Then, the '''clean'''ed structure were optimised by '''HF/3-21G'''. The point group and the energy of each conformer were found and compared to locate the low-energy minima. The optimised structures from '''HF/3-21G''' were then reoptimised by '''B3LYP/6-31G*'''. The point group of each conformer was checked and confirmed. Also, the comparison of the same conformer under different calculation method '''HF/3-21G''' and '''B3LYP/6-31G''' was carried out by looking into energy, bond lengths and bond angles. Furthermore, the '''Thermochemistry''' using job type '''Frequency''' was found in both '''HF/3-21G '''and''' B3LYP/6-31G* '''optimised anti conformers.

The boat and chair transition structures were also drawn and '''clean'''ed. The point group of each transition state was found.

Firstly, the chair transition structure was '''optimised to TS (Berny)''' in '''Opt + Freq '''using the basis of '''HF/3-21G'''. Force constant was calculated '''once'''. The frequency of vibration was checked to make sure there is one imaginary vibrational frequency. Then, '''freeze''' '''coordinate''' of the molecule by freezing the carbon-carbon bond to be made as 2.20000 Å. After that, the frozen coordinate was relaxed so the carbon-carbon bond to be made no longer be restricted to 2.20000 Å. The geometry of the transition state was then compared.

Secondly, at the same time, the boat transition structure was optimised by '''QST2''' method by specifying the reactants and products of the reaction under the basis of '''HF/3-21G'''. Labelling the atoms in
the reactant and product, and adjusting the central '''C-C-C-C '''dihedral angle to 0o plus the two inside '''C-C-C''' angles to 100o, the reactant and product could now be optimised by '''QST2''' method.

Comparing the optimised chair and boat transition structures, the connecting conformer of 1,5-hexadiene was found. The reaction energy profile was then calculated by '''IRC''' under '''HF/3-21G''' with 50 points and force constant as always for every small steps. With that, the mechanism of the reaction, as well as the whole reaction energy profile, could be observed clearly. Take the last point on the '''IRC''' and run a normal '''optimisation''' under '''HF/3-21G''' to obtain a minimized geometry.

Eventually, re'''optimise''' the structures of the two transition states with '''Opt + Freq '''under the basis of '''B3LYP/6-31G*'''. The geometries and energies of the transition structure under two different basis were compared. With that, these computed values were also compared against experimental values.

==== The Diels-Alder Cycloadditions ====
[[File:Yll113DA1.jpg|thumb|'''Figure 3. '''The Diels-Alder Cycloadditions between ethylene and butadiene]]

===== Ethylene and butadiene =====
The structure of cis-butadiene was first optimised under the basis of '''semiempirical AM1'''. The HOMO and LUMO of cis butadiene were visualised and its symmetry was determined.

The transition state of the reaction was drawn and optimised under the basis of '''semiempirical AM1'''. The '''IRC''' was carried out in order to confirm the transition structure postulated is of the correct one. Furthermore, the HOMO of the transition structure was visualised and the nodal
planes and properties of the system were interpreted.

===== Cyclohexa-1,3-diene and maleic anhydride =====
[[File:Yll113DA2.jpg|thumb|'''Figure 4. '''The Diels-Alder Cycloadditions between cyclohexa-1,3-diene and maleic anhydride]]
The transition states of the exo and endo products were drawn and optimised under the basis of '''semiempirical AM1'''. The '''IRC''' was carried out in order to confirm the transition structure postulated is of the correct one. For the number of points, 21 points were used for exo transition states and 24 for endo. This is because the energy was too shallow and the slopes tend to zero after the number of points specified above and ''GaussView 5.0'' cannot predict which direction should it goes on to calculate. Furthermore,
the bond lengths, orientation and the HOMO of the transition structures were investigated.

=== '''Results and Discussion''' ===

==== The Cope Rearrangement ====

===== Optimisation of Reactant =====
1,5-hexadiene has three free rotating carbon-carbon bonds. Each of them has three rotational minima. This gives 27 conformations of the 1,5-hexadiene molecule. Yet, only ten of them were energetically distinct due to symmetry and enantiomeric relationships.<ref>B. G. Rocque, J. M. Gonzales and H. F. Schaefer, ''Molecular Physics'', 2002, '''100''' (4), 441</ref> Two of them, the ''Ci anti ''and ''C1 gauche ''structure in here'' ''were drawn and optimizied as shown in Figure A and B and their energies were calculated as shown in Table 1.
{| class="wikitable"
!Conformation
!Molecule
!Point Group
!Energy/ Hartrees*
HF/3-21G
!RMS Gradient Norm/Hartrees*
HF/3-21G
!Relative Energy#/ kcal/mol
!Newman Projections
|-
|Gauche3
|<jmol>
<jmolApplet>
<title>Figure A: Gauge3 conformer of 1,5-hexadiene</title><color>white</color>
<size>250</size>
<uploadedFileContents> yll113CR_GAUGE_PART1.LOG </uploadedFileContents>
</jmolApplet>
</jmol>
|C1
|<nowiki>-231.69266120</nowiki>
|0.00001556
|0.00
|[[File:Yll113 torsion gauche.jpg|centre|frame|'''Figure 5.''' Newman Projection of C1 gauche3 1,5-hexadiene]]
|-
|Anti2
|<jmol>
<jmolApplet>
<title>Figure B: Anti2 conformer of 1,5-hexadiene</title><color>white</color>
<size>250</size>
<uploadedFileContents>YLL113CR ANTI PART1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|Ci
|<nowiki>-231.69253528</nowiki>
|0.00001891
|0.08
|[[File:Yll113 torsion anti.jpg|centre|frame|'''Figure 6.''' Newman Projection of Ci anti2 1,5-hexadiene]]
|}
<nowiki>*</nowiki>1 hartree = 627.509 kcal/mol

#The difference in energy between the conformer and the lowest energy conformer, in here, which is Gauche3. Then convert Hartree to kcal/mol by *

'''Table 1. '''Conformational analysis of anti2 and gauche3 of 1,5-hexadiene

As shown in Table 1, the energy of Gauche3 is surprisingly lower than the anti2 conformation of 1,5-hexadiene. In most cases, the antiperiplanar conformation of a molecule, such as anti2, is more favourable as it has the least steric clashes. Therefore, usually the antiperiplanar conformation is of the lowest energy. However, here, apart from sterics, the stereoelectroncs concept has also been taken into account. The vinyl proton, in a through space manner, can interact with the π or π* orbital on the sp2 carbon which is separated by four bonds from it.<ref>M. Nishio and M. Hirota, ''Tetrahedron'', 1989, '''45 '''(23), 7201</ref> This is so-called CH-π interaction. The Newman Projection in Figure 5 gives us a closer look on how they are close in space and interact; and the Newman projection in Figure 6 tells us why the vinyl proton cannot interact with the π or π* system through space. Therefore, the gauche3 conformation is more stable than anti2 and of lower energy in 1,5-hexadiene.

Focusing on anti2 conformer of the 1,5-hexadiene, the '''HF/3-21G''' optimised molecule was reoptimised by '''B3LYP/6-31G'''. The energies and the geometries of the anti2 1,5-hexadiene under two basis of calculation method were compared and shown in Table 2.
[[File:Yll113Anti2.png|thumb|'''Figure 7. '''Anti2 conformer of 1,5-hexadiene with labelled atoms]]
{| class="wikitable"
! rowspan="2" |Calculation Method
! rowspan="2" |Point Group
! rowspan="2" |Energy/ Hartrees
! rowspan="2" |RMS Gradient Norm/Hartrees
!Bond length between/ Å
!Bond length between/ Å
!Bond length between/ Å
!Bond angle between
!Bond angle between
|-
|C14-C12 and C4-C1
|C12-C9 and C6-C4
|C6-C9
|C14-C12-C9 and
C1-C4-C6
|C12-C9-C6 and
C4-C6-C9
|-

|'''HF/3-21G'''
|Ci
|<nowiki>-231.69253528</nowiki>
|0.00001891
|1.31613
|1.50891
|1.55275
|124.80579o
|111.34878o
|-
|'''B3LYP/6-31G*'''
|Ci
|<nowiki>-234.61171063</nowiki>
|0.00001249
|1.33350
|1.50419
|1.54816
|125.29968o
|112.67081o
|}
'''Table 2. '''Comparison with the anti2 conformer of 1,5-hexadiene with two calculation method basis ('''HF/3-21G '''and '''B3LYP/6-31G*''')

As seen in Table 2, the point group of the same conformer does not change no matter which method of calculation was carried out. The energy of the conformer with '''B3LYP/6-31G*''' is lower than that of '''HF/3-21G'''. Also, the bond lengths and bond angles changed with a certain amount as shown.

Furthermore, by submitting job type as '''Frequency''' in the optimised structure of anti2 conformer of 1,5-hexadiene under both calculation basis '''HF/3-21G '''and '''B3LYP/6-31G*''', the electronic energy, sum of electronic and zero-point energies at 0 K and sum of electronic and thermal energies at 298.15 K could be determined as shown in Table 3.

{| border="1" cellspacing="1" cellpadding="10"
|-
|rowspan="3"| ''' '''
!colspan="4" |'''HF/3-21G a'''
! colspan="4" |'''B3LYP/6-31G* b'''
|-
! rowspan="2" width="125" align="center" | '''Electronic energy'''
! width="125" align="center" | '''Sum of electronic and zero-point energies'''
! width="125" align="center" | '''Sum of electronic and thermal energies'''
! rowspan="2" width="125" align="center" | '''IR spectrum'''
! rowspan="2" width="125" align="center" | '''Electronic energy'''
! width="125" align="center" | '''Sum of electronic and zero-point energies'''
! width="125" align="center" | '''Sum of electronic and thermal energies'''
! rowspan="2" width="125" align="center" | '''IR spectrum'''
|-
! width="125" align="center" | '''at 0 K'''
! width="125" align="center" | '''at 298.15 K'''
! width="125" align="center" | '''at 0 K'''
! width="125" align="center" | '''at 298.15 K'''
|-
| '''Reactant (anti2)'''
| align="center" | -231.692534
| align="center" |-231.539539
| align="center" | -231.532565
|[[File:Yll113ANTI3-21IR.png|thumb|'''Figure 8. '''IR spectrum of anti2 reactant 1,5-hexadiene calculated by '''HF/3-21G''']]
| align="center" | -234.611703
| align="center" | -234.469213
| align="center" | -234.461857
|[[File:Yll113ANTI6-31IR.png|thumb|'''Figure 9. '''IR spectrum of anti2 reactant 1,5-hexadiene calculated by '''B3LYP/6-31G*''']]
|}

a [https://wiki.ch.ic.ac.uk/wiki/images/5/52/Yll113CR_ANTI_PART4.LOG File]; b [https://wiki.ch.ic.ac.uk/wiki/images/5/54/Yll113_CR_ANTI_PART3.LOG File]

'''Table 3.''' Summary of energies (in hartree) of the reactant (anti2) Comparing Figure 8 and 9, both IR spectra are similar which shows that they have similar stretching/bending mode even under different methods of calculations. However, the strongest peak in Figure 8 is at 1112 cm-1 and that in Figure 9 is 940 cm-1. Yet, they correspond to the same mode of vibration, which is the =C-H bending. Therefore, according to the equation, the wavenumber of absorbance, ν can be calculated:

:<math>\nu = \frac{1}{2 \pi c} \sqrt{\frac{k}{\mu}}</math>

where ''μ'' is the reduced mass, ''k'' is the spring constant for the bond and ''c'' is the light speed,

we can therefore know that the spring constant of the molecule under '''B3LYP/6-31G* '''basis reduced.

Now, focusing on gauche3 conformer of the 1,5-hexadiene, similarly, the '''HF/3-21G''' optimised molecule was reoptimised by '''B3LYP/6-31G'''. The energies and the geometries of the gauche3 1,5-hexadiene under two basis of calculation method were compared and shown in Table 4.
[[File:Yll113Gauche3.png|thumb|'''Figure 10. '''Gauche3 conformer of 1,5-hexadiene with labelled atoms]]
{| class="wikitable"
! rowspan="2" |Calculation Method
! rowspan="2" |Point Group
! rowspan="2" |Energy/ Hartrees
! rowspan="2" |RMS Gradient Norm/Hartrees
! colspan="3" |Bond length between/ Å
! colspan="2" |Bond angle between
|-
|C14-C12 and C4-C1
|C12-C9 and C6-C4
|C6-C9
|C14-C12-C9 and
C1-C4-C6
|C12-C9-C6 and
C4-C6-C9
|-

|'''HF/3-21G'''
|C1
|<nowiki>-231.69266120</nowiki>
|0.00001555
|1.31646
|1.50929
|1.55314
|125.02428o
|111.80728o
|-
|'''B3LYP/6-31G*'''
|C1
|<nowiki>-234.61132605</nowiki>
|0.00000360
|1.33382
|1.50491
|1.55007
|125.49464o
|113.46225o
|}
'''Table 4. '''Comparison with the anti2 conformer of 1,5-hexadiene with two calculation method basis ('''HF/3-21G '''and '''B3LYP/6-31G*''')

As seen in Table 4, the point group of the same conformer, again, does not change no matter which method of calculation was carried out. The energy of the conformer with '''B3LYP/6-31G*''' is lower than that of '''HF/3-21G'''. Also, the bond lengths and bond angles changed with a certain amount as shown.

Furthermore, by submitting job type as '''Frequency''' in the optimised structure of gauche3 conformer of 1,5-hexadiene under both calculation basis '''HF/3-21G '''and '''B3LYP/6-31G*''', the electronic energy, sum of electronic and zero-point energies at 0 K and sum of electronic and thermal energies at 298.15 K could be determined as shown in Table 5.

{| border="1" cellspacing="1" cellpadding="10"
|-
| rowspan="3" | ''' '''
! colspan="4" |'''HF/3-21G c'''
! colspan="4" |'''B3LYP/6-31G* d'''
|-
! rowspan="2" width="125" align="center" | '''Electronic energy'''
! width="125" align="center" | '''Sum of electronic and zero-point energies'''
! width="125" align="center" | '''Sum of electronic and thermal energies'''
! rowspan="2" |IR spectrum
! rowspan="2" width="125" align="center" | '''Electronic energy'''
! width="125" align="center" | '''Sum of electronic and zero-point energies'''
! width="125" align="center" | '''Sum of electronic and thermal energies'''
! rowspan="2" |IR spectrum
|-
! width="125" align="center" | '''at 0 K'''
! width="125" align="center" | '''at 298.15 K'''
! width="125" align="center" | '''at 0 K'''
! width="125" align="center" | '''at 298.15 K'''
|-
| '''Reactant (Gauche 3)'''
| align="center" |-231.692692
| align="center" |-231.539486
| align="center" |-231.532646
|[[File:Yll113GAUCHE3-21IR.png|thumb|'''Figure 11. '''IR spectrum of gauche3 conformer of 1,5-hexadiene calculated by '''HF/3-21G''']]
| align="center" | -234.611326
| align="center" |-234.468719
| align="center" |-234.461477
|[[File:Yll113GAUCHE6-31IR.png|thumb|'''Figure 12.''' IR spectrum of gauche3 conformer of 1,5-hexadiene calculated by '''B3LYP/6-31G*''']]
|}

c [https://wiki.ch.ic.ac.uk/wiki/images/9/98/Yll113CR_GAUGE_PART4.LOG File] ; d [https://wiki.ch.ic.ac.uk/wiki/images/c/ca/Yll113CR_GAUGE_PART3.LOG File]

'''Table 5.''' Summary of energies (in hartree) of the reactant (Gauche3) Comparing Figure 11 and 12, both IR spectra are similar which shows that they have similar stretching/bending mode even under different methods of calculations. However, the strongest peak in Figure 12 is at 939 cm-1 and that in Figure 11 is 1111 cm-1. Yet, they correspond to the same mode of vibration, which is also the =C-H bending. Therefore, similar to the anti2 conformer's case as mentioned above, we can therefore know that the spring constant of the molecule under '''B3LYP/6-31G* '''basis reduced.

===== Optimisation of transition state =====

After optimising the reactants, the chair and boat transition states were optimised accordingly using mainly two different methods. But before that, an allyl fragment (CH2CHCH2) was optimised using '''HF/3-21G''' level of theory for the sake of convenience in constructing the chair and boat transition states. A brief summary was shown in Table 6.
{| class="wikitable"
!
! colspan="2" |Molecule
!Point Group
!Energy/Hartree
!RMS Gradient Norm/ Hartrees
|-
|Allyl fragment
CH2CHCH2
|<jmol>
<jmolApplet>
<title>Figure C: Allyl Fragment</title><color>white</color>
<size>250</size>
<uploadedFileContents> Yll113CR TS 1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|[[File:Yll11313.jpg|thumb|'''Figure 13. '''Optimised Structure of the allyl fragment]]
|C2v
|<nowiki>-115.82304010</nowiki>
|0.00002945
|}
'''Table 6. '''Summary of the optimised allyl fragment

Then, both chair and boat transition state were drawn and optimised using the '''optimisation to TS (Berny)''' under '''HF/3-21G'''. Figure 14 and Figure C show the optimized structure of the chair transition state while Figure 15 and Figure D show the optimized structure of the boat transition state. Table 7 shows the summary of results.
{| class="wikitable"
! rowspan="2" |
! colspan="2" rowspan="2" |Molecule
! rowspan="2" |Point Group
! rowspan="2" |Energy/ Hartree
! rowspan="2" |RMS Gradient Norm/Hartrees
! colspan="6" |Partial C-C bond length between/ Å
|-
!C14-C6
!C11-C3
!C14-C9
!C6-C1
!C9-C11
!C1-C3
|-
|Optimised Chair Transition State
|JSMOL
|[[File:Yll113CHAIR3-21.png|thumb|'''Figure 14. '''Optimised Chair Structure by '''HF/3-21G''' with atoms labelled ]]
|C2h
|<nowiki>-231.61932238</nowiki>
|0.00002645
|2.02016
|2.02016
|1.38929
|1.38930
|1.38930
|1.38929
|-
|Optimised Boat Transition State
|JSMOL
|[[File:Yll11315.jpg|thumb|'''Figure 15. '''Optimised Boat Structure by '''HF/3-21G''' with atoms labelled]]
|C2v
|<nowiki>-231.60280235</nowiki>
|0.00003872
|2.14060
|2.14060
|1.38138
|1.38138
|1.38138
|1.38138
|}

'''Table 7. '''Summary of the optimised chair and boat transition states by '''optimisation to TS (Berny) '''under '''HF/3-21G''' basis

Furthermore, the transition structures’ '''Frequencies''' were calculated as shown in Table 8.
{| class="wikitable"
!
!Molecule
!IR spectrum
!Imaginary Frequency/ cm-1
|-
|Boat Transition State
|
|[[File:Yll11317.jpg|thumb|'''Figure 16. '''IR spectrum of the optimised Boat Structure by '''HF/3-21G''']]
|<nowiki>-840</nowiki>
|-
|Chair Transition State
|[[File: Yll113Chair Part 1 imag freq.gif]]
|[[File:Yll11316.jpg|thumb|'''Figure 17. '''IR spectrum of the optimised Chair Structure by '''HF/3-21G''']]
|<nowiki>-818</nowiki>
|}
'''Table 8.''' IR spectra and imaginary frequencies of the boat and chair transition states

As you may notice that, the
imaginary frequency comes up when calculating with the transition states. This
is common, in other words, this should appear to let us know the transition
structure we postulated is correct.

A transition state is the first
order saddle point on the potential energy surface. Therefore, the force
applied to the saddle point against to the displacement. As force and
displacement are vectors, the force constant will be a negative number.Therefore, according to
:<math>\nu = \frac{1}{2 \pi c} \sqrt{\frac{k}{\mu}}</math>

where ''μ'' is the reduced mass, ''k'' is the spring constant for the bond and ''c'' is the light speed,

the square root of a negative
force constant k gives an imaginary wave number/frequency v. In other words,
the appearance of an imaginary frequency tells us that the structure is a
saddle point of the reaction pathway.

The chair transition state
was followed by first 'frozen' then 'relaxed'. The boat transition structure
was followed by '''QST2''' calculation method.

====== Chair Transition State ======
After the above '''optimisation''', the chair transition
state was reoptimised again with another method. This method first freezes the
coordinate of the molecule, in this case, freeze the bond to be made in the
Cope Rearrangement of 1,5-hexadiene as 2.20000 Å. The molecule then optimised with the frozen
coordinate. Details of this optimisation was summarized in Table 9.
{| class="wikitable"
! rowspan="2" |
! colspan="2" rowspan="2" |Molecule
! rowspan="2" |Point Group
! rowspan="2" |
{| class="MsoTableGrid"

|
<nowiki> </nowiki>Energy/ Hartree

|}
! rowspan="2" |RMS Gradient Norm/ Hartree
! colspan="3" |Bond length between/ Å
! rowspan="2" |Imaginary Frequency/ cm-1
! rowspan="2" |IR Spectrum
|-
!C6-C14 and C3-11
!C1-C3 and C9-C14
!C1-C6 and C9-C11
|-
|Optimised Chair Transition Structure with frozen coordinate
|JSMOL
|[[File:Yll11318.jpg|thumb|'''Figure 18. '''The optimised chair transition structure with frozen coordinate and atoms labelling]]
|
{| class="MsoTableGrid"

|
<nowiki> </nowiki>C2h

|}
|<nowiki>-231.61518510</nowiki>
|0.00325573
|2.20000
|1.38135
|1.38128
|<nowiki>-765</nowiki>
|[[File:Yll11319.jpg|thumb|'''Figure 19. '''IR spectrum of the optimised chair transition structure with frozen coordinate]]
|}

'''Table 9. '''Summary of the optimisation of the chair transition structure with
frozen coordinate(s)

From Table 9, we may notice
that the RMS Gradient Norm value is quite far off from zero. Also, the
imaginary frequency becomes much higher than -818 cm-1 (Shown in
Table 8). With these two pieces of information, we can deduce that the frozen
coordinate(s) affect(s) the force constant of the transition state which does
not give a good optimisation of transition structure. With that, after applying
the frozen coordinate to the molecule, the molecule was reoptimised again with
a degree of '''Derivative '''to the '''Bond'''. Details of the reoptimisation
were presented in Table 10.
{| class="wikitable"
! rowspan="2" |
! colspan="2" rowspan="2" |Molecule
! rowspan="2" |Point Group
! rowspan="2" |Energy/ Hartree
! rowspan="2" |RMS Gradient Norm/ Hartree
! colspan="4" |Bond length between/ Å
! rowspan="2" |Imaginary Frequency/ cm-1
! rowspan="2" |
{| class="MsoTableGrid"

|
<nowiki> </nowiki>IR Spectrum

|}
|-
!C14-C6
!C11-C3
!C14-C9 and C6-C1
!C9-C11 and C1-C3
|-
|Optimised Chair Transition
Structure with a degree of '''Derivative'''
to the '''Bond'''
|JSMOL
|[[File:Yll11320.jpg|thumb|'''Figure 20. '''The optimised chair transition structure with a degree of '''Derivative''' to the '''Bond''' and atoms labelling]]
|C2h
|<nowiki>-231.61932233</nowiki>
|0.00002127
|2.02075
|2.02071
|1.38929
|1.38930
|<nowiki>-818</nowiki>
|[[File:Yll11321.jpg|thumb|'''Figure 21. '''IR spectrum of the optimised chair transition structure with a degree of '''Derivative''' to the '''Bond''']]
|}
'''Table 10. '''Summary of the reoptimisation of the chair transition
structure with a degree of '''Derivative''' to the '''Bond'''

Now, in Table 10, the RMS
Gradient Norm value is close to zero. Also, the imaginary frequency goes back
to -818 cm-1, indicating that the coordinates no longer be frozen
and the stretching/bending mode of the transition state is able to undergo
freely.

Comparing the bond lengths
in Table 7 and 10, we can see that the difference between bond lengths of the
single bond to be made/ broken calculated in two methods is just less than
0.0006 Å. And also, there is no difference in bond length of the double bond to be make/broken ‘inside’ the system. This tells us that the two optimisation
methods are rather similar under the consideration on the Cope Rearrangement
Reaction.

====== Boat Transition State ======
Instead of using the frozen
coordinate method as for the chair transition state above, another method, '''QST2''', was applied to the boat
transition state under the '''HF/3-21G'''
basis. In order to use this method, without any ‘Link died’, the reactant and
product have to be drawn and labelled carefully. '''QST2''' is a method which interpolates the reactant and product to
give a transition state. Therefore, it will fall if the structure of the
reactant and product are not close to the transition state. And therefore, all
molecules have to be carefully labelled and adjusted.

[[File:Yll11322.png|500px|center]]

'''Figure 22. '''The drawings and adjustments of angles of the reactant (left)
and product (right) for '''QST2''' Method,
i.e. the central C-C-C-C dihedral angle was changed to 0o and inside
C-C-C were reduced to 100o

After the adjustment, the job was run and the optimized molecule converge to the boat transition structure. Summary was shown in Table 11.
{| class="wikitable"
! rowspan="2" |
! colspan="2" rowspan="2" |Molecule
! rowspan="2" |Point Group
! rowspan="2" |Energy/Hartree
! rowspan="2" |RMS Gradient Norm/ Hartree
! colspan="3" |Bond length between/ Å
! rowspan="2" |Imaginary Frequency/ cm-1
! rowspan="2" |IR spectrum
|-
!C1-C6
!C3-C4
!C5-C6, C4-C5, C3-C2 and C1-C2
|-
|Boat transition structure
under '''QST2''' method
|JSMOL
|[[File:Yll11323.jpg|thumb|'''Figure 23. '''The optimised Boat transition structure with atom labelling]]
|C2v
|<nowiki>-231.60280241</nowiki>
|0.00002436
|2.13994
|2.14019
|1.38149
|<nowiki>-840</nowiki>
|[[File:Yll11324.jpg|thumb|'''Figure 24. '''IR spectrum of the optimised boat transition structure]]
|}
'''Table 11. '''Summary of the boat transition structure under '''QST2 '''method

====== Intrinsic Reaction Coordinate''' '''('''IRC''') ======
In order to confirm that our transition state is of the
correct one, '''Intrinsic Reaction
Coordinate '''('''IRC''') will be carried
out.

As mentioned above, transition state is the first order
saddle point of the reaction pathway. Therefore, it will start to go to the
product or back to the reactant with it falls off. It resembles that a ball is
at the tip of the mountain, which starts to roll off the mountain on the side
with the steepest slope. Also, when we are doing '''IRC''', we can determine whether the reaction goes forward, backward
or both sides. Also, the number of points, which means the number of little
steps that the geometry of the molecule changes, can be adjusted. A low number
of points will just give us a very rough idea that tell us a little bit about
our transition state. Also, the last point on the '''IRC''' is far from the minimum geometry. A high number of points gives
us more accurate results, however two problems could be raised. First, the time
for calculation will be long and Most importantly, as it goes down the slope
and reaches the minimum geometry, i.e. the plateau of energy, the slope will
become very small or even zero again. However, as the energy difference of the
next or previous geometry compared to the geometry of itself is too small, ''GaussView 5.0'' may not able to know which
direction the molecule should proceed to. And this, therefore, results in ‘Link
died’. Therefore, the most common technique is to have a good number of points,
then take the last point on the IRC and run it with a normal optimisation.

Here, as we know that the
Cope Rearrangement has a symmetric reaction pathway, taking the chair
transition structure, we will run '''IRC'''
on it with 50 points.
{| class="wikitable"
![[File:Yll113hlj29825.jpg|thumb|'''Figure 25. '''Total energy along '''IRC '''of the chair transition structure]]
![[File:Yll11326.jpg|thumb|'''Figure 26. '''RMS Gradient Norm of '''IRC '''of the chair transition structure]]
!JSMOL
|}

[[File:Yll11327.jpg|500px]]

'''Figure 27. '''The product of the Cope Rearrangement after optimisation

The first point on Figure 25 is -231.61932233 Hartree and the last point is -231.69157881 Hartree. Then, we take the last point and optimise it, we get the structure shown in Figure 27.

The structure is of C2
symmetry and the energy calculated is -231.69166702 Hartree. This matches with
Gauche2 C2 on Appendix 1. And therefore, this is how the conformer
of 1,5-hexadiene connects with the chair transition structure.

====== Activation Energy of the Cope Rearrangement ======
Finally, we optimise the chair and boat transition states we got from above, reoptimise it with job Opt + Freq
under a more advanced calculation '''B3LYP/6-31G*'''. And from that, the thermochemistry data were given and we can know the
activation energy of the reaction by comparing to Table 3, which anti2 is used
as a local minimum rather than gauche3 as a global minimum.
{| border="1" cellspacing="1" cellpadding="10"
|-
| rowspan="3" | ''' '''
! colspan="3" |'''HF/3-21G'''
! colspan="3" |'''B3LYP/6-31G*'''
|-
| rowspan="2" width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| rowspan="2" width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619323
| align="center" | -231.466698
| align="center" | -231.461339
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409009
|-
| '''Boat TS'''
| align="center" | -231.602803
| align="center" | -231.450932
| align="center" | -231.445303
| align="center" | -234.543094
| align="center" | -234.402342
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692534
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.611703
| align="center" | -234.469213
| align="center" | -234.461857
|}
'''Table 11'''. Summary of energies of chair, boat and reactant (anti2) structure
{| border="1" cellspacing="1" cellpadding="10"
|-
| rowspan="2" | ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.69
| align="center" | 34.06
| align="center" | 33.16
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.96
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
'''Table 12'''. Summary of activation energies in kcal/mol

==== The Diels-Alder Cycloadditions ====

===== Ethylene and Cis-Butadiene =====
First, the structures of the reactants, ethylene and cis-butadiene, were '''optimised to minimum''' under '''semiempirical AM1''' method. For the butadiene, in order to be in the cis conformer, the dihedral angle was adjusted to be 0o. Details are listed in Table 13.
{| class="wikitable"
!
!Molecule
!Point Group
!HOMO
!LUMO
!Energy/Hartree
!RMS Gradient Norm/ Hartree
!IR spectrum
|-
|Ethylene
|JSMOL
|D2h
|[[File:Yll11331.jpg|thumb|'''Figure 31. '''HOMO of Ethylene]]
|[[File:Yll11330.jpg|thumb|'''Figure 30.''' LUMO of ethylene]]
|0.02619029
|0.00008755
|[[File:Yll11328.jpg|thumb|'''Figure 28. '''IR spectrum of Ethylene]]
|-
|Cis-Butadiene
|JSMOL
|C2v
|[[File:Yll11332.jpg|thumb|'''Figure 32. '''HOMO of cis-butadiene]]
|[[File:Yll11333.jpg|thumb|'''Figure 33. '''LUMO of cis-butadiene]]
|0.04878534
|0.00000087
|[[File:Yll11329.jpg|thumb|'''Figure 29.''' IR spectrum of cis-butadiene]]
|}
'''Table 13.''' Summary of optimised ethylene and cis-butadiene

Looking into Figure 30-33, as we know that the plane is perpendicular to the molecule, the HOMO of Ethylene is symmetric while that of LUMO is antisymmetric.

Also, the HOMO of cis-butadiene is antisymmetric and that of LUMO is symmetric.

Then, the transition state of the reaction was able to constructed using the optimised structure of the reactants made above. By making the bond lengths of the bonds to be made in Diels-Alder [4+2] cycloaddition to 2.20000 Å in the transition state, which is the 'dash bond', the transition structure was optimized to '''Opt + Freq minimisation to''' '''TS (Berny)''' under the same method of calculation - '''semiempitical AM1'''. Details are listed in Table 14.
{| class="wikitable"
!
! colspan="2" |Molecule
!Point Group
!HOMO
!LUMO
!Energy/Hartree
!RMS Gradient Norm/ Hartree
!Imaginary Frequency/ cm-1
!IR spectrum
|-
|Transition state
|JSMOL
|[[File:Yll11334.jpg|thumb|'''Figure 34. '''Transition state of ethylene + cis-butadiene]]
|C1
|[[File:Yll11336.jpg|thumb|'''Figure 36. '''HOMO of transition state]]
|[[File:Yll11337.jpg|thumb|'''Figure 37. '''LUMO of transition state]]
|0.11165467
|0.00002792
|<nowiki>-956</nowiki>
|[[File:Yll11335.jpg|thumb|'''Figure 35. '''IR spectrum of the transition state]]
|}
'''Table 14.''' Summary of optimised transition state

From Figure 36, we can see that the HOMO of the transition state is antisymmetric whilst the LUMO of the transition state is symmetric. By making very careful comparison between Figure 36, Figure 37 and Figure 30-33, we can see that the HOMO of the transition state in Figure 36 is a combination of Figure 32 and 30; the LUMO of the transition state in Figure 37 is a combination of Figure 31 and 33. We can clearly see that the HOMO and LUMO of the transition state have a complementary combination of HOMO and LUMO of the reactants.

Taking a closer look to HOMO of the transition state. Recalling Woodward Hoffmann’s Rule, (4q+2)s+(4r)a = odd for thermally allowed reaction, we have both π2s and π4s. Therefore, the reaction is thermally allowed by letting q = 0, which gives the value of 1 which is odd.

Furthermore, from Table 14, we notice that there is an imaginary frequency reported at -956 cm-1. As explained above, the transition state should have one imaginary frequency to account for the negative force constant. With that, this imaginary frequency confirms that the transition structure we postulated from the optimised reactants is valid, i.e. it is really a transition state. The animation of where the imaginary frequency originates from, which shows the motion of the transition state - how the two reactants approach to each other and bonds are formed, is shown below.

JSMOL

From the above figure, we can see that the bond formation from the reactant to the product happens at the same time, i.e. synchronous, on both sides of the transition structure. Therefore, we can say that this Diels-Alder cycloaddition is a concerted [4+2] pericyclic cycloaddition, which matches with what we learnt in Pericyclic Reaction course.

On top of that, the geometry of the transition structure was investigated by looking into the optimised bond lengths between carbon atoms Details are shown in Figure 38 and Table 15.[[File:Yll11338.jpg|thumb|'''Figure 38. '''Transition state of ethylene + cis-butadiene with atoms labelled]]
{| class="wikitable"
!
!Bond length/ Å
|-
|C7-C9
|2.11938
|-
|C12-C5
|2.11944
|-
|C12-C9
|1.38284
|-
|C3-C7
|1.38187
|-
|C3-C1
|1.39750
|-
|C5-C1
|1.38175
|}
'''Table 15. '''Geometry analysis of the transition state

According to the literature <ref>M. A. Fox and J. K. Whitesell, ''Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen'', Springer, 1995</ref>, C-C carbon-carbon single bond is 1.54 Å, and C=C carbon-carbon double bond is 1.34 Å. Also, the Van der Waals radius of carbon is 1.70 Å,<ref>A. Bondi,(1964), ''J. Phys. Chem.'', 1964, '''68''' (3), 441</ref>
According to the reaction scheme shown in Figure 3, a single bond is forming between C7 and C9, also another single bond is forming between C12-C5. Comparing the data in Table 15 with the literature, we can see that the bond length of two bonds to be made is longer than C-C, but shorter than the twice of carbon's Van der Waals radius. This tells us some hints that the terminal carbons between two reactants are attracting with each other to a certain degree. Furthermore, taking a look to other carbon-carbon bonds within the system, we notice that they are somewhere between C-C and C=C. This tells us that either C-C is becoming C=C or C=C is becoming C-C in the transition state.
After that, the '''IRC''' of the above optimised transition state was carried out with both direction and force constant calculated always for 50 points to see the reaction profile.
{| class="wikitable"
|[[File:Yll11339.jpg|thumb|'''Figure 39.''' IRC of the transition state of ethylene + cis-butadiene]]
|[[File:Yll11340.jpg|thumb|'''Figure 40. '''RMS Gradient Norm of transition state of ethylene + cis-butadiene]]
|JSMOL
|}
In Figure 39, we can clearly see that the reactants was first passed through the energy barrier to get the transition state and it went down the slope to give the product.
Finally, the activation energy for this reaction was calculated in Table 16.
{| class="wikitable"
!
!Ethylene
!Cis-butadiene
!Transition state
!Activation Energy
|-
|Energy/Hartree
|0.02619029
|0.04878534
|0.11165467
|0.03667904
(23.02 kcal/mol)
|}
'''Table 16. '''Activation energy analysis of Diels-Alder Reaction between ethylene and cis-butadiene
===== Cyclohexa-1,3-diene and Maleic Anhydride =====
First, the structures of
the reactants, ethylene and cis-butadiene, were '''optimised to minimum''' under '''semiempirical AM1''' method. Details are listed in Table 17.
{| class="wikitable"
!Optimised Structure
!Molecule
!Point Group
!Energy/Hartree
|-
|Cyclohexa-1,3-diene
|[[File:Yll113Cycloopt.jpg|thumb|'''Figure 41. '''Optimised structure of cyclohexa-1,3-diene]]
|C2
|0.02771130
|-
|Maleic Anhydride
|[[File:Yll113Maopt.jpg|thumb|'''Figure 42. '''Optimised structure of Maleic Anhydride]]
|Cs
|<nowiki>-0.12182418</nowiki>
|}
'''Table 17. '''Summary
of the optimized reactant

Then, the two transition states, exo and endo, of the reaction were able to construct. By making the bond lengths of the bonds to be made in Diels-Alder [4+2] cycloaddition to 2.20000 Å in the transition state, which is the 'dash bond', the transition structure was optimized to '''Opt + Freq minimisation to''' '''TS (Berny)''' under the same method of calculation - '''semiempitical AM1'''. Details are listed in Table 18.
{| class="wikitable"
!Transition Structure
!Molecule
!Point Group
!HOMO
!LUMO
!Energy/Hartree
!'''RMS Gradient Norm/ Hartree'''
!'''Imaginary Frequency/ cm-1'''
!IR spectrum
|-
|Exo
|[[File:Yll113Exots.jpg|thumb|'''Figure 43.''' Optimised Structure of Exo Transition State]]
|C1
|[[File:Yll113Exo homo.jpg|thumb|'''Figure 45. '''HOMO of Exo Transition State]]
|[[File:Yll113Exo lumo.jpg|thumb|'''Figure 47.''' LUMO of Exo Transition State]]
|<nowiki>-0.05041984</nowiki>
|0.00000883
|<nowiki>-812</nowiki>
|[[File:Yll113Exo ir.jpg|thumb|'''Figure 49. '''Exo Transition State IR spectrum]]
|-
|Endo
|[[File:Yll113Endots.jpg|thumb|'''Figure 44. '''Optimised Structure of Enfo Transition State]]
|C1
|[[File:Yll113Endo homo.jpg|thumb|'''Figure 46'''. HOMO of Endo Transition State]]
|[[File:Yll113Endo lumo.jpg|thumb|'''Figure 48.''' LUMO of Endo Transition States]]
|<nowiki>-0.05150469</nowiki>
|0.00004308
|<nowiki>-807</nowiki>
|[[File:Yll113Endo ir.jpg|thumb|'''Figure 50.''' Endo Transition State IR Spectrum]]
|}
'''Table 18. '''Summary of the optimized transition structures

In Table 18, we can see that both HOMO and LUMO of the exo
transition state are antisymmetric with respect to the plane. Also, both HOMO and LUMO of the endo transition state are antisymmetric as well.

Also, we notice that the energy of exo is higher than that of endo. This can be explained by the poorer overlap between the C=C π and C=O π* compared to that of endo. This is called secondary orbital effect, which will be further discussed below.

There is one imaginary frequency in both transition structure. This confirms that both of them are valid postulated transition structures.
{| class="wikitable"
|JSMOL
'''Imaginary frequency of exo transition state animation'''
|JSMOL
'''Imaginary frequency of exo transition state animation'''
|}
Again, from the above animations, we can see that the bonds are formed in a synchronous fashion, which means the Diels-Alder Cycloaddition reaction is concerted.

Furthermore, the geometries of both transition states were examined carefully in Table 19.
{| class="wikitable"
!Geometry summary of Exo Transition State (Please refer to Figure 43 for atom labelling)
!Geometry summary of Endo Transition State (Please refer to Figure 44 for atom labelling)
|-
|
{| class="wikitable"
!
!Bond length/ Å
!
!Bond length/ Å
|-
|C5-C13
|2.17032
|C6-C14
|2.17035
|-
|C5-C3
|1.39440
|C3-C1
|1.39674
|-
|C1-C6
|1.39440
|C14-C13
|1.4612
|-
|C7-C6
|1.48976
|C7-C10
|1.52208
|-
|C10-C5
|1.48976
|C13-C15
|1.48822
|-
|C14-C16
|1.48822
| colspan="2" |N/A
|-
|C7-C16
(Through space)
|2.94507
|C10-C15
(Through space)
|2.94484
|-
|C1-C16
(Through space)
|3.78172
|C3-C15
(Through Space)
|3.78155
|}
|
{| class="wikitable"
!
!Bond length/ Å
!
!Bond length/ Å
|-
|C15-C7
|2.16230
|C16-C5
|2.16229
|-
|C1-C3
|1.39726
|C3-C7
|1.39296
|-
|C1-C5
|1.39308
|C7-C9
|1.49503
|-
|C9-C12
|1.52300
|C5-C12
|1.49054
|-
|C16-C18
|1.48918
|C15-C17
|1.48903
|-
|C15-C16
|1.40863
| colspan="2" |N/A
|-
|C1-C18
(Through space)
|2.89232
|C3-C17
(Through space)
|2.89203
|}
|}
'''Table 19.''' Geometry analysis of exo and endo transition states

According to the reaction scheme shown in Figure 4, a single bond is forming between C5 and C13, also another single bond is forming between C6-C14 for exo; C15 and C7 plus C16 and C5 for endo, which is what the first row in the two tables in the left and right in Table 19 shows. the single bond to be made Comparing these values with literature, we find that they are longer than C-C but shorter than twice of carbon's Van der Waals' radius. This tells us some hints that these pairs of carbons between two reactants are attracting with each other to a certain degree. Furthermore, taking a look to other carbon-carbon bonds within the system, i.e. except row 1 and those labelled with (through space), we notice that they are somewhere between C-C and C=C. This tells us that either C-C is becoming C=C or C=C is becoming C-C in the transition state.

Now, looking at the through space bond length. In the exo transition structure, the through space distance -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH2-CH2- is smaller than the twice of the Van der Waals radius but larger than C-C which gives us a hint that they might interact. In the endo transition structure, the through space distance -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH=CH- is smaller than the twice of the Van der Waals radius but larger than C-C which gives us a hint that they might interact. However, according to the definition of secondary orbital effect, it is looking for the interaction between the C=C π of the diene and C=O π* of the dienophile. Endo clearly shows that as explained, but exo seems to just demonstrate the sterics clash between the -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH2-CH2- fragment of diene. In order to further confirm that exo has no secondary orbital effect, a measurement of bond length was carried out between -(C=O)-O-(C=O)- fragment of the maleic anhydride and the -CH=CH- in diene in the exo transition state. The result was shown in the last row on the left table in Table 19. This shows that they are too far away which means they are not possible to interact.

Now, looking back to the HOMO of exo and endo transition states in Figure 45 and 46 respectively. We can definitely see that the overlap between the two reactants is relatively smaller in exo. From these two pieces of information, we can conclude that the endo is kinetically controlled, while exo is thermodynamically controlled.

After that, the '''IRC''' of the both optimised transition state was carried out with both direction and force constant calculated always for the reaction profile. 21 points were used for exo transition states and 24 for endo (reasons explained under '''Introduction)''' to see the reaction profiles.
{| class="wikitable"
! colspan="3" |Exo Transition State
! colspan="3" |Endo Transition State
|-
|[[File:Yll113Exo irc.jpg|thumb|'''Figure 51.''' IRC of the exo transtion state]]
|[[File:Yll113Exo rms.jpg|thumb|'''Figure 52. '''RMS of the exo transition structure]]
|JSMOL
|[[File:Yll113Endo irc.jpg|thumb|'''Figure 53. '''IRC of the endo transition state]]
|[[File:Yll113Endo rms.jpg|thumb|'''Figure 54.''' RMS of the endo transition state]]
|JSMOL
|}
And eventually, the activation energies of the reaction via different transition structures were summarised in Table 20.
{| class="wikitable"
!
!Cyclohexa-1,3-diene
!Maleic Anhydride
!Endo Transition State
!ExoTransition State
!Activation Energy via endo
!Activation Energy via exo
|-
|Energy/Hartree
|0.02771130
|<nowiki>-0.12182418</nowiki>
|<nowiki>-0.05150469</nowiki>
|<nowiki>-0.05041984</nowiki>
|0.04260819
(26.74 kcal/mol)
|0.04369304
(27.42 kcal/mol)
|}
'''Table 20.''' Activation energy analysis

=== '''Conclusion''' ===

=== '''Reference''' ===

<gallery>
File:
</gallery>

File:Yll113CR TS 1.LOG

2015-12-17T07:34:03Z

Yll113:

Rep:Mod:hlj298

2015-12-17T07:30:25Z

Yll113: /* Optimisation of Reactant */

== '''Study of the reaction profiles of the Cope Rearrangement and the Diels-Alder Cycloadditions''' ==
'''Y. L. J. Lam'''

''Department of Chemistry, South Kensington Campus, Imperial College London, London SW7 2AZ, United Kingdom''

''Received 18 December, 2015''

=== Abstract ===
The reactants, products and transition states of the Cope
Rearrangement of 1,5-hexadiene were investigated by ''GaussView 5.0'' at the '''HF/3-21G''' and '''B3LYP/6-31G*'''levels
of theories respectively. With that, the point groups, vibrational frequencies and different energies at different temperatures of the reactants, products and transition states were calculated. Also, by optimizing the transition structures with different methods, i.e. computing the force constants at the
beginning of the calculations, using the redundant coordinate editor and '''QST2''', at the '''HF/3-21G''' level of theory, closer views of the geometries of the transition states can be observed. Furthermore, by using the '''IRC''' method, the reaction profiles can be
obtained and the activation energies can therefore be calculated. Plus, using '''IRC''' method, all reaction intermediates
can now be observed, which helps us to understand the mechanism of the Cope Rearrangement. Similarly, for Diels-Alder Cycloadditions between ethene and
butadiene and Diels-Alder Cycloadditions between cyclohexa-1,3-diene and maleic anhydride, the reactants, products and transition states were optimized and
their respective energies were calculated at '''AM1 semi-empirical molecular orbital method'''. Furthermore, the symmetries of the
molecular orbitals were visualized and the reaction profiles calculated by '''IRC''' method were obtained.

=== '''Introduction''' ===
Chemical reactions are happening around the world in every second. Some reactions are fast, whilst some are slow. The most common and general reason for that is on the kinetic and thermodynamic aspects. On the kinetic aspect, we might argue that the energy barrier(s) form the reactant(s) to the product(s) is/are huge, and therefore, the reactant(s) cannot overcome the barrier(s) and the reaction is slow or does not proceed. The transformation between crude carbon and diamond is a good example. The energy difference between crude carbon and diamond is just few kcal/mol, however, the energy barrier for the transformation is huge. Hence, the transformation is extremely slow, or even does not proceed. With that, diamond symbolizes eternity. On the other hand, on the thermodynamic aspect, we might argue that the reaction is endothermic, i.e. absorbing/requiring heat from the surroundings in order to proceed. In fact, these two aspects just provide us with a little bit of the story and therefore, chemists, or scientists in general, study the mechanism of the reactions to find out the full story. Unfortunately, some reactions are spontaneous, such as the thiocyanation of the iron complex. Also, some intermediates of the reactions are unstable, which cannot be separated or detected even using very advanced analytical instruments, such as nuclear magnetic resonance (NMR) spectromenter. Therefore, scientists devised some programs and computational methods to find out the mechanism of the reactions. Here we use ''GaussView 5.0'' for our investigation.

==== Computational Theory ====
[[File:Yll113 AM1 and HF.jpg|thumb|463x463px|'''Figure 1.''' HOMO and LUMO (highlighted in yellow) of cis-butadiene under the basis of calculation '''HF/3-21G '''(left) and '''AM1''' (right)]]
In ''GaussView 5.0'', there are numerous methods for calculation, such as '''HF/3-21G''', '''B3LYP/6-31G*''', '''semiempirical AM1''', '''MP4 '''and '''MP2'''. Here, the first two calculation method, namely, '''HF/3-21G''' and '''B3LYP/6-31G* '''were applied for calculation of the Cope Rearrangement Reaction, while '''semiempirical AM1''' was used for the investigation of the two Diels-Alder reactions.

N.B. No matter which method applied, the RMS Gradient Norm in hartress would also be computed. This is a measure of how well does the optimisation go during the calculation of the
structure drawn. The closer to zero, the better the structure is optimised.

===== Hartree-Fock ('''HF''') Method =====
Hartree-Fock theory ('''HF''') is the fundamentals of electronic structure theory. It gives a good starting point for more elaborate theoretical methods which can approximate the electronic Schrödinger equation better. It is the basis of the molecular orbital (MO) theory that assumes the motion of each electron can be described by a single-particle function/orbital and it does not depend on/interact with the instantaneous motions of the other electrons.<ref>C. D. Sherrill, ''An Introduction to Hartree-Fock Molecular Orbital Theory'', 2000</ref>

===== Becke, 3-parameter, Lee-Yeang-Parr ('''B3LYP''') Method =====
Beeke, 3-parameter, Lee-Yang-Parr ('''B3LYP''') is one of the most commonly used hybrid functionals. Hybrid functionals are a class of approximation of the exchange-correlation energy functional in density functional theory.<ref>What is B3LYP?, https://www.quora.com/What-is-B3LYP (accessed December 2015)</ref> '''B3LYP''' contains an '''HF''' exchange with the weight of 0.2, which can be regarded as a uniform screening of
exchange by 80 %.<ref>C. H. Patterson, ''International Journal of Quantum Chemistry'', 2006, '''106 '''(15), 3383</ref> '''B3LYP''' also takes a set of atomization
and ionization energies, proton affinities and total atomic energies into account.<ref>A. D. Becke, ''The Journal of Chemical Physics'', 1993, '''98''', 5648</ref>

===== Semiempirical Austin Model 1 ('''AM1''') =====
Semiempirical Austin Model 1 ('''AM1''') based on the Neglect of Differential Diatomic Overlap (NDDO) integral approximation.<ref>M.
J. S. Dewar, E. G. Zoebisch, E. F. Healy and J. J. P. Stewart, ''J. Am. Chem. Soc.'', 1985, '''107''' (13), 3902</ref> Therefore, when taking the same molecule for '''AM1''' and '''HF/3-21G''', you would find that the numbers of HOMO and LUMO are different, which '''AM1''' gives smaller numbers as shown in Figure 1. This is due to the neglect of the low-lying orbitals overlapping. With that, '''AM1''' proceeds much faster than '''HF/3-21G''' for the sake of time.

==== The Cope Rearrangement ====
The Cope Rearrangement is an organic reaction involving [3,3]-sigmatropic rearrangement of 1,5-dienes, which resembles the Claisen Rearrangement.<ref>A. C. Cope and E. M. Hardy, ''J. Am. Chem. Soc.'', 1940, '''62''' (2), 441</ref> The mechanism of the Rearrangement has sparked a controversy – whether it is concerted, dissociative or stepwise.<ref>O. Wiest, K. A. Black and K. N. Houk, ''J. Am. Chem. Soc.'', 1994, '''116''', 10336</ref> With that, first, each conformer of the reactant, 1,5-hexadiene, was optimised by '''HF/3-21G'''. The lowest energy conformer of 1,5-hexadiene was found. Then, as we know, the Rearrangement undergoes either a chair or boat transition state. So, each transition state was optimised by '''HF/3-21G '''as well. By looking into the energy difference between the transition states and the reactant, the activation energy of the Cope Rearrangement with 1,5-hexadiene was found. In order to find the reaction profile and see how the 1,5-diene rearranges, i.e. the mechanism, the transition state was optimised again with
mainly two methods. The coordinate of the chair transition state was first frozen, with the bond to be made set as 2.20000 Å. 2.20000 Å is a good bond length for partially C-C bond as suggested by the chemists’ observations in the literature. <ref>N. H. Kendall, Y. Li and J. D. Evanseck, ''Angew. Chem. Int. Ed. Engl.'', 1992, '''31''' (6), 682</ref> Then, after the optimization of the frozen coordinate, the partly form 2.20000 Å can be relaxed and the structure was then reoptimised. This methods skips the process of computing the whole force constant matrix i.e. Hessian, which saves time and costs. Furthermore, the boat transition state was optimised by '''QST2'''. '''QST2''' has a higher constrains in which requires a more accurate transition state structure to be put in. In this case, the dihedral angle plays an important role in order to be calculated by ''GaussView'' 5.0. Hence, this method is more expensive and time-consuming. From the optimised transition states, an '''IRC''' can be run for the optimised structure to see the full reaction profile. Also, the intermediates of the reaction can be observed. And finally, the reactant and two transition states
were optimised with '''B3LYP/6-31G*''' similarly. Hence, the two calculation methods can be compared by looking into the numbers obtained. Also, the numbers can be compared against the
experimental values. As explained above, '''B3LYP''' takes a more in-depth consideration, the numbers got from this method should be closer to the reality.

==== The Diels-Alder Cycloaddition ====
The Diels-Alder cycloaddition is a [4+2] cycloaddition between a dienophile and a conjugated alkene to give a cyclohexane system. Here, calculations on two Diels-Alder cycloaddition reactions are reported. They are (1) ethylene and butadiene and (2) cyclohexa-1,3-diene and maleic anhydride.

For Diels-Alder cycloaddition reaction, it is well-known that the reaction gives exo and/or endo product. Exo product implies the reaction pathway is thermodynamically controlled to give more stable product; endo product implies
the reaction pathway is kinetically controlled to give a relatively less stable product. In other words, the activation energy to form the exo product is higher than that of endo, however, the endo product is higher in energy than exo. This can usually be explained by the secondary orbital effects. In our cases, both the exo and endo products were investigated undoubtedly. This time, as you may notice, the molecule is more large in size and there are two reactants instead of just one reactant in the Cope Rearrangement, a simpler method of calculation was implemented, which is '''AM1'''. Also, the electronic distributions and orbitals of the HOMO and LUMO of the transition states were computed and visualised.

=== '''Computational Method''' ===
''All calculations were performed by GaussView 5.0. Relevant JSmol files were uploaded here, however, due to some technique glitches, some bonds, especially double bonds, might not come up properly. Yet, the structures of the molecules are generally correctly shown.''

==== The Cope Rearrangement ====
[[File:Yll113 CR.png|thumb|'''Figure 2.''' The Cope Rearrangement of 1,5-hexadiene]]
An anti and gauche conformation of the 1,5-hexadiene were drawn respectively. The drawn structures were first optimised by a not very accurate technique, i.e. '''Clean'''. Then, the '''clean'''ed structure were optimised by '''HF/3-21G'''. The point group and the energy of each conformer were found and compared to locate the low-energy minima. The optimised structures from '''HF/3-21G''' were then reoptimised by '''B3LYP/6-31G*'''. The point group of each conformer was checked and confirmed. Also, the comparison of the same conformer under different calculation method '''HF/3-21G''' and '''B3LYP/6-31G''' was carried out by looking into energy, bond lengths and bond angles. Furthermore, the '''Thermochemistry''' using job type '''Frequency''' was found in both '''HF/3-21G '''and''' B3LYP/6-31G* '''optimised anti conformers.

The boat and chair transition structures were also drawn and '''clean'''ed. The point group of each transition state was found.

Firstly, the chair transition structure was '''optimised to TS (Berny)''' in '''Opt + Freq '''using the basis of '''HF/3-21G'''. Force constant was calculated '''once'''. The frequency of vibration was checked to make sure there is one imaginary vibrational frequency. Then, '''freeze''' '''coordinate''' of the molecule by freezing the carbon-carbon bond to be made as 2.20000 Å. After that, the frozen coordinate was relaxed so the carbon-carbon bond to be made no longer be restricted to 2.20000 Å. The geometry of the transition state was then compared.

Secondly, at the same time, the boat transition structure was optimised by '''QST2''' method by specifying the reactants and products of the reaction under the basis of '''HF/3-21G'''. Labelling the atoms in
the reactant and product, and adjusting the central '''C-C-C-C '''dihedral angle to 0o plus the two inside '''C-C-C''' angles to 100o, the reactant and product could now be optimised by '''QST2''' method.

Comparing the optimised chair and boat transition structures, the connecting conformer of 1,5-hexadiene was found. The reaction energy profile was then calculated by '''IRC''' under '''HF/3-21G''' with 50 points and force constant as always for every small steps. With that, the mechanism of the reaction, as well as the whole reaction energy profile, could be observed clearly. Take the last point on the '''IRC''' and run a normal '''optimisation''' under '''HF/3-21G''' to obtain a minimized geometry.

Eventually, re'''optimise''' the structures of the two transition states with '''Opt + Freq '''under the basis of '''B3LYP/6-31G*'''. The geometries and energies of the transition structure under two different basis were compared. With that, these computed values were also compared against experimental values.

==== The Diels-Alder Cycloadditions ====
[[File:Yll113DA1.jpg|thumb|'''Figure 3. '''The Diels-Alder Cycloadditions between ethylene and butadiene]]

===== Ethylene and butadiene =====
The structure of cis-butadiene was first optimised under the basis of '''semiempirical AM1'''. The HOMO and LUMO of cis butadiene were visualised and its symmetry was determined.

The transition state of the reaction was drawn and optimised under the basis of '''semiempirical AM1'''. The '''IRC''' was carried out in order to confirm the transition structure postulated is of the correct one. Furthermore, the HOMO of the transition structure was visualised and the nodal
planes and properties of the system were interpreted.

===== Cyclohexa-1,3-diene and maleic anhydride =====
[[File:Yll113DA2.jpg|thumb|'''Figure 4. '''The Diels-Alder Cycloadditions between cyclohexa-1,3-diene and maleic anhydride]]
The transition states of the exo and endo products were drawn and optimised under the basis of '''semiempirical AM1'''. The '''IRC''' was carried out in order to confirm the transition structure postulated is of the correct one. For the number of points, 21 points were used for exo transition states and 24 for endo. This is because the energy was too shallow and the slopes tend to zero after the number of points specified above and ''GaussView 5.0'' cannot predict which direction should it goes on to calculate. Furthermore,
the bond lengths, orientation and the HOMO of the transition structures were investigated.

=== '''Results and Discussion''' ===

==== The Cope Rearrangement ====

===== Optimisation of Reactant =====
1,5-hexadiene has three free rotating carbon-carbon bonds. Each of them has three rotational minima. This gives 27 conformations of the 1,5-hexadiene molecule. Yet, only ten of them were energetically distinct due to symmetry and enantiomeric relationships.<ref>B. G. Rocque, J. M. Gonzales and H. F. Schaefer, ''Molecular Physics'', 2002, '''100''' (4), 441</ref> Two of them, the ''Ci anti ''and ''C1 gauche ''structure in here'' ''were drawn and optimizied as shown in Figure A and B and their energies were calculated as shown in Table 1.
{| class="wikitable"
!Conformation
!Molecule
!Point Group
!Energy/ Hartrees*
HF/3-21G
!RMS Gradient Norm/Hartrees*
HF/3-21G
!Relative Energy#/ kcal/mol
!Newman Projections
|-
|Gauche3
|<jmol>
<jmolApplet>
<title>Figure A: Gauge3 conformer of 1,5-hexadiene</title><color>white</color>
<size>250</size>
<uploadedFileContents> yll113CR_GAUGE_PART1.LOG </uploadedFileContents>
</jmolApplet>
</jmol>
|C1
|<nowiki>-231.69266120</nowiki>
|0.00001556
|0.00
|[[File:Yll113 torsion gauche.jpg|centre|frame|'''Figure 5.''' Newman Projection of C1 gauche3 1,5-hexadiene]]
|-
|Anti2
|<jmol>
<jmolApplet>
<title>Figure B: Anti2 conformer of 1,5-hexadiene</title><color>white</color>
<size>250</size>
<uploadedFileContents>YLL113CR ANTI PART1.LOG</uploadedFileContents>
</jmolApplet>
</jmol>
|Ci
|<nowiki>-231.69253528</nowiki>
|0.00001891
|0.08
|[[File:Yll113 torsion anti.jpg|centre|frame|'''Figure 6.''' Newman Projection of Ci anti2 1,5-hexadiene]]
|}
<nowiki>*</nowiki>1 hartree = 627.509 kcal/mol

#The difference in energy between the conformer and the lowest energy conformer, in here, which is Gauche3. Then convert Hartree to kcal/mol by *

'''Table 1. '''Conformational analysis of anti2 and gauche3 of 1,5-hexadiene

As shown in Table 1, the energy of Gauche3 is surprisingly lower than the anti2 conformation of 1,5-hexadiene. In most cases, the antiperiplanar conformation of a molecule, such as anti2, is more favourable as it has the least steric clashes. Therefore, usually the antiperiplanar conformation is of the lowest energy. However, here, apart from sterics, the stereoelectroncs concept has also been taken into account. The vinyl proton, in a through space manner, can interact with the π or π* orbital on the sp2 carbon which is separated by four bonds from it.<ref>M. Nishio and M. Hirota, ''Tetrahedron'', 1989, '''45 '''(23), 7201</ref> This is so-called CH-π interaction. The Newman Projection in Figure 5 gives us a closer look on how they are close in space and interact; and the Newman projection in Figure 6 tells us why the vinyl proton cannot interact with the π or π* system through space. Therefore, the gauche3 conformation is more stable than anti2 and of lower energy in 1,5-hexadiene.

Focusing on anti2 conformer of the 1,5-hexadiene, the '''HF/3-21G''' optimised molecule was reoptimised by '''B3LYP/6-31G'''. The energies and the geometries of the anti2 1,5-hexadiene under two basis of calculation method were compared and shown in Table 2.
[[File:Yll113Anti2.png|thumb|'''Figure 7. '''Anti2 conformer of 1,5-hexadiene with labelled atoms]]
{| class="wikitable"
! rowspan="2" |Calculation Method
! rowspan="2" |Point Group
! rowspan="2" |Energy/ Hartrees
! rowspan="2" |RMS Gradient Norm/Hartrees
!Bond length between/ Å
!Bond length between/ Å
!Bond length between/ Å
!Bond angle between
!Bond angle between
|-
|C14-C12 and C4-C1
|C12-C9 and C6-C4
|C6-C9
|C14-C12-C9 and
C1-C4-C6
|C12-C9-C6 and
C4-C6-C9
|-

|'''HF/3-21G'''
|Ci
|<nowiki>-231.69253528</nowiki>
|0.00001891
|1.31613
|1.50891
|1.55275
|124.80579o
|111.34878o
|-
|'''B3LYP/6-31G*'''
|Ci
|<nowiki>-234.61171063</nowiki>
|0.00001249
|1.33350
|1.50419
|1.54816
|125.29968o
|112.67081o
|}
'''Table 2. '''Comparison with the anti2 conformer of 1,5-hexadiene with two calculation method basis ('''HF/3-21G '''and '''B3LYP/6-31G*''')

As seen in Table 2, the point group of the same conformer does not change no matter which method of calculation was carried out. The energy of the conformer with '''B3LYP/6-31G*''' is lower than that of '''HF/3-21G'''. Also, the bond lengths and bond angles changed with a certain amount as shown.

Furthermore, by submitting job type as '''Frequency''' in the optimised structure of anti2 conformer of 1,5-hexadiene under both calculation basis '''HF/3-21G '''and '''B3LYP/6-31G*''', the electronic energy, sum of electronic and zero-point energies at 0 K and sum of electronic and thermal energies at 298.15 K could be determined as shown in Table 3.

{| border="1" cellspacing="1" cellpadding="10"
|-
|rowspan="3"| ''' '''
!colspan="4" |'''HF/3-21G a'''
! colspan="4" |'''B3LYP/6-31G* b'''
|-
! rowspan="2" width="125" align="center" | '''Electronic energy'''
! width="125" align="center" | '''Sum of electronic and zero-point energies'''
! width="125" align="center" | '''Sum of electronic and thermal energies'''
! rowspan="2" width="125" align="center" | '''IR spectrum'''
! rowspan="2" width="125" align="center" | '''Electronic energy'''
! width="125" align="center" | '''Sum of electronic and zero-point energies'''
! width="125" align="center" | '''Sum of electronic and thermal energies'''
! rowspan="2" width="125" align="center" | '''IR spectrum'''
|-
! width="125" align="center" | '''at 0 K'''
! width="125" align="center" | '''at 298.15 K'''
! width="125" align="center" | '''at 0 K'''
! width="125" align="center" | '''at 298.15 K'''
|-
| '''Reactant (anti2)'''
| align="center" | -231.692534
| align="center" |-231.539539
| align="center" | -231.532565
|[[File:Yll113ANTI3-21IR.png|thumb|'''Figure 8. '''IR spectrum of anti2 reactant 1,5-hexadiene calculated by '''HF/3-21G''']]
| align="center" | -234.611703
| align="center" | -234.469213
| align="center" | -234.461857
|[[File:Yll113ANTI6-31IR.png|thumb|'''Figure 9. '''IR spectrum of anti2 reactant 1,5-hexadiene calculated by '''B3LYP/6-31G*''']]
|}

a [https://wiki.ch.ic.ac.uk/wiki/images/5/52/Yll113CR_ANTI_PART4.LOG File]; b [https://wiki.ch.ic.ac.uk/wiki/images/5/54/Yll113_CR_ANTI_PART3.LOG File]

'''Table 3.''' Summary of energies (in hartree) of the reactant (anti2) Comparing Figure 8 and 9, both IR spectra are similar which shows that they have similar stretching/bending mode even under different methods of calculations. However, the strongest peak in Figure 8 is at 1112 cm-1 and that in Figure 9 is 940 cm-1. Yet, they correspond to the same mode of vibration, which is the =C-H bending. Therefore, according to the equation, the wavenumber of absorbance, ν can be calculated:

:<math>\nu = \frac{1}{2 \pi c} \sqrt{\frac{k}{\mu}}</math>

where ''μ'' is the reduced mass, ''k'' is the spring constant for the bond and ''c'' is the light speed,

we can therefore know that the spring constant of the molecule under '''B3LYP/6-31G* '''basis reduced.

Now, focusing on gauche3 conformer of the 1,5-hexadiene, similarly, the '''HF/3-21G''' optimised molecule was reoptimised by '''B3LYP/6-31G'''. The energies and the geometries of the gauche3 1,5-hexadiene under two basis of calculation method were compared and shown in Table 4.
[[File:Yll113Gauche3.png|thumb|'''Figure 10. '''Gauche3 conformer of 1,5-hexadiene with labelled atoms]]
{| class="wikitable"
! rowspan="2" |Calculation Method
! rowspan="2" |Point Group
! rowspan="2" |Energy/ Hartrees
! rowspan="2" |RMS Gradient Norm/Hartrees
! colspan="3" |Bond length between/ Å
! colspan="2" |Bond angle between
|-
|C14-C12 and C4-C1
|C12-C9 and C6-C4
|C6-C9
|C14-C12-C9 and
C1-C4-C6
|C12-C9-C6 and
C4-C6-C9
|-

|'''HF/3-21G'''
|C1
|<nowiki>-231.69266120</nowiki>
|0.00001555
|1.31646
|1.50929
|1.55314
|125.02428o
|111.80728o
|-
|'''B3LYP/6-31G*'''
|C1
|<nowiki>-234.61132605</nowiki>
|0.00000360
|1.33382
|1.50491
|1.55007
|125.49464o
|113.46225o
|}
'''Table 4. '''Comparison with the anti2 conformer of 1,5-hexadiene with two calculation method basis ('''HF/3-21G '''and '''B3LYP/6-31G*''')

As seen in Table 4, the point group of the same conformer, again, does not change no matter which method of calculation was carried out. The energy of the conformer with '''B3LYP/6-31G*''' is lower than that of '''HF/3-21G'''. Also, the bond lengths and bond angles changed with a certain amount as shown.

Furthermore, by submitting job type as '''Frequency''' in the optimised structure of gauche3 conformer of 1,5-hexadiene under both calculation basis '''HF/3-21G '''and '''B3LYP/6-31G*''', the electronic energy, sum of electronic and zero-point energies at 0 K and sum of electronic and thermal energies at 298.15 K could be determined as shown in Table 5.

{| border="1" cellspacing="1" cellpadding="10"
|-
| rowspan="3" | ''' '''
! colspan="4" |'''HF/3-21G c'''
! colspan="4" |'''B3LYP/6-31G* d'''
|-
! rowspan="2" width="125" align="center" | '''Electronic energy'''
! width="125" align="center" | '''Sum of electronic and zero-point energies'''
! width="125" align="center" | '''Sum of electronic and thermal energies'''
! rowspan="2" |IR spectrum
! rowspan="2" width="125" align="center" | '''Electronic energy'''
! width="125" align="center" | '''Sum of electronic and zero-point energies'''
! width="125" align="center" | '''Sum of electronic and thermal energies'''
! rowspan="2" |IR spectrum
|-
! width="125" align="center" | '''at 0 K'''
! width="125" align="center" | '''at 298.15 K'''
! width="125" align="center" | '''at 0 K'''
! width="125" align="center" | '''at 298.15 K'''
|-
| '''Reactant (Gauche 3)'''
| align="center" |-231.692692
| align="center" |-231.539486
| align="center" |-231.532646
|[[File:Yll113GAUCHE3-21IR.png|thumb|'''Figure 11. '''IR spectrum of gauche3 conformer of 1,5-hexadiene calculated by '''HF/3-21G''']]
| align="center" | -234.611326
| align="center" |-234.468719
| align="center" |-234.461477
|[[File:Yll113GAUCHE6-31IR.png|thumb|'''Figure 12.''' IR spectrum of gauche3 conformer of 1,5-hexadiene calculated by '''B3LYP/6-31G*''']]
|}

c [https://wiki.ch.ic.ac.uk/wiki/images/9/98/Yll113CR_GAUGE_PART4.LOG File] ; d [https://wiki.ch.ic.ac.uk/wiki/images/c/ca/Yll113CR_GAUGE_PART3.LOG File]

'''Table 5.''' Summary of energies (in hartree) of the reactant (Gauche3) Comparing Figure 11 and 12, both IR spectra are similar which shows that they have similar stretching/bending mode even under different methods of calculations. However, the strongest peak in Figure 12 is at 939 cm-1 and that in Figure 11 is 1111 cm-1. Yet, they correspond to the same mode of vibration, which is also the =C-H bending. Therefore, similar to the anti2 conformer's case as mentioned above, we can therefore know that the spring constant of the molecule under '''B3LYP/6-31G* '''basis reduced.

===== Optimisation of transition state =====

After optimising the reactants, the chair and boat transition states were optimised accordingly using mainly two different methods. But before that, an allyl fragment (CH2CHCH2) was optimised using '''HF/3-21G''' level of theory for the sake of convenience in constructing the chair and boat transition states. A brief summary was shown in Table 6.
{| class="wikitable"
!
! colspan="2" |Molecule
!Point Group
!Energy/Hartree
!RMS Gradient Norm/ Hartrees
|-
|Allyl fragment
CH2CHCH2
|JSMOL
|[[File:Yll11313.jpg|thumb|'''Figure 13. '''Optimised Structure of the allyl fragment]]
|C2v
|<nowiki>-115.82304010</nowiki>
|0.00002945
|}
'''Table 6. '''Summary of the optimised allyl fragment

Then, both chair and boat transition state were drawn and optimised using the '''optimisation to TS (Berny)''' under '''HF/3-21G'''. Figure 14 and Figure C show the optimized structure of the chair transition state while Figure 15 and Figure D show the optimized structure of the boat transition state. Table 7 shows the summary of results.
{| class="wikitable"
! rowspan="2" |
! colspan="2" rowspan="2" |Molecule
! rowspan="2" |Point Group
! rowspan="2" |Energy/ Hartree
! rowspan="2" |RMS Gradient Norm/Hartrees
! colspan="6" |Partial C-C bond length between/ Å
|-
!C14-C6
!C11-C3
!C14-C9
!C6-C1
!C9-C11
!C1-C3
|-
|Optimised Chair Transition State
|JSMOL
|[[File:Yll113CHAIR3-21.png|thumb|'''Figure 14. '''Optimised Chair Structure by '''HF/3-21G''' with atoms labelled ]]
|C2h
|<nowiki>-231.61932238</nowiki>
|0.00002645
|2.02016
|2.02016
|1.38929
|1.38930
|1.38930
|1.38929
|-
|Optimised Boat Transition State
|JSMOL
|[[File:Yll11315.jpg|thumb|'''Figure 15. '''Optimised Boat Structure by '''HF/3-21G''' with atoms labelled]]
|C2v
|<nowiki>-231.60280235</nowiki>
|0.00003872
|2.14060
|2.14060
|1.38138
|1.38138
|1.38138
|1.38138
|}

'''Table 7. '''Summary of the optimised chair and boat transition states by '''optimisation to TS (Berny) '''under '''HF/3-21G''' basis

Furthermore, the transition structures’ '''Frequencies''' were calculated as shown in Table 8.
{| class="wikitable"
!
!Molecule
!IR spectrum
!Imaginary Frequency/ cm-1
|-
|Boat Transition State
|
|[[File:Yll11317.jpg|thumb|'''Figure 16. '''IR spectrum of the optimised Boat Structure by '''HF/3-21G''']]
|<nowiki>-840</nowiki>
|-
|Chair Transition State
|[[File: Yll113Chair Part 1 imag freq.gif]]
|[[File:Yll11316.jpg|thumb|'''Figure 17. '''IR spectrum of the optimised Chair Structure by '''HF/3-21G''']]
|<nowiki>-818</nowiki>
|}
'''Table 8.''' IR spectra and imaginary frequencies of the boat and chair transition states

As you may notice that, the
imaginary frequency comes up when calculating with the transition states. This
is common, in other words, this should appear to let us know the transition
structure we postulated is correct.

A transition state is the first
order saddle point on the potential energy surface. Therefore, the force
applied to the saddle point against to the displacement. As force and
displacement are vectors, the force constant will be a negative number.Therefore, according to
:<math>\nu = \frac{1}{2 \pi c} \sqrt{\frac{k}{\mu}}</math>

where ''μ'' is the reduced mass, ''k'' is the spring constant for the bond and ''c'' is the light speed,

the square root of a negative
force constant k gives an imaginary wave number/frequency v. In other words,
the appearance of an imaginary frequency tells us that the structure is a
saddle point of the reaction pathway.

The chair transition state
was followed by first 'frozen' then 'relaxed'. The boat transition structure
was followed by '''QST2''' calculation method.

====== Chair Transition State ======
After the above '''optimisation''', the chair transition
state was reoptimised again with another method. This method first freezes the
coordinate of the molecule, in this case, freeze the bond to be made in the
Cope Rearrangement of 1,5-hexadiene as 2.20000 Å. The molecule then optimised with the frozen
coordinate. Details of this optimisation was summarized in Table 9.
{| class="wikitable"
! rowspan="2" |
! colspan="2" rowspan="2" |Molecule
! rowspan="2" |Point Group
! rowspan="2" |
{| class="MsoTableGrid"

|
<nowiki> </nowiki>Energy/ Hartree

|}
! rowspan="2" |RMS Gradient Norm/ Hartree
! colspan="3" |Bond length between/ Å
! rowspan="2" |Imaginary Frequency/ cm-1
! rowspan="2" |IR Spectrum
|-
!C6-C14 and C3-11
!C1-C3 and C9-C14
!C1-C6 and C9-C11
|-
|Optimised Chair Transition Structure with frozen coordinate
|JSMOL
|[[File:Yll11318.jpg|thumb|'''Figure 18. '''The optimised chair transition structure with frozen coordinate and atoms labelling]]
|
{| class="MsoTableGrid"

|
<nowiki> </nowiki>C2h

|}
|<nowiki>-231.61518510</nowiki>
|0.00325573
|2.20000
|1.38135
|1.38128
|<nowiki>-765</nowiki>
|[[File:Yll11319.jpg|thumb|'''Figure 19. '''IR spectrum of the optimised chair transition structure with frozen coordinate]]
|}

'''Table 9. '''Summary of the optimisation of the chair transition structure with
frozen coordinate(s)

From Table 9, we may notice
that the RMS Gradient Norm value is quite far off from zero. Also, the
imaginary frequency becomes much higher than -818 cm-1 (Shown in
Table 8). With these two pieces of information, we can deduce that the frozen
coordinate(s) affect(s) the force constant of the transition state which does
not give a good optimisation of transition structure. With that, after applying
the frozen coordinate to the molecule, the molecule was reoptimised again with
a degree of '''Derivative '''to the '''Bond'''. Details of the reoptimisation
were presented in Table 10.
{| class="wikitable"
! rowspan="2" |
! colspan="2" rowspan="2" |Molecule
! rowspan="2" |Point Group
! rowspan="2" |Energy/ Hartree
! rowspan="2" |RMS Gradient Norm/ Hartree
! colspan="4" |Bond length between/ Å
! rowspan="2" |Imaginary Frequency/ cm-1
! rowspan="2" |
{| class="MsoTableGrid"

|
<nowiki> </nowiki>IR Spectrum

|}
|-
!C14-C6
!C11-C3
!C14-C9 and C6-C1
!C9-C11 and C1-C3
|-
|Optimised Chair Transition
Structure with a degree of '''Derivative'''
to the '''Bond'''
|JSMOL
|[[File:Yll11320.jpg|thumb|'''Figure 20. '''The optimised chair transition structure with a degree of '''Derivative''' to the '''Bond''' and atoms labelling]]
|C2h
|<nowiki>-231.61932233</nowiki>
|0.00002127
|2.02075
|2.02071
|1.38929
|1.38930
|<nowiki>-818</nowiki>
|[[File:Yll11321.jpg|thumb|'''Figure 21. '''IR spectrum of the optimised chair transition structure with a degree of '''Derivative''' to the '''Bond''']]
|}
'''Table 10. '''Summary of the reoptimisation of the chair transition
structure with a degree of '''Derivative''' to the '''Bond'''

Now, in Table 10, the RMS
Gradient Norm value is close to zero. Also, the imaginary frequency goes back
to -818 cm-1, indicating that the coordinates no longer be frozen
and the stretching/bending mode of the transition state is able to undergo
freely.

Comparing the bond lengths
in Table 7 and 10, we can see that the difference between bond lengths of the
single bond to be made/ broken calculated in two methods is just less than
0.0006 Å. And also, there is no difference in bond length of the double bond to be make/broken ‘inside’ the system. This tells us that the two optimisation
methods are rather similar under the consideration on the Cope Rearrangement
Reaction.

====== Boat Transition State ======
Instead of using the frozen
coordinate method as for the chair transition state above, another method, '''QST2''', was applied to the boat
transition state under the '''HF/3-21G'''
basis. In order to use this method, without any ‘Link died’, the reactant and
product have to be drawn and labelled carefully. '''QST2''' is a method which interpolates the reactant and product to
give a transition state. Therefore, it will fall if the structure of the
reactant and product are not close to the transition state. And therefore, all
molecules have to be carefully labelled and adjusted.

[[File:Yll11322.png|500px|center]]

'''Figure 22. '''The drawings and adjustments of angles of the reactant (left)
and product (right) for '''QST2''' Method,
i.e. the central C-C-C-C dihedral angle was changed to 0o and inside
C-C-C were reduced to 100o

After the adjustment, the job was run and the optimized molecule converge to the boat transition structure. Summary was shown in Table 11.
{| class="wikitable"
! rowspan="2" |
! colspan="2" rowspan="2" |Molecule
! rowspan="2" |Point Group
! rowspan="2" |Energy/Hartree
! rowspan="2" |RMS Gradient Norm/ Hartree
! colspan="3" |Bond length between/ Å
! rowspan="2" |Imaginary Frequency/ cm-1
! rowspan="2" |IR spectrum
|-
!C1-C6
!C3-C4
!C5-C6, C4-C5, C3-C2 and C1-C2
|-
|Boat transition structure
under '''QST2''' method
|JSMOL
|[[File:Yll11323.jpg|thumb|'''Figure 23. '''The optimised Boat transition structure with atom labelling]]
|C2v
|<nowiki>-231.60280241</nowiki>
|0.00002436
|2.13994
|2.14019
|1.38149
|<nowiki>-840</nowiki>
|[[File:Yll11324.jpg|thumb|'''Figure 24. '''IR spectrum of the optimised boat transition structure]]
|}
'''Table 11. '''Summary of the boat transition structure under '''QST2 '''method

====== Intrinsic Reaction Coordinate''' '''('''IRC''') ======
In order to confirm that our transition state is of the
correct one, '''Intrinsic Reaction
Coordinate '''('''IRC''') will be carried
out.

As mentioned above, transition state is the first order
saddle point of the reaction pathway. Therefore, it will start to go to the
product or back to the reactant with it falls off. It resembles that a ball is
at the tip of the mountain, which starts to roll off the mountain on the side
with the steepest slope. Also, when we are doing '''IRC''', we can determine whether the reaction goes forward, backward
or both sides. Also, the number of points, which means the number of little
steps that the geometry of the molecule changes, can be adjusted. A low number
of points will just give us a very rough idea that tell us a little bit about
our transition state. Also, the last point on the '''IRC''' is far from the minimum geometry. A high number of points gives
us more accurate results, however two problems could be raised. First, the time
for calculation will be long and Most importantly, as it goes down the slope
and reaches the minimum geometry, i.e. the plateau of energy, the slope will
become very small or even zero again. However, as the energy difference of the
next or previous geometry compared to the geometry of itself is too small, ''GaussView 5.0'' may not able to know which
direction the molecule should proceed to. And this, therefore, results in ‘Link
died’. Therefore, the most common technique is to have a good number of points,
then take the last point on the IRC and run it with a normal optimisation.

Here, as we know that the
Cope Rearrangement has a symmetric reaction pathway, taking the chair
transition structure, we will run '''IRC'''
on it with 50 points.
{| class="wikitable"
![[File:Yll113hlj29825.jpg|thumb|'''Figure 25. '''Total energy along '''IRC '''of the chair transition structure]]
![[File:Yll11326.jpg|thumb|'''Figure 26. '''RMS Gradient Norm of '''IRC '''of the chair transition structure]]
!JSMOL
|}

[[File:Yll11327.jpg|500px]]

'''Figure 27. '''The product of the Cope Rearrangement after optimisation

The first point on Figure 25 is -231.61932233 Hartree and the last point is -231.69157881 Hartree. Then, we take the last point and optimise it, we get the structure shown in Figure 27.

The structure is of C2
symmetry and the energy calculated is -231.69166702 Hartree. This matches with
Gauche2 C2 on Appendix 1. And therefore, this is how the conformer
of 1,5-hexadiene connects with the chair transition structure.

====== Activation Energy of the Cope Rearrangement ======
Finally, we optimise the chair and boat transition states we got from above, reoptimise it with job Opt + Freq
under a more advanced calculation '''B3LYP/6-31G*'''. And from that, the thermochemistry data were given and we can know the
activation energy of the reaction by comparing to Table 3, which anti2 is used
as a local minimum rather than gauche3 as a global minimum.
{| border="1" cellspacing="1" cellpadding="10"
|-
| rowspan="3" | ''' '''
! colspan="3" |'''HF/3-21G'''
! colspan="3" |'''B3LYP/6-31G*'''
|-
| rowspan="2" width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| rowspan="2" width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619323
| align="center" | -231.466698
| align="center" | -231.461339
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409009
|-
| '''Boat TS'''
| align="center" | -231.602803
| align="center" | -231.450932
| align="center" | -231.445303
| align="center" | -234.543094
| align="center" | -234.402342
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692534
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.611703
| align="center" | -234.469213
| align="center" | -234.461857
|}
'''Table 11'''. Summary of energies of chair, boat and reactant (anti2) structure
{| border="1" cellspacing="1" cellpadding="10"
|-
| rowspan="2" | ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.69
| align="center" | 34.06
| align="center" | 33.16
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.96
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
'''Table 12'''. Summary of activation energies in kcal/mol

==== The Diels-Alder Cycloadditions ====

===== Ethylene and Cis-Butadiene =====
First, the structures of the reactants, ethylene and cis-butadiene, were '''optimised to minimum''' under '''semiempirical AM1''' method. For the butadiene, in order to be in the cis conformer, the dihedral angle was adjusted to be 0o. Details are listed in Table 13.
{| class="wikitable"
!
!Molecule
!Point Group
!HOMO
!LUMO
!Energy/Hartree
!RMS Gradient Norm/ Hartree
!IR spectrum
|-
|Ethylene
|JSMOL
|D2h
|[[File:Yll11331.jpg|thumb|'''Figure 31. '''HOMO of Ethylene]]
|[[File:Yll11330.jpg|thumb|'''Figure 30.''' LUMO of ethylene]]
|0.02619029
|0.00008755
|[[File:Yll11328.jpg|thumb|'''Figure 28. '''IR spectrum of Ethylene]]
|-
|Cis-Butadiene
|JSMOL
|C2v
|[[File:Yll11332.jpg|thumb|'''Figure 32. '''HOMO of cis-butadiene]]
|[[File:Yll11333.jpg|thumb|'''Figure 33. '''LUMO of cis-butadiene]]
|0.04878534
|0.00000087
|[[File:Yll11329.jpg|thumb|'''Figure 29.''' IR spectrum of cis-butadiene]]
|}
'''Table 13.''' Summary of optimised ethylene and cis-butadiene

Looking into Figure 30-33, as we know that the plane is perpendicular to the molecule, the HOMO of Ethylene is symmetric while that of LUMO is antisymmetric.

Also, the HOMO of cis-butadiene is antisymmetric and that of LUMO is symmetric.

Then, the transition state of the reaction was able to constructed using the optimised structure of the reactants made above. By making the bond lengths of the bonds to be made in Diels-Alder [4+2] cycloaddition to 2.20000 Å in the transition state, which is the 'dash bond', the transition structure was optimized to '''Opt + Freq minimisation to''' '''TS (Berny)''' under the same method of calculation - '''semiempitical AM1'''. Details are listed in Table 14.
{| class="wikitable"
!
! colspan="2" |Molecule
!Point Group
!HOMO
!LUMO
!Energy/Hartree
!RMS Gradient Norm/ Hartree
!Imaginary Frequency/ cm-1
!IR spectrum
|-
|Transition state
|JSMOL
|[[File:Yll11334.jpg|thumb|'''Figure 34. '''Transition state of ethylene + cis-butadiene]]
|C1
|[[File:Yll11336.jpg|thumb|'''Figure 36. '''HOMO of transition state]]
|[[File:Yll11337.jpg|thumb|'''Figure 37. '''LUMO of transition state]]
|0.11165467
|0.00002792
|<nowiki>-956</nowiki>
|[[File:Yll11335.jpg|thumb|'''Figure 35. '''IR spectrum of the transition state]]
|}
'''Table 14.''' Summary of optimised transition state

From Figure 36, we can see that the HOMO of the transition state is antisymmetric whilst the LUMO of the transition state is symmetric. By making very careful comparison between Figure 36, Figure 37 and Figure 30-33, we can see that the HOMO of the transition state in Figure 36 is a combination of Figure 32 and 30; the LUMO of the transition state in Figure 37 is a combination of Figure 31 and 33. We can clearly see that the HOMO and LUMO of the transition state have a complementary combination of HOMO and LUMO of the reactants.

Taking a closer look to HOMO of the transition state. Recalling Woodward Hoffmann’s Rule, (4q+2)s+(4r)a = odd for thermally allowed reaction, we have both π2s and π4s. Therefore, the reaction is thermally allowed by letting q = 0, which gives the value of 1 which is odd.

Furthermore, from Table 14, we notice that there is an imaginary frequency reported at -956 cm-1. As explained above, the transition state should have one imaginary frequency to account for the negative force constant. With that, this imaginary frequency confirms that the transition structure we postulated from the optimised reactants is valid, i.e. it is really a transition state. The animation of where the imaginary frequency originates from, which shows the motion of the transition state - how the two reactants approach to each other and bonds are formed, is shown below.

JSMOL

From the above figure, we can see that the bond formation from the reactant to the product happens at the same time, i.e. synchronous, on both sides of the transition structure. Therefore, we can say that this Diels-Alder cycloaddition is a concerted [4+2] pericyclic cycloaddition, which matches with what we learnt in Pericyclic Reaction course.

On top of that, the geometry of the transition structure was investigated by looking into the optimised bond lengths between carbon atoms Details are shown in Figure 38 and Table 15.[[File:Yll11338.jpg|thumb|'''Figure 38. '''Transition state of ethylene + cis-butadiene with atoms labelled]]
{| class="wikitable"
!
!Bond length/ Å
|-
|C7-C9
|2.11938
|-
|C12-C5
|2.11944
|-
|C12-C9
|1.38284
|-
|C3-C7
|1.38187
|-
|C3-C1
|1.39750
|-
|C5-C1
|1.38175
|}
'''Table 15. '''Geometry analysis of the transition state

According to the literature <ref>M. A. Fox and J. K. Whitesell, ''Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen'', Springer, 1995</ref>, C-C carbon-carbon single bond is 1.54 Å, and C=C carbon-carbon double bond is 1.34 Å. Also, the Van der Waals radius of carbon is 1.70 Å,<ref>A. Bondi,(1964), ''J. Phys. Chem.'', 1964, '''68''' (3), 441</ref>
According to the reaction scheme shown in Figure 3, a single bond is forming between C7 and C9, also another single bond is forming between C12-C5. Comparing the data in Table 15 with the literature, we can see that the bond length of two bonds to be made is longer than C-C, but shorter than the twice of carbon's Van der Waals radius. This tells us some hints that the terminal carbons between two reactants are attracting with each other to a certain degree. Furthermore, taking a look to other carbon-carbon bonds within the system, we notice that they are somewhere between C-C and C=C. This tells us that either C-C is becoming C=C or C=C is becoming C-C in the transition state.
After that, the '''IRC''' of the above optimised transition state was carried out with both direction and force constant calculated always for 50 points to see the reaction profile.
{| class="wikitable"
|[[File:Yll11339.jpg|thumb|'''Figure 39.''' IRC of the transition state of ethylene + cis-butadiene]]
|[[File:Yll11340.jpg|thumb|'''Figure 40. '''RMS Gradient Norm of transition state of ethylene + cis-butadiene]]
|JSMOL
|}
In Figure 39, we can clearly see that the reactants was first passed through the energy barrier to get the transition state and it went down the slope to give the product.
Finally, the activation energy for this reaction was calculated in Table 16.
{| class="wikitable"
!
!Ethylene
!Cis-butadiene
!Transition state
!Activation Energy
|-
|Energy/Hartree
|0.02619029
|0.04878534
|0.11165467
|0.03667904
(23.02 kcal/mol)
|}
'''Table 16. '''Activation energy analysis of Diels-Alder Reaction between ethylene and cis-butadiene
===== Cyclohexa-1,3-diene and Maleic Anhydride =====
First, the structures of
the reactants, ethylene and cis-butadiene, were '''optimised to minimum''' under '''semiempirical AM1''' method. Details are listed in Table 17.
{| class="wikitable"
!Optimised Structure
!Molecule
!Point Group
!Energy/Hartree
|-
|Cyclohexa-1,3-diene
|[[File:Yll113Cycloopt.jpg|thumb|'''Figure 41. '''Optimised structure of cyclohexa-1,3-diene]]
|C2
|0.02771130
|-
|Maleic Anhydride
|[[File:Yll113Maopt.jpg|thumb|'''Figure 42. '''Optimised structure of Maleic Anhydride]]
|Cs
|<nowiki>-0.12182418</nowiki>
|}
'''Table 17. '''Summary
of the optimized reactant

Then, the two transition states, exo and endo, of the reaction were able to construct. By making the bond lengths of the bonds to be made in Diels-Alder [4+2] cycloaddition to 2.20000 Å in the transition state, which is the 'dash bond', the transition structure was optimized to '''Opt + Freq minimisation to''' '''TS (Berny)''' under the same method of calculation - '''semiempitical AM1'''. Details are listed in Table 18.
{| class="wikitable"
!Transition Structure
!Molecule
!Point Group
!HOMO
!LUMO
!Energy/Hartree
!'''RMS Gradient Norm/ Hartree'''
!'''Imaginary Frequency/ cm-1'''
!IR spectrum
|-
|Exo
|[[File:Yll113Exots.jpg|thumb|'''Figure 43.''' Optimised Structure of Exo Transition State]]
|C1
|[[File:Yll113Exo homo.jpg|thumb|'''Figure 45. '''HOMO of Exo Transition State]]
|[[File:Yll113Exo lumo.jpg|thumb|'''Figure 47.''' LUMO of Exo Transition State]]
|<nowiki>-0.05041984</nowiki>
|0.00000883
|<nowiki>-812</nowiki>
|[[File:Yll113Exo ir.jpg|thumb|'''Figure 49. '''Exo Transition State IR spectrum]]
|-
|Endo
|[[File:Yll113Endots.jpg|thumb|'''Figure 44. '''Optimised Structure of Enfo Transition State]]
|C1
|[[File:Yll113Endo homo.jpg|thumb|'''Figure 46'''. HOMO of Endo Transition State]]
|[[File:Yll113Endo lumo.jpg|thumb|'''Figure 48.''' LUMO of Endo Transition States]]
|<nowiki>-0.05150469</nowiki>
|0.00004308
|<nowiki>-807</nowiki>
|[[File:Yll113Endo ir.jpg|thumb|'''Figure 50.''' Endo Transition State IR Spectrum]]
|}
'''Table 18. '''Summary of the optimized transition structures

In Table 18, we can see that both HOMO and LUMO of the exo
transition state are antisymmetric with respect to the plane. Also, both HOMO and LUMO of the endo transition state are antisymmetric as well.

Also, we notice that the energy of exo is higher than that of endo. This can be explained by the poorer overlap between the C=C π and C=O π* compared to that of endo. This is called secondary orbital effect, which will be further discussed below.

There is one imaginary frequency in both transition structure. This confirms that both of them are valid postulated transition structures.
{| class="wikitable"
|JSMOL
'''Imaginary frequency of exo transition state animation'''
|JSMOL
'''Imaginary frequency of exo transition state animation'''
|}
Again, from the above animations, we can see that the bonds are formed in a synchronous fashion, which means the Diels-Alder Cycloaddition reaction is concerted.

Furthermore, the geometries of both transition states were examined carefully in Table 19.
{| class="wikitable"
!Geometry summary of Exo Transition State (Please refer to Figure 43 for atom labelling)
!Geometry summary of Endo Transition State (Please refer to Figure 44 for atom labelling)
|-
|
{| class="wikitable"
!
!Bond length/ Å
!
!Bond length/ Å
|-
|C5-C13
|2.17032
|C6-C14
|2.17035
|-
|C5-C3
|1.39440
|C3-C1
|1.39674
|-
|C1-C6
|1.39440
|C14-C13
|1.4612
|-
|C7-C6
|1.48976
|C7-C10
|1.52208
|-
|C10-C5
|1.48976
|C13-C15
|1.48822
|-
|C14-C16
|1.48822
| colspan="2" |N/A
|-
|C7-C16
(Through space)
|2.94507
|C10-C15
(Through space)
|2.94484
|-
|C1-C16
(Through space)
|3.78172
|C3-C15
(Through Space)
|3.78155
|}
|
{| class="wikitable"
!
!Bond length/ Å
!
!Bond length/ Å
|-
|C15-C7
|2.16230
|C16-C5
|2.16229
|-
|C1-C3
|1.39726
|C3-C7
|1.39296
|-
|C1-C5
|1.39308
|C7-C9
|1.49503
|-
|C9-C12
|1.52300
|C5-C12
|1.49054
|-
|C16-C18
|1.48918
|C15-C17
|1.48903
|-
|C15-C16
|1.40863
| colspan="2" |N/A
|-
|C1-C18
(Through space)
|2.89232
|C3-C17
(Through space)
|2.89203
|}
|}
'''Table 19.''' Geometry analysis of exo and endo transition states

According to the reaction scheme shown in Figure 4, a single bond is forming between C5 and C13, also another single bond is forming between C6-C14 for exo; C15 and C7 plus C16 and C5 for endo, which is what the first row in the two tables in the left and right in Table 19 shows. the single bond to be made Comparing these values with literature, we find that they are longer than C-C but shorter than twice of carbon's Van der Waals' radius. This tells us some hints that these pairs of carbons between two reactants are attracting with each other to a certain degree. Furthermore, taking a look to other carbon-carbon bonds within the system, i.e. except row 1 and those labelled with (through space), we notice that they are somewhere between C-C and C=C. This tells us that either C-C is becoming C=C or C=C is becoming C-C in the transition state.

Now, looking at the through space bond length. In the exo transition structure, the through space distance -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH2-CH2- is smaller than the twice of the Van der Waals radius but larger than C-C which gives us a hint that they might interact. In the endo transition structure, the through space distance -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH=CH- is smaller than the twice of the Van der Waals radius but larger than C-C which gives us a hint that they might interact. However, according to the definition of secondary orbital effect, it is looking for the interaction between the C=C π of the diene and C=O π* of the dienophile. Endo clearly shows that as explained, but exo seems to just demonstrate the sterics clash between the -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH2-CH2- fragment of diene. In order to further confirm that exo has no secondary orbital effect, a measurement of bond length was carried out between -(C=O)-O-(C=O)- fragment of the maleic anhydride and the -CH=CH- in diene in the exo transition state. The result was shown in the last row on the left table in Table 19. This shows that they are too far away which means they are not possible to interact.

Now, looking back to the HOMO of exo and endo transition states in Figure 45 and 46 respectively. We can definitely see that the overlap between the two reactants is relatively smaller in exo. From these two pieces of information, we can conclude that the endo is kinetically controlled, while exo is thermodynamically controlled.

After that, the '''IRC''' of the both optimised transition state was carried out with both direction and force constant calculated always for the reaction profile. 21 points were used for exo transition states and 24 for endo (reasons explained under '''Introduction)''' to see the reaction profiles.
{| class="wikitable"
! colspan="3" |Exo Transition State
! colspan="3" |Endo Transition State
|-
|[[File:Yll113Exo irc.jpg|thumb|'''Figure 51.''' IRC of the exo transtion state]]
|[[File:Yll113Exo rms.jpg|thumb|'''Figure 52. '''RMS of the exo transition structure]]
|JSMOL
|[[File:Yll113Endo irc.jpg|thumb|'''Figure 53. '''IRC of the endo transition state]]
|[[File:Yll113Endo rms.jpg|thumb|'''Figure 54.''' RMS of the endo transition state]]
|JSMOL
|}
And eventually, the activation energies of the reaction via different transition structures were summarised in Table 20.
{| class="wikitable"
!
!Cyclohexa-1,3-diene
!Maleic Anhydride
!Endo Transition State
!ExoTransition State
!Activation Energy via endo
!Activation Energy via exo
|-
|Energy/Hartree
|0.02771130
|<nowiki>-0.12182418</nowiki>
|<nowiki>-0.05150469</nowiki>
|<nowiki>-0.05041984</nowiki>
|0.04260819
(26.74 kcal/mol)
|0.04369304
(27.42 kcal/mol)
|}
'''Table 20.''' Activation energy analysis

=== '''Conclusion''' ===

=== '''Reference''' ===

<gallery>
File:
</gallery>

Rep:Mod:hlj298

2015-12-17T07:27:15Z

Yll113: /* Cyclohexa-1,3-diene and Maleic Anhydride */

== '''Study of the reaction profiles of the Cope Rearrangement and the Diels-Alder Cycloadditions''' ==
'''Y. L. J. Lam'''

''Department of Chemistry, South Kensington Campus, Imperial College London, London SW7 2AZ, United Kingdom''

''Received 18 December, 2015''

=== Abstract ===
The reactants, products and transition states of the Cope
Rearrangement of 1,5-hexadiene were investigated by ''GaussView 5.0'' at the '''HF/3-21G''' and '''B3LYP/6-31G*'''levels
of theories respectively. With that, the point groups, vibrational frequencies and different energies at different temperatures of the reactants, products and transition states were calculated. Also, by optimizing the transition structures with different methods, i.e. computing the force constants at the
beginning of the calculations, using the redundant coordinate editor and '''QST2''', at the '''HF/3-21G''' level of theory, closer views of the geometries of the transition states can be observed. Furthermore, by using the '''IRC''' method, the reaction profiles can be
obtained and the activation energies can therefore be calculated. Plus, using '''IRC''' method, all reaction intermediates
can now be observed, which helps us to understand the mechanism of the Cope Rearrangement. Similarly, for Diels-Alder Cycloadditions between ethene and
butadiene and Diels-Alder Cycloadditions between cyclohexa-1,3-diene and maleic anhydride, the reactants, products and transition states were optimized and
their respective energies were calculated at '''AM1 semi-empirical molecular orbital method'''. Furthermore, the symmetries of the
molecular orbitals were visualized and the reaction profiles calculated by '''IRC''' method were obtained.

=== '''Introduction''' ===
Chemical reactions are happening around the world in every second. Some reactions are fast, whilst some are slow. The most common and general reason for that is on the kinetic and thermodynamic aspects. On the kinetic aspect, we might argue that the energy barrier(s) form the reactant(s) to the product(s) is/are huge, and therefore, the reactant(s) cannot overcome the barrier(s) and the reaction is slow or does not proceed. The transformation between crude carbon and diamond is a good example. The energy difference between crude carbon and diamond is just few kcal/mol, however, the energy barrier for the transformation is huge. Hence, the transformation is extremely slow, or even does not proceed. With that, diamond symbolizes eternity. On the other hand, on the thermodynamic aspect, we might argue that the reaction is endothermic, i.e. absorbing/requiring heat from the surroundings in order to proceed. In fact, these two aspects just provide us with a little bit of the story and therefore, chemists, or scientists in general, study the mechanism of the reactions to find out the full story. Unfortunately, some reactions are spontaneous, such as the thiocyanation of the iron complex. Also, some intermediates of the reactions are unstable, which cannot be separated or detected even using very advanced analytical instruments, such as nuclear magnetic resonance (NMR) spectromenter. Therefore, scientists devised some programs and computational methods to find out the mechanism of the reactions. Here we use ''GaussView 5.0'' for our investigation.

==== Computational Theory ====
[[File:Yll113 AM1 and HF.jpg|thumb|463x463px|'''Figure 1.''' HOMO and LUMO (highlighted in yellow) of cis-butadiene under the basis of calculation '''HF/3-21G '''(left) and '''AM1''' (right)]]
In ''GaussView 5.0'', there are numerous methods for calculation, such as '''HF/3-21G''', '''B3LYP/6-31G*''', '''semiempirical AM1''', '''MP4 '''and '''MP2'''. Here, the first two calculation method, namely, '''HF/3-21G''' and '''B3LYP/6-31G* '''were applied for calculation of the Cope Rearrangement Reaction, while '''semiempirical AM1''' was used for the investigation of the two Diels-Alder reactions.

N.B. No matter which method applied, the RMS Gradient Norm in hartress would also be computed. This is a measure of how well does the optimisation go during the calculation of the
structure drawn. The closer to zero, the better the structure is optimised.

===== Hartree-Fock ('''HF''') Method =====
Hartree-Fock theory ('''HF''') is the fundamentals of electronic structure theory. It gives a good starting point for more elaborate theoretical methods which can approximate the electronic Schrödinger equation better. It is the basis of the molecular orbital (MO) theory that assumes the motion of each electron can be described by a single-particle function/orbital and it does not depend on/interact with the instantaneous motions of the other electrons.<ref>C. D. Sherrill, ''An Introduction to Hartree-Fock Molecular Orbital Theory'', 2000</ref>

===== Becke, 3-parameter, Lee-Yeang-Parr ('''B3LYP''') Method =====
Beeke, 3-parameter, Lee-Yang-Parr ('''B3LYP''') is one of the most commonly used hybrid functionals. Hybrid functionals are a class of approximation of the exchange-correlation energy functional in density functional theory.<ref>What is B3LYP?, https://www.quora.com/What-is-B3LYP (accessed December 2015)</ref> '''B3LYP''' contains an '''HF''' exchange with the weight of 0.2, which can be regarded as a uniform screening of
exchange by 80 %.<ref>C. H. Patterson, ''International Journal of Quantum Chemistry'', 2006, '''106 '''(15), 3383</ref> '''B3LYP''' also takes a set of atomization
and ionization energies, proton affinities and total atomic energies into account.<ref>A. D. Becke, ''The Journal of Chemical Physics'', 1993, '''98''', 5648</ref>

===== Semiempirical Austin Model 1 ('''AM1''') =====
Semiempirical Austin Model 1 ('''AM1''') based on the Neglect of Differential Diatomic Overlap (NDDO) integral approximation.<ref>M.
J. S. Dewar, E. G. Zoebisch, E. F. Healy and J. J. P. Stewart, ''J. Am. Chem. Soc.'', 1985, '''107''' (13), 3902</ref> Therefore, when taking the same molecule for '''AM1''' and '''HF/3-21G''', you would find that the numbers of HOMO and LUMO are different, which '''AM1''' gives smaller numbers as shown in Figure 1. This is due to the neglect of the low-lying orbitals overlapping. With that, '''AM1''' proceeds much faster than '''HF/3-21G''' for the sake of time.

==== The Cope Rearrangement ====
The Cope Rearrangement is an organic reaction involving [3,3]-sigmatropic rearrangement of 1,5-dienes, which resembles the Claisen Rearrangement.<ref>A. C. Cope and E. M. Hardy, ''J. Am. Chem. Soc.'', 1940, '''62''' (2), 441</ref> The mechanism of the Rearrangement has sparked a controversy – whether it is concerted, dissociative or stepwise.<ref>O. Wiest, K. A. Black and K. N. Houk, ''J. Am. Chem. Soc.'', 1994, '''116''', 10336</ref> With that, first, each conformer of the reactant, 1,5-hexadiene, was optimised by '''HF/3-21G'''. The lowest energy conformer of 1,5-hexadiene was found. Then, as we know, the Rearrangement undergoes either a chair or boat transition state. So, each transition state was optimised by '''HF/3-21G '''as well. By looking into the energy difference between the transition states and the reactant, the activation energy of the Cope Rearrangement with 1,5-hexadiene was found. In order to find the reaction profile and see how the 1,5-diene rearranges, i.e. the mechanism, the transition state was optimised again with
mainly two methods. The coordinate of the chair transition state was first frozen, with the bond to be made set as 2.20000 Å. 2.20000 Å is a good bond length for partially C-C bond as suggested by the chemists’ observations in the literature. <ref>N. H. Kendall, Y. Li and J. D. Evanseck, ''Angew. Chem. Int. Ed. Engl.'', 1992, '''31''' (6), 682</ref> Then, after the optimization of the frozen coordinate, the partly form 2.20000 Å can be relaxed and the structure was then reoptimised. This methods skips the process of computing the whole force constant matrix i.e. Hessian, which saves time and costs. Furthermore, the boat transition state was optimised by '''QST2'''. '''QST2''' has a higher constrains in which requires a more accurate transition state structure to be put in. In this case, the dihedral angle plays an important role in order to be calculated by ''GaussView'' 5.0. Hence, this method is more expensive and time-consuming. From the optimised transition states, an '''IRC''' can be run for the optimised structure to see the full reaction profile. Also, the intermediates of the reaction can be observed. And finally, the reactant and two transition states
were optimised with '''B3LYP/6-31G*''' similarly. Hence, the two calculation methods can be compared by looking into the numbers obtained. Also, the numbers can be compared against the
experimental values. As explained above, '''B3LYP''' takes a more in-depth consideration, the numbers got from this method should be closer to the reality.

==== The Diels-Alder Cycloaddition ====
The Diels-Alder cycloaddition is a [4+2] cycloaddition between a dienophile and a conjugated alkene to give a cyclohexane system. Here, calculations on two Diels-Alder cycloaddition reactions are reported. They are (1) ethylene and butadiene and (2) cyclohexa-1,3-diene and maleic anhydride.

For Diels-Alder cycloaddition reaction, it is well-known that the reaction gives exo and/or endo product. Exo product implies the reaction pathway is thermodynamically controlled to give more stable product; endo product implies
the reaction pathway is kinetically controlled to give a relatively less stable product. In other words, the activation energy to form the exo product is higher than that of endo, however, the endo product is higher in energy than exo. This can usually be explained by the secondary orbital effects. In our cases, both the exo and endo products were investigated undoubtedly. This time, as you may notice, the molecule is more large in size and there are two reactants instead of just one reactant in the Cope Rearrangement, a simpler method of calculation was implemented, which is '''AM1'''. Also, the electronic distributions and orbitals of the HOMO and LUMO of the transition states were computed and visualised.

=== '''Computational Method''' ===
''All calculations were performed by GaussView 5.0. Relevant JSmol files were uploaded here, however, due to some technique glitches, some bonds, especially double bonds, might not come up properly. Yet, the structures of the molecules are generally correctly shown.''

==== The Cope Rearrangement ====
[[File:Yll113 CR.png|thumb|'''Figure 2.''' The Cope Rearrangement of 1,5-hexadiene]]
An anti and gauche conformation of the 1,5-hexadiene were drawn respectively. The drawn structures were first optimised by a not very accurate technique, i.e. '''Clean'''. Then, the '''clean'''ed structure were optimised by '''HF/3-21G'''. The point group and the energy of each conformer were found and compared to locate the low-energy minima. The optimised structures from '''HF/3-21G''' were then reoptimised by '''B3LYP/6-31G*'''. The point group of each conformer was checked and confirmed. Also, the comparison of the same conformer under different calculation method '''HF/3-21G''' and '''B3LYP/6-31G''' was carried out by looking into energy, bond lengths and bond angles. Furthermore, the '''Thermochemistry''' using job type '''Frequency''' was found in both '''HF/3-21G '''and''' B3LYP/6-31G* '''optimised anti conformers.

The boat and chair transition structures were also drawn and '''clean'''ed. The point group of each transition state was found.

Firstly, the chair transition structure was '''optimised to TS (Berny)''' in '''Opt + Freq '''using the basis of '''HF/3-21G'''. Force constant was calculated '''once'''. The frequency of vibration was checked to make sure there is one imaginary vibrational frequency. Then, '''freeze''' '''coordinate''' of the molecule by freezing the carbon-carbon bond to be made as 2.20000 Å. After that, the frozen coordinate was relaxed so the carbon-carbon bond to be made no longer be restricted to 2.20000 Å. The geometry of the transition state was then compared.

Secondly, at the same time, the boat transition structure was optimised by '''QST2''' method by specifying the reactants and products of the reaction under the basis of '''HF/3-21G'''. Labelling the atoms in
the reactant and product, and adjusting the central '''C-C-C-C '''dihedral angle to 0o plus the two inside '''C-C-C''' angles to 100o, the reactant and product could now be optimised by '''QST2''' method.

Comparing the optimised chair and boat transition structures, the connecting conformer of 1,5-hexadiene was found. The reaction energy profile was then calculated by '''IRC''' under '''HF/3-21G''' with 50 points and force constant as always for every small steps. With that, the mechanism of the reaction, as well as the whole reaction energy profile, could be observed clearly. Take the last point on the '''IRC''' and run a normal '''optimisation''' under '''HF/3-21G''' to obtain a minimized geometry.

Eventually, re'''optimise''' the structures of the two transition states with '''Opt + Freq '''under the basis of '''B3LYP/6-31G*'''. The geometries and energies of the transition structure under two different basis were compared. With that, these computed values were also compared against experimental values.

==== The Diels-Alder Cycloadditions ====
[[File:Yll113DA1.jpg|thumb|'''Figure 3. '''The Diels-Alder Cycloadditions between ethylene and butadiene]]

===== Ethylene and butadiene =====
The structure of cis-butadiene was first optimised under the basis of '''semiempirical AM1'''. The HOMO and LUMO of cis butadiene were visualised and its symmetry was determined.

The transition state of the reaction was drawn and optimised under the basis of '''semiempirical AM1'''. The '''IRC''' was carried out in order to confirm the transition structure postulated is of the correct one. Furthermore, the HOMO of the transition structure was visualised and the nodal
planes and properties of the system were interpreted.

===== Cyclohexa-1,3-diene and maleic anhydride =====
[[File:Yll113DA2.jpg|thumb|'''Figure 4. '''The Diels-Alder Cycloadditions between cyclohexa-1,3-diene and maleic anhydride]]
The transition states of the exo and endo products were drawn and optimised under the basis of '''semiempirical AM1'''. The '''IRC''' was carried out in order to confirm the transition structure postulated is of the correct one. For the number of points, 21 points were used for exo transition states and 24 for endo. This is because the energy was too shallow and the slopes tend to zero after the number of points specified above and ''GaussView 5.0'' cannot predict which direction should it goes on to calculate. Furthermore,
the bond lengths, orientation and the HOMO of the transition structures were investigated.

=== '''Results and Discussion''' ===

==== The Cope Rearrangement ====

===== Optimisation of Reactant =====
1,5-hexadiene has three free rotating carbon-carbon bonds. Each of them has three rotational minima. This gives 27 conformations of the 1,5-hexadiene molecule. Yet, only ten of them were energetically distinct due to symmetry and enantiomeric relationships.<ref>B. G. Rocque, J. M. Gonzales and H. F. Schaefer, ''Molecular Physics'', 2002, '''100''' (4), 441</ref> Two of them, the ''Ci anti ''and ''C1 gauche ''structure in here'' ''were drawn and optimizied as shown in Figure A and B and their energies were calculated as shown in Table 1.
{| class="wikitable"
!Conformation
!Molecule
!Point Group
!Energy/ Hartrees*
HF/3-21G
!RMS Gradient Norm/Hartrees*
HF/3-21G
!Relative Energy#/ kcal/mol
!Newman Projections
|-
|Gauche3
|JSMOL
|C1
|<nowiki>-231.69266120</nowiki>
|0.00001556
|0.00
|[[File:Yll113 torsion gauche.jpg|centre|frame|'''Figure 5.''' Newman Projection of C1 gauche3 1,5-hexadiene]]
|-
|Anti2
|JSMOL
|Ci
|<nowiki>-231.69253528</nowiki>
|0.00001891
|0.08
|[[File:Yll113 torsion anti.jpg|centre|frame|'''Figure 6.''' Newman Projection of Ci anti2 1,5-hexadiene]]
|}
<nowiki>*</nowiki>1 hartree = 627.509 kcal/mol

#The difference in energy between the conformer and the lowest energy conformer, in here, which is Gauche3. Then convert Hartree to kcal/mol by *

'''Table 1. '''Conformational analysis of anti2 and gauche3 of 1,5-hexadiene

As shown in Table 1, the energy of Gauche3 is surprisingly lower than the anti2 conformation of 1,5-hexadiene. In most cases, the antiperiplanar conformation of a molecule, such as anti2, is more favourable as it has the least steric clashes. Therefore, usually the antiperiplanar conformation is of the lowest energy. However, here, apart from sterics, the stereoelectroncs concept has also been taken into account. The vinyl proton, in a through space manner, can interact with the π or π* orbital on the sp2 carbon which is separated by four bonds from it.<ref>M. Nishio and M. Hirota, ''Tetrahedron'', 1989, '''45 '''(23), 7201</ref> This is so-called CH-π interaction. The Newman Projection in Figure 5 gives us a closer look on how they are close in space and interact; and the Newman projection in Figure 6 tells us why the vinyl proton cannot interact with the π or π* system through space. Therefore, the gauche3 conformation is more stable than anti2 and of lower energy in 1,5-hexadiene.

Focusing on anti2 conformer of the 1,5-hexadiene, the '''HF/3-21G''' optimised molecule was reoptimised by '''B3LYP/6-31G'''. The energies and the geometries of the anti2 1,5-hexadiene under two basis of calculation method were compared and shown in Table 2.
[[File:Yll113Anti2.png|thumb|'''Figure 7. '''Anti2 conformer of 1,5-hexadiene with labelled atoms]]
{| class="wikitable"
! rowspan="2" |Calculation Method
! rowspan="2" |Point Group
! rowspan="2" |Energy/ Hartrees
! rowspan="2" |RMS Gradient Norm/Hartrees
!Bond length between/ Å
!Bond length between/ Å
!Bond length between/ Å
!Bond angle between
!Bond angle between
|-
|C14-C12 and C4-C1
|C12-C9 and C6-C4
|C6-C9
|C14-C12-C9 and
C1-C4-C6
|C12-C9-C6 and
C4-C6-C9
|-

|'''HF/3-21G'''
|Ci
|<nowiki>-231.69253528</nowiki>
|0.00001891
|1.31613
|1.50891
|1.55275
|124.80579o
|111.34878o
|-
|'''B3LYP/6-31G*'''
|Ci
|<nowiki>-234.61171063</nowiki>
|0.00001249
|1.33350
|1.50419
|1.54816
|125.29968o
|112.67081o
|}
'''Table 2. '''Comparison with the anti2 conformer of 1,5-hexadiene with two calculation method basis ('''HF/3-21G '''and '''B3LYP/6-31G*''')

As seen in Table 2, the point group of the same conformer does not change no matter which method of calculation was carried out. The energy of the conformer with '''B3LYP/6-31G*''' is lower than that of '''HF/3-21G'''. Also, the bond lengths and bond angles changed with a certain amount as shown.

Furthermore, by submitting job type as '''Frequency''' in the optimised structure of anti2 conformer of 1,5-hexadiene under both calculation basis '''HF/3-21G '''and '''B3LYP/6-31G*''', the electronic energy, sum of electronic and zero-point energies at 0 K and sum of electronic and thermal energies at 298.15 K could be determined as shown in Table 3.

{| border="1" cellspacing="1" cellpadding="10"
|-
|rowspan="3"| ''' '''
!colspan="4" |'''HF/3-21G a'''
! colspan="4" |'''B3LYP/6-31G* b'''
|-
! rowspan="2" width="125" align="center" | '''Electronic energy'''
! width="125" align="center" | '''Sum of electronic and zero-point energies'''
! width="125" align="center" | '''Sum of electronic and thermal energies'''
! rowspan="2" width="125" align="center" | '''IR spectrum'''
! rowspan="2" width="125" align="center" | '''Electronic energy'''
! width="125" align="center" | '''Sum of electronic and zero-point energies'''
! width="125" align="center" | '''Sum of electronic and thermal energies'''
! rowspan="2" width="125" align="center" | '''IR spectrum'''
|-
! width="125" align="center" | '''at 0 K'''
! width="125" align="center" | '''at 298.15 K'''
! width="125" align="center" | '''at 0 K'''
! width="125" align="center" | '''at 298.15 K'''
|-
| '''Reactant (anti2)'''
| align="center" | -231.692534
| align="center" |-231.539539
| align="center" | -231.532565
|[[File:Yll113ANTI3-21IR.png|thumb|'''Figure 8. '''IR spectrum of anti2 reactant 1,5-hexadiene calculated by '''HF/3-21G''']]
| align="center" | -234.611703
| align="center" | -234.469213
| align="center" | -234.461857
|[[File:Yll113ANTI6-31IR.png|thumb|'''Figure 9. '''IR spectrum of anti2 reactant 1,5-hexadiene calculated by '''B3LYP/6-31G*''']]
|}

a [https://wiki.ch.ic.ac.uk/wiki/images/5/52/Yll113CR_ANTI_PART4.LOG File]; b [https://wiki.ch.ic.ac.uk/wiki/images/5/54/Yll113_CR_ANTI_PART3.LOG File]

'''Table 3.''' Summary of energies (in hartree) of the reactant (anti2) Comparing Figure 8 and 9, both IR spectra are similar which shows that they have similar stretching/bending mode even under different methods of calculations. However, the strongest peak in Figure 8 is at 1112 cm-1 and that in Figure 9 is 940 cm-1. Yet, they correspond to the same mode of vibration, which is the =C-H bending. Therefore, according to the equation, the wavenumber of absorbance, ν can be calculated:

:<math>\nu = \frac{1}{2 \pi c} \sqrt{\frac{k}{\mu}}</math>

where ''μ'' is the reduced mass, ''k'' is the spring constant for the bond and ''c'' is the light speed,

we can therefore know that the spring constant of the molecule under '''B3LYP/6-31G* '''basis reduced.

Now, focusing on gauche3 conformer of the 1,5-hexadiene, similarly, the '''HF/3-21G''' optimised molecule was reoptimised by '''B3LYP/6-31G'''. The energies and the geometries of the gauche3 1,5-hexadiene under two basis of calculation method were compared and shown in Table 4.
[[File:Yll113Gauche3.png|thumb|'''Figure 10. '''Gauche3 conformer of 1,5-hexadiene with labelled atoms]]
{| class="wikitable"
! rowspan="2" |Calculation Method
! rowspan="2" |Point Group
! rowspan="2" |Energy/ Hartrees
! rowspan="2" |RMS Gradient Norm/Hartrees
! colspan="3" |Bond length between/ Å
! colspan="2" |Bond angle between
|-
|C14-C12 and C4-C1
|C12-C9 and C6-C4
|C6-C9
|C14-C12-C9 and
C1-C4-C6
|C12-C9-C6 and
C4-C6-C9
|-

|'''HF/3-21G'''
|C1
|<nowiki>-231.69266120</nowiki>
|0.00001555
|1.31646
|1.50929
|1.55314
|125.02428o
|111.80728o
|-
|'''B3LYP/6-31G*'''
|C1
|<nowiki>-234.61132605</nowiki>
|0.00000360
|1.33382
|1.50491
|1.55007
|125.49464o
|113.46225o
|}
'''Table 4. '''Comparison with the anti2 conformer of 1,5-hexadiene with two calculation method basis ('''HF/3-21G '''and '''B3LYP/6-31G*''')

As seen in Table 4, the point group of the same conformer, again, does not change no matter which method of calculation was carried out. The energy of the conformer with '''B3LYP/6-31G*''' is lower than that of '''HF/3-21G'''. Also, the bond lengths and bond angles changed with a certain amount as shown.

Furthermore, by submitting job type as '''Frequency''' in the optimised structure of gauche3 conformer of 1,5-hexadiene under both calculation basis '''HF/3-21G '''and '''B3LYP/6-31G*''', the electronic energy, sum of electronic and zero-point energies at 0 K and sum of electronic and thermal energies at 298.15 K could be determined as shown in Table 5.

{| border="1" cellspacing="1" cellpadding="10"
|-
| rowspan="3" | ''' '''
! colspan="4" |'''HF/3-21G c'''
! colspan="4" |'''B3LYP/6-31G* d'''
|-
! rowspan="2" width="125" align="center" | '''Electronic energy'''
! width="125" align="center" | '''Sum of electronic and zero-point energies'''
! width="125" align="center" | '''Sum of electronic and thermal energies'''
! rowspan="2" |IR spectrum
! rowspan="2" width="125" align="center" | '''Electronic energy'''
! width="125" align="center" | '''Sum of electronic and zero-point energies'''
! width="125" align="center" | '''Sum of electronic and thermal energies'''
! rowspan="2" |IR spectrum
|-
! width="125" align="center" | '''at 0 K'''
! width="125" align="center" | '''at 298.15 K'''
! width="125" align="center" | '''at 0 K'''
! width="125" align="center" | '''at 298.15 K'''
|-
| '''Reactant (Gauche 3)'''
| align="center" |-231.692692
| align="center" |-231.539486
| align="center" |-231.532646
|[[File:Yll113GAUCHE3-21IR.png|thumb|'''Figure 11. '''IR spectrum of gauche3 conformer of 1,5-hexadiene calculated by '''HF/3-21G''']]
| align="center" | -234.611326
| align="center" |-234.468719
| align="center" |-234.461477
|[[File:Yll113GAUCHE6-31IR.png|thumb|'''Figure 12.''' IR spectrum of gauche3 conformer of 1,5-hexadiene calculated by '''B3LYP/6-31G*''']]
|}

c [https://wiki.ch.ic.ac.uk/wiki/images/9/98/Yll113CR_GAUGE_PART4.LOG File] ; d [https://wiki.ch.ic.ac.uk/wiki/images/c/ca/Yll113CR_GAUGE_PART3.LOG File]

'''Table 5.''' Summary of energies (in hartree) of the reactant (Gauche3) Comparing Figure 11 and 12, both IR spectra are similar which shows that they have similar stretching/bending mode even under different methods of calculations. However, the strongest peak in Figure 12 is at 939 cm-1 and that in Figure 11 is 1111 cm-1. Yet, they correspond to the same mode of vibration, which is also the =C-H bending. Therefore, similar to the anti2 conformer's case as mentioned above, we can therefore know that the spring constant of the molecule under '''B3LYP/6-31G* '''basis reduced.

===== Optimisation of transition state =====

After optimising the reactants, the chair and boat transition states were optimised accordingly using mainly two different methods. But before that, an allyl fragment (CH2CHCH2) was optimised using '''HF/3-21G''' level of theory for the sake of convenience in constructing the chair and boat transition states. A brief summary was shown in Table 6.
{| class="wikitable"
!
! colspan="2" |Molecule
!Point Group
!Energy/Hartree
!RMS Gradient Norm/ Hartrees
|-
|Allyl fragment
CH2CHCH2
|JSMOL
|[[File:Yll11313.jpg|thumb|'''Figure 13. '''Optimised Structure of the allyl fragment]]
|C2v
|<nowiki>-115.82304010</nowiki>
|0.00002945
|}
'''Table 6. '''Summary of the optimised allyl fragment

Then, both chair and boat transition state were drawn and optimised using the '''optimisation to TS (Berny)''' under '''HF/3-21G'''. Figure 14 and Figure C show the optimized structure of the chair transition state while Figure 15 and Figure D show the optimized structure of the boat transition state. Table 7 shows the summary of results.
{| class="wikitable"
! rowspan="2" |
! colspan="2" rowspan="2" |Molecule
! rowspan="2" |Point Group
! rowspan="2" |Energy/ Hartree
! rowspan="2" |RMS Gradient Norm/Hartrees
! colspan="6" |Partial C-C bond length between/ Å
|-
!C14-C6
!C11-C3
!C14-C9
!C6-C1
!C9-C11
!C1-C3
|-
|Optimised Chair Transition State
|JSMOL
|[[File:Yll113CHAIR3-21.png|thumb|'''Figure 14. '''Optimised Chair Structure by '''HF/3-21G''' with atoms labelled ]]
|C2h
|<nowiki>-231.61932238</nowiki>
|0.00002645
|2.02016
|2.02016
|1.38929
|1.38930
|1.38930
|1.38929
|-
|Optimised Boat Transition State
|JSMOL
|[[File:Yll11315.jpg|thumb|'''Figure 15. '''Optimised Boat Structure by '''HF/3-21G''' with atoms labelled]]
|C2v
|<nowiki>-231.60280235</nowiki>
|0.00003872
|2.14060
|2.14060
|1.38138
|1.38138
|1.38138
|1.38138
|}

'''Table 7. '''Summary of the optimised chair and boat transition states by '''optimisation to TS (Berny) '''under '''HF/3-21G''' basis

Furthermore, the transition structures’ '''Frequencies''' were calculated as shown in Table 8.
{| class="wikitable"
!
!Molecule
!IR spectrum
!Imaginary Frequency/ cm-1
|-
|Boat Transition State
|
|[[File:Yll11317.jpg|thumb|'''Figure 16. '''IR spectrum of the optimised Boat Structure by '''HF/3-21G''']]
|<nowiki>-840</nowiki>
|-
|Chair Transition State
|[[File: Yll113Chair Part 1 imag freq.gif]]
|[[File:Yll11316.jpg|thumb|'''Figure 17. '''IR spectrum of the optimised Chair Structure by '''HF/3-21G''']]
|<nowiki>-818</nowiki>
|}
'''Table 8.''' IR spectra and imaginary frequencies of the boat and chair transition states

As you may notice that, the
imaginary frequency comes up when calculating with the transition states. This
is common, in other words, this should appear to let us know the transition
structure we postulated is correct.

A transition state is the first
order saddle point on the potential energy surface. Therefore, the force
applied to the saddle point against to the displacement. As force and
displacement are vectors, the force constant will be a negative number.Therefore, according to
:<math>\nu = \frac{1}{2 \pi c} \sqrt{\frac{k}{\mu}}</math>

where ''μ'' is the reduced mass, ''k'' is the spring constant for the bond and ''c'' is the light speed,

the square root of a negative
force constant k gives an imaginary wave number/frequency v. In other words,
the appearance of an imaginary frequency tells us that the structure is a
saddle point of the reaction pathway.

The chair transition state
was followed by first 'frozen' then 'relaxed'. The boat transition structure
was followed by '''QST2''' calculation method.

====== Chair Transition State ======
After the above '''optimisation''', the chair transition
state was reoptimised again with another method. This method first freezes the
coordinate of the molecule, in this case, freeze the bond to be made in the
Cope Rearrangement of 1,5-hexadiene as 2.20000 Å. The molecule then optimised with the frozen
coordinate. Details of this optimisation was summarized in Table 9.
{| class="wikitable"
! rowspan="2" |
! colspan="2" rowspan="2" |Molecule
! rowspan="2" |Point Group
! rowspan="2" |
{| class="MsoTableGrid"

|
<nowiki> </nowiki>Energy/ Hartree

|}
! rowspan="2" |RMS Gradient Norm/ Hartree
! colspan="3" |Bond length between/ Å
! rowspan="2" |Imaginary Frequency/ cm-1
! rowspan="2" |IR Spectrum
|-
!C6-C14 and C3-11
!C1-C3 and C9-C14
!C1-C6 and C9-C11
|-
|Optimised Chair Transition Structure with frozen coordinate
|JSMOL
|[[File:Yll11318.jpg|thumb|'''Figure 18. '''The optimised chair transition structure with frozen coordinate and atoms labelling]]
|
{| class="MsoTableGrid"

|
<nowiki> </nowiki>C2h

|}
|<nowiki>-231.61518510</nowiki>
|0.00325573
|2.20000
|1.38135
|1.38128
|<nowiki>-765</nowiki>
|[[File:Yll11319.jpg|thumb|'''Figure 19. '''IR spectrum of the optimised chair transition structure with frozen coordinate]]
|}

'''Table 9. '''Summary of the optimisation of the chair transition structure with
frozen coordinate(s)

From Table 9, we may notice
that the RMS Gradient Norm value is quite far off from zero. Also, the
imaginary frequency becomes much higher than -818 cm-1 (Shown in
Table 8). With these two pieces of information, we can deduce that the frozen
coordinate(s) affect(s) the force constant of the transition state which does
not give a good optimisation of transition structure. With that, after applying
the frozen coordinate to the molecule, the molecule was reoptimised again with
a degree of '''Derivative '''to the '''Bond'''. Details of the reoptimisation
were presented in Table 10.
{| class="wikitable"
! rowspan="2" |
! colspan="2" rowspan="2" |Molecule
! rowspan="2" |Point Group
! rowspan="2" |Energy/ Hartree
! rowspan="2" |RMS Gradient Norm/ Hartree
! colspan="4" |Bond length between/ Å
! rowspan="2" |Imaginary Frequency/ cm-1
! rowspan="2" |
{| class="MsoTableGrid"

|
<nowiki> </nowiki>IR Spectrum

|}
|-
!C14-C6
!C11-C3
!C14-C9 and C6-C1
!C9-C11 and C1-C3
|-
|Optimised Chair Transition
Structure with a degree of '''Derivative'''
to the '''Bond'''
|JSMOL
|[[File:Yll11320.jpg|thumb|'''Figure 20. '''The optimised chair transition structure with a degree of '''Derivative''' to the '''Bond''' and atoms labelling]]
|C2h
|<nowiki>-231.61932233</nowiki>
|0.00002127
|2.02075
|2.02071
|1.38929
|1.38930
|<nowiki>-818</nowiki>
|[[File:Yll11321.jpg|thumb|'''Figure 21. '''IR spectrum of the optimised chair transition structure with a degree of '''Derivative''' to the '''Bond''']]
|}
'''Table 10. '''Summary of the reoptimisation of the chair transition
structure with a degree of '''Derivative''' to the '''Bond'''

Now, in Table 10, the RMS
Gradient Norm value is close to zero. Also, the imaginary frequency goes back
to -818 cm-1, indicating that the coordinates no longer be frozen
and the stretching/bending mode of the transition state is able to undergo
freely.

Comparing the bond lengths
in Table 7 and 10, we can see that the difference between bond lengths of the
single bond to be made/ broken calculated in two methods is just less than
0.0006 Å. And also, there is no difference in bond length of the double bond to be make/broken ‘inside’ the system. This tells us that the two optimisation
methods are rather similar under the consideration on the Cope Rearrangement
Reaction.

====== Boat Transition State ======
Instead of using the frozen
coordinate method as for the chair transition state above, another method, '''QST2''', was applied to the boat
transition state under the '''HF/3-21G'''
basis. In order to use this method, without any ‘Link died’, the reactant and
product have to be drawn and labelled carefully. '''QST2''' is a method which interpolates the reactant and product to
give a transition state. Therefore, it will fall if the structure of the
reactant and product are not close to the transition state. And therefore, all
molecules have to be carefully labelled and adjusted.

[[File:Yll11322.png|500px|center]]

'''Figure 22. '''The drawings and adjustments of angles of the reactant (left)
and product (right) for '''QST2''' Method,
i.e. the central C-C-C-C dihedral angle was changed to 0o and inside
C-C-C were reduced to 100o

After the adjustment, the job was run and the optimized molecule converge to the boat transition structure. Summary was shown in Table 11.
{| class="wikitable"
! rowspan="2" |
! colspan="2" rowspan="2" |Molecule
! rowspan="2" |Point Group
! rowspan="2" |Energy/Hartree
! rowspan="2" |RMS Gradient Norm/ Hartree
! colspan="3" |Bond length between/ Å
! rowspan="2" |Imaginary Frequency/ cm-1
! rowspan="2" |IR spectrum
|-
!C1-C6
!C3-C4
!C5-C6, C4-C5, C3-C2 and C1-C2
|-
|Boat transition structure
under '''QST2''' method
|JSMOL
|[[File:Yll11323.jpg|thumb|'''Figure 23. '''The optimised Boat transition structure with atom labelling]]
|C2v
|<nowiki>-231.60280241</nowiki>
|0.00002436
|2.13994
|2.14019
|1.38149
|<nowiki>-840</nowiki>
|[[File:Yll11324.jpg|thumb|'''Figure 24. '''IR spectrum of the optimised boat transition structure]]
|}
'''Table 11. '''Summary of the boat transition structure under '''QST2 '''method

====== Intrinsic Reaction Coordinate''' '''('''IRC''') ======
In order to confirm that our transition state is of the
correct one, '''Intrinsic Reaction
Coordinate '''('''IRC''') will be carried
out.

As mentioned above, transition state is the first order
saddle point of the reaction pathway. Therefore, it will start to go to the
product or back to the reactant with it falls off. It resembles that a ball is
at the tip of the mountain, which starts to roll off the mountain on the side
with the steepest slope. Also, when we are doing '''IRC''', we can determine whether the reaction goes forward, backward
or both sides. Also, the number of points, which means the number of little
steps that the geometry of the molecule changes, can be adjusted. A low number
of points will just give us a very rough idea that tell us a little bit about
our transition state. Also, the last point on the '''IRC''' is far from the minimum geometry. A high number of points gives
us more accurate results, however two problems could be raised. First, the time
for calculation will be long and Most importantly, as it goes down the slope
and reaches the minimum geometry, i.e. the plateau of energy, the slope will
become very small or even zero again. However, as the energy difference of the
next or previous geometry compared to the geometry of itself is too small, ''GaussView 5.0'' may not able to know which
direction the molecule should proceed to. And this, therefore, results in ‘Link
died’. Therefore, the most common technique is to have a good number of points,
then take the last point on the IRC and run it with a normal optimisation.

Here, as we know that the
Cope Rearrangement has a symmetric reaction pathway, taking the chair
transition structure, we will run '''IRC'''
on it with 50 points.
{| class="wikitable"
![[File:Yll113hlj29825.jpg|thumb|'''Figure 25. '''Total energy along '''IRC '''of the chair transition structure]]
![[File:Yll11326.jpg|thumb|'''Figure 26. '''RMS Gradient Norm of '''IRC '''of the chair transition structure]]
!JSMOL
|}

[[File:Yll11327.jpg|500px]]

'''Figure 27. '''The product of the Cope Rearrangement after optimisation

The first point on Figure 25 is -231.61932233 Hartree and the last point is -231.69157881 Hartree. Then, we take the last point and optimise it, we get the structure shown in Figure 27.

The structure is of C2
symmetry and the energy calculated is -231.69166702 Hartree. This matches with
Gauche2 C2 on Appendix 1. And therefore, this is how the conformer
of 1,5-hexadiene connects with the chair transition structure.

====== Activation Energy of the Cope Rearrangement ======
Finally, we optimise the chair and boat transition states we got from above, reoptimise it with job Opt + Freq
under a more advanced calculation '''B3LYP/6-31G*'''. And from that, the thermochemistry data were given and we can know the
activation energy of the reaction by comparing to Table 3, which anti2 is used
as a local minimum rather than gauche3 as a global minimum.
{| border="1" cellspacing="1" cellpadding="10"
|-
| rowspan="3" | ''' '''
! colspan="3" |'''HF/3-21G'''
! colspan="3" |'''B3LYP/6-31G*'''
|-
| rowspan="2" width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
| rowspan="2" width="125" align="center" | '''Electronic energy'''
| width="125" align="center" | '''Sum of electronic and zero-point energies'''
| width="125" align="center" | '''Sum of electronic and thermal energies'''
|-
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619323
| align="center" | -231.466698
| align="center" | -231.461339
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409009
|-
| '''Boat TS'''
| align="center" | -231.602803
| align="center" | -231.450932
| align="center" | -231.445303
| align="center" | -234.543094
| align="center" | -234.402342
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692534
| align="center" | -231.539539
| align="center" | -231.532565
| align="center" | -234.611703
| align="center" | -234.469213
| align="center" | -234.461857
|}
'''Table 11'''. Summary of energies of chair, boat and reactant (anti2) structure
{| border="1" cellspacing="1" cellpadding="10"
|-
| rowspan="2" | ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE (Chair)'''
| align="center" | 45.71
| align="center" | 44.69
| align="center" | 34.06
| align="center" | 33.16
| align="center" | 33.5 ± 0.5
|-
| '''ΔE (Boat)'''
| align="center" | 55.60
| align="center" | 54.76
| align="center" | 41.96
| align="center" | 41.32
| align="center" | 44.7 ± 2.0
|}
'''Table 12'''. Summary of activation energies in kcal/mol

==== The Diels-Alder Cycloadditions ====

===== Ethylene and Cis-Butadiene =====
First, the structures of the reactants, ethylene and cis-butadiene, were '''optimised to minimum''' under '''semiempirical AM1''' method. For the butadiene, in order to be in the cis conformer, the dihedral angle was adjusted to be 0o. Details are listed in Table 13.
{| class="wikitable"
!
!Molecule
!Point Group
!HOMO
!LUMO
!Energy/Hartree
!RMS Gradient Norm/ Hartree
!IR spectrum
|-
|Ethylene
|JSMOL
|D2h
|[[File:Yll11331.jpg|thumb|'''Figure 31. '''HOMO of Ethylene]]
|[[File:Yll11330.jpg|thumb|'''Figure 30.''' LUMO of ethylene]]
|0.02619029
|0.00008755
|[[File:Yll11328.jpg|thumb|'''Figure 28. '''IR spectrum of Ethylene]]
|-
|Cis-Butadiene
|JSMOL
|C2v
|[[File:Yll11332.jpg|thumb|'''Figure 32. '''HOMO of cis-butadiene]]
|[[File:Yll11333.jpg|thumb|'''Figure 33. '''LUMO of cis-butadiene]]
|0.04878534
|0.00000087
|[[File:Yll11329.jpg|thumb|'''Figure 29.''' IR spectrum of cis-butadiene]]
|}
'''Table 13.''' Summary of optimised ethylene and cis-butadiene

Looking into Figure 30-33, as we know that the plane is perpendicular to the molecule, the HOMO of Ethylene is symmetric while that of LUMO is antisymmetric.

Also, the HOMO of cis-butadiene is antisymmetric and that of LUMO is symmetric.

Then, the transition state of the reaction was able to constructed using the optimised structure of the reactants made above. By making the bond lengths of the bonds to be made in Diels-Alder [4+2] cycloaddition to 2.20000 Å in the transition state, which is the 'dash bond', the transition structure was optimized to '''Opt + Freq minimisation to''' '''TS (Berny)''' under the same method of calculation - '''semiempitical AM1'''. Details are listed in Table 14.
{| class="wikitable"
!
! colspan="2" |Molecule
!Point Group
!HOMO
!LUMO
!Energy/Hartree
!RMS Gradient Norm/ Hartree
!Imaginary Frequency/ cm-1
!IR spectrum
|-
|Transition state
|JSMOL
|[[File:Yll11334.jpg|thumb|'''Figure 34. '''Transition state of ethylene + cis-butadiene]]
|C1
|[[File:Yll11336.jpg|thumb|'''Figure 36. '''HOMO of transition state]]
|[[File:Yll11337.jpg|thumb|'''Figure 37. '''LUMO of transition state]]
|0.11165467
|0.00002792
|<nowiki>-956</nowiki>
|[[File:Yll11335.jpg|thumb|'''Figure 35. '''IR spectrum of the transition state]]
|}
'''Table 14.''' Summary of optimised transition state

From Figure 36, we can see that the HOMO of the transition state is antisymmetric whilst the LUMO of the transition state is symmetric. By making very careful comparison between Figure 36, Figure 37 and Figure 30-33, we can see that the HOMO of the transition state in Figure 36 is a combination of Figure 32 and 30; the LUMO of the transition state in Figure 37 is a combination of Figure 31 and 33. We can clearly see that the HOMO and LUMO of the transition state have a complementary combination of HOMO and LUMO of the reactants.

Taking a closer look to HOMO of the transition state. Recalling Woodward Hoffmann’s Rule, (4q+2)s+(4r)a = odd for thermally allowed reaction, we have both π2s and π4s. Therefore, the reaction is thermally allowed by letting q = 0, which gives the value of 1 which is odd.

Furthermore, from Table 14, we notice that there is an imaginary frequency reported at -956 cm-1. As explained above, the transition state should have one imaginary frequency to account for the negative force constant. With that, this imaginary frequency confirms that the transition structure we postulated from the optimised reactants is valid, i.e. it is really a transition state. The animation of where the imaginary frequency originates from, which shows the motion of the transition state - how the two reactants approach to each other and bonds are formed, is shown below.

JSMOL

From the above figure, we can see that the bond formation from the reactant to the product happens at the same time, i.e. synchronous, on both sides of the transition structure. Therefore, we can say that this Diels-Alder cycloaddition is a concerted [4+2] pericyclic cycloaddition, which matches with what we learnt in Pericyclic Reaction course.

On top of that, the geometry of the transition structure was investigated by looking into the optimised bond lengths between carbon atoms Details are shown in Figure 38 and Table 15.[[File:Yll11338.jpg|thumb|'''Figure 38. '''Transition state of ethylene + cis-butadiene with atoms labelled]]
{| class="wikitable"
!
!Bond length/ Å
|-
|C7-C9
|2.11938
|-
|C12-C5
|2.11944
|-
|C12-C9
|1.38284
|-
|C3-C7
|1.38187
|-
|C3-C1
|1.39750
|-
|C5-C1
|1.38175
|}
'''Table 15. '''Geometry analysis of the transition state

According to the literature <ref>M. A. Fox and J. K. Whitesell, ''Organische Chemie: Grundlagen, Mechanismen, Bioorganische Anwendungen'', Springer, 1995</ref>, C-C carbon-carbon single bond is 1.54 Å, and C=C carbon-carbon double bond is 1.34 Å. Also, the Van der Waals radius of carbon is 1.70 Å,<ref>A. Bondi,(1964), ''J. Phys. Chem.'', 1964, '''68''' (3), 441</ref>
According to the reaction scheme shown in Figure 3, a single bond is forming between C7 and C9, also another single bond is forming between C12-C5. Comparing the data in Table 15 with the literature, we can see that the bond length of two bonds to be made is longer than C-C, but shorter than the twice of carbon's Van der Waals radius. This tells us some hints that the terminal carbons between two reactants are attracting with each other to a certain degree. Furthermore, taking a look to other carbon-carbon bonds within the system, we notice that they are somewhere between C-C and C=C. This tells us that either C-C is becoming C=C or C=C is becoming C-C in the transition state.
After that, the '''IRC''' of the above optimised transition state was carried out with both direction and force constant calculated always for 50 points to see the reaction profile.
{| class="wikitable"
|[[File:Yll11339.jpg|thumb|'''Figure 39.''' IRC of the transition state of ethylene + cis-butadiene]]
|[[File:Yll11340.jpg|thumb|'''Figure 40. '''RMS Gradient Norm of transition state of ethylene + cis-butadiene]]
|JSMOL
|}
In Figure 39, we can clearly see that the reactants was first passed through the energy barrier to get the transition state and it went down the slope to give the product.
Finally, the activation energy for this reaction was calculated in Table 16.
{| class="wikitable"
!
!Ethylene
!Cis-butadiene
!Transition state
!Activation Energy
|-
|Energy/Hartree
|0.02619029
|0.04878534
|0.11165467
|0.03667904
(23.02 kcal/mol)
|}
'''Table 16. '''Activation energy analysis of Diels-Alder Reaction between ethylene and cis-butadiene
===== Cyclohexa-1,3-diene and Maleic Anhydride =====
First, the structures of
the reactants, ethylene and cis-butadiene, were '''optimised to minimum''' under '''semiempirical AM1''' method. Details are listed in Table 17.
{| class="wikitable"
!Optimised Structure
!Molecule
!Point Group
!Energy/Hartree
|-
|Cyclohexa-1,3-diene
|[[File:Yll113Cycloopt.jpg|thumb|'''Figure 41. '''Optimised structure of cyclohexa-1,3-diene]]
|C2
|0.02771130
|-
|Maleic Anhydride
|[[File:Yll113Maopt.jpg|thumb|'''Figure 42. '''Optimised structure of Maleic Anhydride]]
|Cs
|<nowiki>-0.12182418</nowiki>
|}
'''Table 17. '''Summary
of the optimized reactant

Then, the two transition states, exo and endo, of the reaction were able to construct. By making the bond lengths of the bonds to be made in Diels-Alder [4+2] cycloaddition to 2.20000 Å in the transition state, which is the 'dash bond', the transition structure was optimized to '''Opt + Freq minimisation to''' '''TS (Berny)''' under the same method of calculation - '''semiempitical AM1'''. Details are listed in Table 18.
{| class="wikitable"
!Transition Structure
!Molecule
!Point Group
!HOMO
!LUMO
!Energy/Hartree
!'''RMS Gradient Norm/ Hartree'''
!'''Imaginary Frequency/ cm-1'''
!IR spectrum
|-
|Exo
|[[File:Yll113Exots.jpg|thumb|'''Figure 43.''' Optimised Structure of Exo Transition State]]
|C1
|[[File:Yll113Exo homo.jpg|thumb|'''Figure 45. '''HOMO of Exo Transition State]]
|[[File:Yll113Exo lumo.jpg|thumb|'''Figure 47.''' LUMO of Exo Transition State]]
|<nowiki>-0.05041984</nowiki>
|0.00000883
|<nowiki>-812</nowiki>
|[[File:Yll113Exo ir.jpg|thumb|'''Figure 49. '''Exo Transition State IR spectrum]]
|-
|Endo
|[[File:Yll113Endots.jpg|thumb|'''Figure 44. '''Optimised Structure of Enfo Transition State]]
|C1
|[[File:Yll113Endo homo.jpg|thumb|'''Figure 46'''. HOMO of Endo Transition State]]
|[[File:Yll113Endo lumo.jpg|thumb|'''Figure 48.''' LUMO of Endo Transition States]]
|<nowiki>-0.05150469</nowiki>
|0.00004308
|<nowiki>-807</nowiki>
|[[File:Yll113Endo ir.jpg|thumb|'''Figure 50.''' Endo Transition State IR Spectrum]]
|}
'''Table 18. '''Summary of the optimized transition structures

In Table 18, we can see that both HOMO and LUMO of the exo
transition state are antisymmetric with respect to the plane. Also, both HOMO and LUMO of the endo transition state are antisymmetric as well.

Also, we notice that the energy of exo is higher than that of endo. This can be explained by the poorer overlap between the C=C π and C=O π* compared to that of endo. This is called secondary orbital effect, which will be further discussed below.

There is one imaginary frequency in both transition structure. This confirms that both of them are valid postulated transition structures.
{| class="wikitable"
|JSMOL
'''Imaginary frequency of exo transition state animation'''
|JSMOL
'''Imaginary frequency of exo transition state animation'''
|}
Again, from the above animations, we can see that the bonds are formed in a synchronous fashion, which means the Diels-Alder Cycloaddition reaction is concerted.

Furthermore, the geometries of both transition states were examined carefully in Table 19.
{| class="wikitable"
!Geometry summary of Exo Transition State (Please refer to Figure 43 for atom labelling)
!Geometry summary of Endo Transition State (Please refer to Figure 44 for atom labelling)
|-
|
{| class="wikitable"
!
!Bond length/ Å
!
!Bond length/ Å
|-
|C5-C13
|2.17032
|C6-C14
|2.17035
|-
|C5-C3
|1.39440
|C3-C1
|1.39674
|-
|C1-C6
|1.39440
|C14-C13
|1.4612
|-
|C7-C6
|1.48976
|C7-C10
|1.52208
|-
|C10-C5
|1.48976
|C13-C15
|1.48822
|-
|C14-C16
|1.48822
| colspan="2" |N/A
|-
|C7-C16
(Through space)
|2.94507
|C10-C15
(Through space)
|2.94484
|-
|C1-C16
(Through space)
|3.78172
|C3-C15
(Through Space)
|3.78155
|}
|
{| class="wikitable"
!
!Bond length/ Å
!
!Bond length/ Å
|-
|C15-C7
|2.16230
|C16-C5
|2.16229
|-
|C1-C3
|1.39726
|C3-C7
|1.39296
|-
|C1-C5
|1.39308
|C7-C9
|1.49503
|-
|C9-C12
|1.52300
|C5-C12
|1.49054
|-
|C16-C18
|1.48918
|C15-C17
|1.48903
|-
|C15-C16
|1.40863
| colspan="2" |N/A
|-
|C1-C18
(Through space)
|2.89232
|C3-C17
(Through space)
|2.89203
|}
|}
'''Table 19.''' Geometry analysis of exo and endo transition states

According to the reaction scheme shown in Figure 4, a single bond is forming between C5 and C13, also another single bond is forming between C6-C14 for exo; C15 and C7 plus C16 and C5 for endo, which is what the first row in the two tables in the left and right in Table 19 shows. the single bond to be made Comparing these values with literature, we find that they are longer than C-C but shorter than twice of carbon's Van der Waals' radius. This tells us some hints that these pairs of carbons between two reactants are attracting with each other to a certain degree. Furthermore, taking a look to other carbon-carbon bonds within the system, i.e. except row 1 and those labelled with (through space), we notice that they are somewhere between C-C and C=C. This tells us that either C-C is becoming C=C or C=C is becoming C-C in the transition state.

Now, looking at the through space bond length. In the exo transition structure, the through space distance -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH2-CH2- is smaller than the twice of the Van der Waals radius but larger than C-C which gives us a hint that they might interact. In the endo transition structure, the through space distance -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH=CH- is smaller than the twice of the Van der Waals radius but larger than C-C which gives us a hint that they might interact. However, according to the definition of secondary orbital effect, it is looking for the interaction between the C=C π of the diene and C=O π* of the dienophile. Endo clearly shows that as explained, but exo seems to just demonstrate the sterics clash between the -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH2-CH2- fragment of diene. In order to further confirm that exo has no secondary orbital effect, a measurement of bond length was carried out between -(C=O)-O-(C=O)- fragment of the maleic anhydride and the -CH=CH- in diene in the exo transition state. The result was shown in the last row on the left table in Table 19. This shows that they are too far away which means they are not possible to interact.

Now, looking back to the HOMO of exo and endo transition states in Figure 45 and 46 respectively. We can definitely see that the overlap between the two reactants is relatively smaller in exo. From these two pieces of information, we can conclude that the endo is kinetically controlled, while exo is thermodynamically controlled.

After that, the '''IRC''' of the both optimised transition state was carried out with both direction and force constant calculated always for the reaction profile. 21 points were used for exo transition states and 24 for endo (reasons explained under '''Introduction)''' to see the reaction profiles.
{| class="wikitable"
! colspan="3" |Exo Transition State
! colspan="3" |Endo Transition State
|-
|[[File:Yll113Exo irc.jpg|thumb|'''Figure 51.''' IRC of the exo transtion state]]
|[[File:Yll113Exo rms.jpg|thumb|'''Figure 52. '''RMS of the exo transition structure]]
|JSMOL
|[[File:Yll113Endo irc.jpg|thumb|'''Figure 53. '''IRC of the endo transition state]]
|[[File:Yll113Endo rms.jpg|thumb|'''Figure 54.''' RMS of the endo transition state]]
|JSMOL
|}
And eventually, the activation energies of the reaction via different transition structures were summarised in Table 20.
{| class="wikitable"
!
!Cyclohexa-1,3-diene
!Maleic Anhydride
!Endo Transition State
!ExoTransition State
!Activation Energy via endo
!Activation Energy via exo
|-
|Energy/Hartree
|0.02771130
|<nowiki>-0.12182418</nowiki>
|<nowiki>-0.05150469</nowiki>
|<nowiki>-0.05041984</nowiki>
|0.04260819
(26.74 kcal/mol)
|0.04369304
(27.42 kcal/mol)
|}
'''Table 20.''' Activation energy analysis

=== '''Conclusion''' ===

=== '''Reference''' ===

<gallery>
File:
</gallery>

Rep:Mod:testhlj298

2015-12-17T07:25:32Z

Yll113: /* Cyclohexa-1,3-diene and Maleic Anhydride */

===== Cyclohexa-1,3-diene and Maleic Anhydride =====
First, the structures of
the reactants, ethylene and cis-butadiene, were '''optimised to minimum''' under '''semiempirical AM1''' method. Details are listed in Table 17.
{| class="wikitable"
!Optimised Structure
!Molecule
!Point Group
!Energy/Hartree
|-
|Cyclohexa-1,3-diene
|[[File:Yll113Cycloopt.jpg|thumb|'''Figure 41. '''Optimised structure of cyclohexa-1,3-diene]]
|C2
|0.02771130
|-
|Maleic Anhydride
|[[File:Yll113Maopt.jpg|thumb|'''Figure 42. '''Optimised structure of Maleic Anhydride]]
|Cs
|<nowiki>-0.12182418</nowiki>
|}
'''Table 17. '''Summary
of the optimized reactant

Then, the two transition states, exo and endo, of the reaction were able to construct. By making the bond lengths of the bonds to be made in Diels-Alder [4+2] cycloaddition to 2.20000 Å in the transition state, which is the 'dash bond', the transition structure was optimized to '''Opt + Freq minimisation to''' '''TS (Berny)''' under the same method of calculation - '''semiempitical AM1'''. Details are listed in Table 18.
{| class="wikitable"
!Transition Structure
!Molecule
!Point Group
!HOMO
!LUMO
!Energy/Hartree
!'''RMS Gradient Norm/ Hartree'''
!'''Imaginary Frequency/ cm-1'''
!IR spectrum
|-
|Exo
|[[File:Yll113Exots.jpg|thumb|'''Figure 43.''' Optimised Structure of Exo Transition State]]
|C1
|[[File:Yll113Exo homo.jpg|thumb|'''Figure 45. '''HOMO of Exo Transition State]]
|[[File:Yll113Exo lumo.jpg|thumb|'''Figure 47.''' LUMO of Exo Transition State]]
|<nowiki>-0.05041984</nowiki>
|0.00000883
|<nowiki>-812</nowiki>
|[[File:Yll113Exo ir.jpg|thumb|'''Figure 49. '''Exo Transition State IR spectrum]]
|-
|Endo
|[[File:Yll113Endots.jpg|thumb|'''Figure 44. '''Optimised Structure of Enfo Transition State]]
|C1
|[[File:Yll113Endo homo.jpg|thumb|'''Figure 46'''. HOMO of Endo Transition State]]
|[[File:Yll113Endo lumo.jpg|thumb|'''Figure 48.''' LUMO of Endo Transition States]]
|<nowiki>-0.05150469</nowiki>
|0.00004308
|<nowiki>-807</nowiki>
|[[File:Yll113Endo ir.jpg|thumb|'''Figure 50.''' Endo Transition State IR Spectrum]]
|}
'''Table 18. '''Summary of the optimized transition structures

In Table 18, we can see that both HOMO and LUMO of the exo
transition state are antisymmetric with respect to the plane. Also, both HOMO and LUMO of the endo transition state are antisymmetric as well.

Also, we notice that the energy of exo is higher than that of endo. This can be explained by the poorer overlap between the C=C π and C=O π* compared to that of endo. This is called secondary orbital effect, which will be further discussed below.

There is one imaginary frequency in both transition structure. This confirms that both of them are valid postulated transition structures.
{| class="wikitable"
|JSMOL
'''Imaginary frequency of exo transition state animation'''
|JSMOL
'''Imaginary frequency of exo transition state animation'''
|}
Again, from the above animations, we can see that the bonds are formed in a synchronous fashion, which means the Diels-Alder Cycloaddition reaction is concerted.

Furthermore, the geometries of both transition states were examined carefully in Table 19.
{| class="wikitable"
!Geometry summary of Exo Transition State (Please refer to Figure 43 for atom labelling)
!Geometry summary of Endo Transition State (Please refer to Figure 44 for atom labelling)
|-
|
{| class="wikitable"
!
!Bond length/ Å
!
!Bond length/ Å
|-
|C5-C13
|2.17032
|C6-C14
|2.17035
|-
|C5-C3
|1.39440
|C3-C1
|1.39674
|-
|C1-C6
|1.39440
|C14-C13
|1.4612
|-
|C7-C6
|1.48976
|C7-C10
|1.52208
|-
|C10-C5
|1.48976
|C13-C15
|1.48822
|-
|C14-C16
|1.48822
| colspan="2" |N/A
|-
|C7-C16
(Through space)
|2.94507
|C10-C15
(Through space)
|2.94484
|-
|C1-C16
(Through space)
|3.78172
|C3-C15
(Through Space)
|3.78155
|}
|
{| class="wikitable"
!
!Bond length/ Å
!
!Bond length/ Å
|-
|C15-C7
|2.16230
|C16-C5
|2.16229
|-
|C1-C3
|1.39726
|C3-C7
|1.39296
|-
|C1-C5
|1.39308
|C7-C9
|1.49503
|-
|C9-C12
|1.52300
|C5-C12
|1.49054
|-
|C16-C18
|1.48918
|C15-C17
|1.48903
|-
|C15-C16
|1.40863
| colspan="2" |N/A
|-
|C1-C18
(Through space)
|2.89232
|C3-C17
(Through space)
|2.89203
|}
|}
'''Table 19.''' Geometry analysis of exo and endo transition states

According to the reaction scheme shown in Figure 4, a single bond is forming between C5 and C13, also another single bond is forming between C6-C14 for exo; C15 and C7 plus C16 and C5 for endo, which is what the first row in the two tables in the left and right in Table 19 shows. the single bond to be made Comparing these values with literature, we find that they are longer than C-C but shorter than twice of carbon's Van der Waals' radius. This tells us some hints that these pairs of carbons between two reactants are attracting with each other to a certain degree. Furthermore, taking a look to other carbon-carbon bonds within the system, i.e. except row 1 and those labelled with (through space), we notice that they are somewhere between C-C and C=C. This tells us that either C-C is becoming C=C or C=C is becoming C-C in the transition state.

Now, looking at the through space bond length. In the exo transition structure, the through space distance -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH2-CH2- is smaller than the twice of the Van der Waals radius but larger than C-C which gives us a hint that they might interact. In the endo transition structure, the through space distance -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH=CH- is smaller than the twice of the Van der Waals radius but larger than C-C which gives us a hint that they might interact. However, according to the definition of secondary orbital effect, it is looking for the interaction between the C=C π of the diene and C=O π* of the dienophile. Endo clearly shows that as explained, but exo seems to just demonstrate the sterics clash between the -(C=O)-O-(C=O)- fragment of the maleic anhydride and the C atoms of the “opposite” -CH2-CH2- fragment of diene. In order to further confirm that exo has no secondary orbital effect, a measurement of bond length was carried out between -(C=O)-O-(C=O)- fragment of the maleic anhydride and the -CH=CH- in diene in the exo transition state. The result was shown in the last row on the left table in Table 19. This shows that they are too far away which means they are not possible to interact.

Now, looking back to the HOMO of exo and endo transition states in Figure 45 and 46 respectively. We can definitely see that the overlap between the two reactants is relatively smaller in exo. From these two pieces of information, we can conclude that the endo is kinetically controlled, while exo is thermodynamically controlled.

After that, the '''IRC''' of the both optimised transition state was carried out with both direction and force constant calculated always for the reaction profile. 21 points were used for exo transition states and 24 for endo (reasons explained under '''Introduction)''' to see the reaction profiles.
{| class="wikitable"
! colspan="3" |Exo Transition State
! colspan="3" |Endo Transition State
|-
|[[File:Yll113Exo irc.jpg|thumb|'''Figure 51.''' IRC of the exo transtion state]]
|[[File:Yll113Exo rms.jpg|thumb|'''Figure 52. '''RMS of the exo transition structure]]
|JSMOL
|[[File:Yll113Endo irc.jpg|thumb|'''Figure 53. '''IRC of the endo transition state]]
|[[File:Yll113Endo rms.jpg|thumb|'''Figure 54.''' RMS of the endo transition state]]
|JSMOL
|}
And eventually, the activation energies of the reaction via different transition structures were summarised in Table 20.
{| class="wikitable"
!
!Cyclohexa-1,3-diene
!Maleic Anhydride
!Endo Transition State
!ExoTransition State
!Activation Energy via endo
!Activation Energy via exo
|-
|Energy/Hartree
|0.02771130
|<nowiki>-0.12182418</nowiki>
|<nowiki>-0.05150469</nowiki>
|<nowiki>-0.05041984</nowiki>
|0.04260819
(26.74 kcal/mol)
|0.04369304
(27.42 kcal/mol)
|}
'''Table 20.''' Activation energy analysis

Rep:Mod:testhlj298

2015-12-17T07:21:57Z

2015-12-17T05:47:17Z

Yll113: /* Cyclohexa-1,3-diene and Maleic Anhydride */

Rep:Mod:testhlj298

2015-12-17T05:46:35Z

Yll113: /* Cyclohexa-1,3-diene and Maleic Anhydride */