Rep:Mod:SXC13

2013-12-06T15:33:15Z

Xs1711: /* Endo transition structure */

=Transition states and reactivity=
In this experiment, the transition structures on potential surfaces for the Cope rearrangement and Diels-Alder cycloaddition reactions are generated.

==The cope rearrangement of 1,5-hexadiene==

In this part, the favored reaction mechanism is determined by finding out the low-energy minima and transition structures on the 1,5-hexadiene potential energy surface.

The structure of 1,5-hexadiene with an "anti" linkage for the central four carbon atoms was optimized at the '''HF/3-21G''' level of the theory. As shown in '''Figure.1''', the molecule has a symmetry of C<sub>i</sub> and the energy of this structure is calculated to be -231.69253528 Ha. Therefore the result structure is determined to be ''anti2'' by comparing this energy with the structures in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>1,5-hexadiene (Anti)</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>1,5 hexadienegauche.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 1.''' ''1,5-hexadiene (anti2)''</div>

The structure of 1,5-hexadiene with a "gauche" linkage for the central four carbon atoms was then optimized also at the '''HF/3-21G''' level of theory. The calculated energy of this structure is -231.69266120 Ha, which is slightly higher than the anti2 structure. Comparing with [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1], the structure is equivalent to ''gauche3'', for which the point group is C<sub>1</sub>.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>1,5-hexadiene (Anti)</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>1-reoptimized.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 2.''' ''1,5-hexadiene (gauche3)''</div>

The lowest energy conformation of a reactant is used as a reference in the calculations of lowest activation energies and enthalpies. In this case, ''gauche3'' structure, shown in '''Figure. 2''', is the lowest energy conformation, with zero relative energy.

The structure of ''anti2'' was reoptimized at the '''B3LYP/3-21G''' level of theory. The energy is calculated to be -234.55970873 Ha.
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>1,5-hexadiene (Anti)</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>1-reoptimized.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 3.''' ''1,5-hexadiene (anti2 re-optimized)''</div>

The computed energies are shown in '''Table 1.'''

{| class="wikitable" border="1"
| align="center" style="background:#f0f0f0;"|'''Energy'''
| align="center" style="background:#f0f0f0;"|'''Description'''
| align="center" style="background:#f0f0f0;"|'''Relative Energy/Hartrees B3LYP/6-31G'''
|-
| Sum of electronic and zero-point Energies||Potential energy at 0 K including the zero-point vibrational energy (E = E<sub>elec</sub> + ZPE)||-234.416224
|-
| Sum of electronic and thermal Energies||Energy contributions from the translational, rotational, and vibrational energy modes (E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>) at 298.15 K and 1 atm||-234.408935
|-
| Sum of electronic and thermal Enthalpies||As above but contains an additional correction for RT (H = E + RT)||-234.407991
|-
| Sum of electronic and thermal Free Energies||As above but includes entropic contribution to the free energy (G = H - TS)||-234.44783
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 1.''' ''Sum of energies of ''anti2'' 1,5-hexadiene optimized at '''B3LYP/6-31G'''''</div>

==Optimization of "Chair" and "Boat" structures==

Computing the force constants at the start of the calculation, using the redundant coordinate editor and using the '''QST2''' are the methods for optimizing the transition structure.

==="Chair" conformation of cope rearrangement===

In this part, to work out the "Chair" rearrangement, Hartree Fock and the default set 3-21G were used.
'''Figure 10.''' shown below is the infrared spectrum simulated by the optimization to a Berny TS. The force constant was calculated once and the frequency calculation carried out spontaneously. The result for this simulation is summarized in '''Table 2.'''.

{| class="wikitable" border="1"
| align="center" style="background:#f0f0f0;"|'''Energy'''
| align="center" style="background:#f0f0f0;"|'''Bond Length'''
| align="center" style="background:#f0f0f0;"|'''Energy/Hartrees'''
| align="center" style="background:#f0f0f0;"|'''Frequency'''
| align="center" style="background:#f0f0f0;"|'''Animation'''
|-
| '''"Chair" TS'''|| [[File:B-bond lenghth.PNG|200px]]|| -231.61932242||Imaginary frequency of -818.07||[[File:B2.gif]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 2.''' ''The result for the simulation of chair structure at '''HF/3-21G'''''</div>

[[File:B-IR.svg|thumb|right|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 4.''' ''Simulated IR Spectrum of the Chair Transition State at '''HF/3-21G'''''' ''</div>]]

The transition structure was then re-optimized by using the frozen coordinate method. The distance between the two terminal ends of the allyl fragments were frozen to 2.2 Å. This gives a similar structure as the '''HF/3-21G''' one (in '''Table 2.''') and the computed energy is -231.61502682 Ha.

{| class="wikitable" border="1"
| align="center" style="background:#f0f0f0;"|''' '''
| align="center" style="background:#f0f0f0;"|'''Bond Length'''
| align="center" style="background:#f0f0f0;"|'''Energy/Hartrees'''
|-
| '''"Chair" TS''' || [[File:C-bond length.PNG|200px]] || -231.61502682
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 3.''' ''The result for the re-optimized chair structure with freezing coordinates''</div>

The structure was then optimized again without the distance between the two fragment ends fixed. This time the energy is calculated to be -231.69166702 Ha and the bond lengths of the two ends are quite different, one 1.55 Å while the other 4.39 Å. Therefore the structure is not a "chair" shape, unlike the optimized structure gives by freezing the coordinates shown above.

{| class="wikitable" border="1"
| align="center" style="background:#f0f0f0;"|''' '''
| align="center" style="background:#f0f0f0;"|'''Bond Length'''
| align="center" style="background:#f0f0f0;"|'''Energy/Hartrees'''
|-
| '''"Chair" TS''' || [[File:D-bond length.PNG|200px]] || -231.69166702
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 4.''' ''The result for the re-optimized chair structure without freezing coordinates''</div>

==="Boat" conformation of cope rearrangement===

'''QST2''', a different method, was used in this part to compute the "boat" rearrangement.

Firstly, the ''anti2'' structure optimized above was used as a template. The two copies of ''anti2'' gives the reactant and the product molecules. The molecules were re-numbered as '''Figure 5.''' shown below.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">[[File:Numbering atoms.JPG|650px|thumb|center|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 5.''' ''The atoms of the reactant and the product were renumbered''</div>]]</div>

Then the optimization was carried out at '''HF/3-21G''' using '''QST2''' method.

The first optimization failed, as shown in '''Figure 6.''' and '''7''', produced a C<sub>2h</sub> symmetry point group. The transition state is dissociated. This leads to an unsuccessful optimization. The frequency calculation gives an imaginary frequency of magnitude of 817.94 cm<sup>-1</sup>. And '''Figure 8.''' displays the simulated IR spectrum of this optimization.

[[File:E-failure.svg|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 8.''' ''Simulated IR spectrum of the failed optimization''</div>]]

[[File:E-failure.PNG|200px|center]]

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 6.''' ''Failure in optimization''</div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>E</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>E.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 7.''' ''Failure in optimization''</div>

Then the bond angles of the central four carbon atoms was set to 0° and the angles of each inside C-C-C bond to 100° therefore the geometry of the reactant and the product were more like the "boat" structure.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">[[File:Numbering atoms2.JPG|500px|thumb|center|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 9.''' ''The atoms of the reactant and the product were renumbered''</div>]]</div>

After the modification of their geometries (modified structure shown in '''Figure 9.'''), the optimization was carried out by using '''QST2''' method again.

[[File:E-success.svg|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 10.''' ''Simulated IR spectrum of the success optimization''</div>]]

[[File:E-success.PNG|200px|center]]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 11.''' ''Success Optimization using QST2''</div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>SXC-E2-QST2</title><color>white</color>
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<uploadedFileContents>SXC-E2-QST2.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 12.''' ''Success Optimization using QST2''</div>

'''Figure 11.''' and '''12''' displays the result "boat" structure. This optimization gives a C<sub>2v</sub> transition structure with an electronic energy of -231.602802 Ha. The IR spectrum of the boat transition structure was also optimized ('''Figure 10.'''). And a magnitude of imaginary frequency was calculated to be 839.34 cm<sup>-1</sup>.

Intrinsic Reaction Coordinate or IRC method, makes the tracking of the minimum energy path from a transition structure down to its local minimum on a potential energy surface possible. Small geometry steps are taken in the direction at the largest gradient of the energy surface so that a series of points are generated. The "chair" transition structure was then optimized by using IRC method, with 50 points specified.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>E IRC</title><color>white</color>
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<uploadedFileContents>E IRC.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 13.''' ''Structure of the last point on the simulated IRC ''</div>

[[File:E IRC.PNG|thumb|right|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 14.''' ''Simulated IRC Spectrum of the Chair Transition State''' ''</div>]]

From the last point (with electronic energy = -231.68298131 Ha) on the IRC spectrum simulated, shown in '''Figure 14.''', it is clear that the structure has not reached the energy minimum yet. Therefore further simulations were carried out in the following three different ways.

<u>Method 1</u> The last point on the IRC was taken and a normal optimization was ran. The resultant electronic energy is -231.68302549 Ha, which slightly lower than the first simulation. The optimized structure is shown in '''Figure.15'''.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>E IRC i</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>E IRC i.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 15.''' ''Simulated structure via method 1''</div>

<u>Method 2</u> The simulation was restarted with specified 100 points. This gives energy of -231.68298131 Ha, which is the same as the first simulation done with 50 points specified. Therefore the re-simulated geometry has not reach a minimum as well. The IRC path, displays in '''Figure. 16''', is quite similar to the first simulation.

[[File:E IRC ii.PNG|thumb|right|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 16.''' ''Re-simulated IRC Spectrum of the Chair Transition State via method 2''' ''</div>]]

<u>Method 3</u> The IRC was carried out again with force constants calculated at every step and no specified number of points. This time the calculated energy is -231.65069991 Ha, the lowest simulated energy among these three methods. '''Figure 17.''' shown below is the IRC spectrum for this effective simulation.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>E IRC iii</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>E IRC iii.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 18.''' ''Simulated structure via method 3''</div>

[[File:E IRC iii.PNG|thumb|right|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 17.''' ''Re-simulated IRC Spectrum of the Chair Transition State via method 3''' ''</div>]]

The chair and boat transition structures are re-optimized by using the '''B3LYP/6-31G''' level of theory and the frequency calculations was carried out. Therefore the energies calculated by using different methods can be compared.

===Comparison of the "Chair" and the "Boat" transition structures===

''' Summary of energies (in hartree) '''

{| class="wikitable" style="margin: 1em auto 1em auto;"
| align="center" style="background:#f0f0f0;"|'''Energy'''
| align="center" style="background:#f0f0f0;"|'''Relative Energy/Hartrees, HF/3-21G'''
| align="center" style="background:#f0f0f0;"|'''Relative Energy/Hartrees, B3LYP/6-31G'''
|-
| Sum of electronic and zero-point Energies||-231.4667||-234.4149
|-
| Sum of electronic and thermal Energies||-231.4613||-234.4090
|-
| Sum of electronic and thermal Enthalpies||-231.4604||-234.4081
|-
| Sum of electronic and thermal Free Energies||-231.4952||-234.4438
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 5.''' ''Sum of energies of the chair transition state optimized at '''HF/3-21G''' and '''B3LYP/6-31G''' ''</div>

{| class="wikitable" style="margin: 1em auto 1em auto;"
| align="center" style="background:#f0f0f0;"|'''Energy'''
| align="center" style="background:#f0f0f0;"|'''Relative Energy/Hartrees, HF/3-21G'''
| align="center" style="background:#f0f0f0;"|'''Relative Energy/Hartrees, B3LYP/6-31G'''
|-
| Sum of electronic and zero-point Energies||-231.4510||-234.4023
|-
| Sum of electronic and thermal Energies||-231.4453||-234.3960
|-
| Sum of electronic and thermal Enthalpies||-231.4444||-234.3951
|-
| Sum of electronic and thermal Free Energies||-231.4798||-234.4318
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 6.''' ''Sum of energies of the boat transition state optimized at '''HF/3-21G''' and '''B3LYP/6-31G''' ''</div>

<br /> *1 hartree = 627.509 kcal/mol <br /><br />

The sum of energies of the chair and the boat transition structures are summarized in Table '''5.''' and '''6.'''. It is noticeable that the energies calculated at the theory of '''B3LYP/6-31G''' are lower than those at '''HF/3-21G'''.

''' Summary of activation energies (in kcal/mol) '''

{| class="wikitable" style="margin: 1em auto 1em auto;"
| align="center" style="background:#f0f0f0;"|'''Transition Structure'''
| align="center" style="background:#f0f0f0;"|'''HF/3-21G at 0 K'''
| align="center" style="background:#f0f0f0;"|'''HF/3-21G at 298.15 K'''
| align="center" style="background:#f0f0f0;"|'''B3LYP/6-31G at 0 K'''
| align="center" style="background:#f0f0f0;"|'''B3LYP/6-31G at 298.15 K'''
|-
| ΔE (Chair)||45.68||44.73||34.07||33.19
|-
| ΔE (Boat)||55.61||54.76||41.96||41.34
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 7.''' ''Activateion energies of the chair and the boat transition state optimized at '''HF/3-21G''' and '''B3LYP/6-31G''' under 0 K and 298.15 K''</div>

The experimental activation energies are given to be 33.5 ± 0.5 kcal/mol and 44.7 ± 0.5 kcal/mol for chair and boat transition structures respectively. Therefore simulations at '''B3LYP/6-31G''' give closer value to the experimental.

[[File:Ii-IR.svg|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 19.''' ''Simulated IR spectrum of the exo transition structure''</div>]]

==The Diels Alder Cycloaddition==
The AM1 semi-empirical molecular orbital method was used for the following calculations.

===Ethylene and cis-butadiene reaction===

The HOMO and LUMO of cis-butadiene were plotted and the symmetry was determined.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
| ''' '''
| width="80" align="center" | '''HOMO'''
| width="100" align="center" | '''LUMO'''
| width="100" align="center" | '''Discussion'''
|-
| '''cis-Butadiene'''
| align="center" | [[File:SXC-Cis butadiene HOMO.PNG|200px]]
| align="center" | [[File:SXC-Cis butadiene LUMO.PNG|200px]]
| align="center" | The HOMO is anti-symmetrical and the LUMO is symmetrical
|-
| '''Ethylene'''
| align="center" | [[File:SXC-Ethylene HOMO.PNG|200px]]
| align="center" | [[File:SXC-Ethylene LUMO.PNG|200px]]
| align="center" | The HOMO is symmetrical and the LUMO is anti-symmetrical
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 8.''' ''The HOMO and LUMO of cis-butadiene and ethylene''</div>

'''Calculation of ethylene + cis-butadiene transition structure'''

As the reaction scheme shown below in '''Figure ?.''', the reaction between ethylene and cis-butadiene gives cyclohexene.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">[[File:Reaction between ethylene and butadiene.PNG|650px|thumb|center|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure ?.''' ''The reaction scheme of reaction between ethylene and cis-butadiene''</div>]]</div>

The MOs of the transition structure were simulated and result displays in '''Table 9.'''. The HOMO at the transition state is anti-symmetrical, thus, it is should be formed by the anti-symmetrical HOMO of the cis-butadiene fragment and the anti-symmetrical LUMO of the ethylene fragment.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
| ''' '''
| width="80" align="center" | '''HOMO'''
| width="100" align="center" | '''LUMO'''
| width="100" align="center" | '''Discussion'''
|-
| '''Ethylene-butadiene transition structure'''
| align="center" | [[File:Ii-HOMO.PNG|200px]]
| align="center" | [[File:Ii-LUMO.PNG|200px]]
| align="center" | The HOMO is anti-symmetrical and the LUMO is symmetrical
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 9.''' ''The HOMO and LUMO of ethylene-butadiene transition structure''</div>

As shown in '''Figure 16.''', The bond length of the partly bonded σ C-C bond of the ethylene-butadiene transition structure are calculated to be 2.12 Å. sp<sup>2</sup> and sp<sup>3</sup> C-C bondlengths are typically 1.476 Å and 1.537 Å respectively.<ref>J M Baranowski 1986 J. Phys. C: Solid State Phys. '''19''' 4613 {{DOI|10.1088/0022-3719/19/24/006}}</ref> And the van der Waals radius of the carbon atom is 1.70 Å.<ref>J. Phys. Chem., '''1996''', 100 (18), pp 7384–7391 {{DOI|10.1021/jp953141+}}</ref> Therefore bond forming interaction should takes place as the simulated bondlength of 2.12 Å is within this literature value.

[[File:Ii bond length.PNG|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 16.''' ''Calculated bond length of partly formed σ C-C bond in ethylene-dibutadiene transition structure''</div>]]

The vibration was animated ('''Figure 17.'''), which indicates that the formation of the two bonds are synchronous. The calculation gave an imaginary frequency of 955.67 cm<sup>-1</sup>.

[[File:Ii ethylene-dibutadiene.gif|center]]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 17.''' ''The animation of ethylene-dibutadiene transition structure''</div>

At 147.28 cm<sup>-1</sup>, the lowest positive frequency, the vibration is quite different ('''Figure. ?'''). It is not vibrating along the bond forming between the two fragments but simply vibrating themselves.

[[File:Ii-lowest frequency.gif|center]]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure ?.''' ''The animation of ethylene-dibutadiene transition structure at the lowest positive frequency''</div>

===Cyclohexa-1,3-diene and maleic anhydride reaction===

Cyclohexa-1,3-diene reacts with maleic anhydride to give two adducts ('''Figure ?.'''), in which the endo one is believed to be the major product. In this part, to study the regiochemistry of the Diels Alder cycloaddition, the transition structures of this reaction was investigated.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">[[File:Reaction scheme cyclohexa-1,3-diene and maleic anhydride.PNG|500px|thumb|center|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure ?.''' ''The reaction scheme of Cyclohexa-1,3-diene and maleic anhydride reaction''</div>]]</div>

====Exo transition structure====

The animation of the vibrating exo transition structure displays in '''Figure 17.''', accounts for the synchronous forming of the two bonds. The magnitude of the imaginary frequency given by the calculation is 812.19 cm<sup>-1</sup>. And the partly formed σ C-C bond length is calculated to be 2.17 Å, as shown in '''Figure 19.'''.

[[File:Iii-exo.svg|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 18.''' ''Simulated IR spectrum of the exo transition structure''</div>]]

[[File:Iii-exo.gif|center]]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 17.''' ''The animation of the exo transition structure''</div>

[[File:Iii-exo-bond length.PNG|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 19.''' ''Calculated bond length of partly formed σ C-C bond in the exo transition structure''</div>]]

The sum of energies are list in '''Table 7.''' below and the electronic energy of this exo structure is -0.05041983 Ha by the simulation.

{| class="wikitable" style="margin: 1em auto 1em auto;"
| align="center" style="background:#f0f0f0;"|'''Energy'''
| align="center" style="background:#f0f0f0;"|'''Energy/Hatrees ''Exo'''''
|-
| Sum of electronic and zero-point Energies||0.134882
|-
| Sum of electronic and thermal Energies ||0.144882
|-
| Sum of electronic and thermal Enthalpies ||0.145826
|-
| Sum of electronic and thermal Free Energies ||0.099118
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 7.''' ''Sum of energies of the exo transition structure''</div>

{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
| ''' '''
| width="80" align="center" | '''HOMO'''
| width="100" align="center" | '''LUMO'''
| width="200" align="center" | '''Discussion'''
|-
| '''Ethylene-butadiene transition structure'''
| align="center" | [[File:Iii-exo-HOMO.PNG|200px]]
| align="center" | [[File:Iii-exo-LUMO.PNG|200px]]
| align="center" | Both of the HOMO and LUMO are anti-symmetrical
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 8.''' ''The HOMO and LUMO of the exo transition structure''</div>

====Endo transition structure====
The animated vibration of the endo transition structure ('''Figure 18.'''), shows the synchronous formation of the bonds. The simulation gave an imaginary frequency of magnitude 806.22 cm<sup>-1</sup>. According to the simulation, the partially formed σ C-C bond of the endo transition structure is 2.16 Å apart ('''Figure 21.'''), which is slightly shorter than the exo one. A infrared spectrum was generated ('''Figure 20.''').

[[File:Iii-endo-final.svg|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 20.''' ''Simulated IR spectrum of the endo transition structure''</div>]]

[[File:Iii-endo-final.gif|center]]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 19.''' ''The animation of the endo transition structure''</div>

[[File:Iii-endo-bond length.PNG|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 21.''' ''Calculated bond length of partly formed σ C-C bond in the endo transition structure''</div>]]

{| class="wikitable" style="margin: 1em auto 1em auto;"
| align="center" style="background:#f0f0f0;"|'''Energy'''
| align="center" style="background:#f0f0f0;"|'''Energy/Hatrees ''Endo'''''
|-
| Sum of electronic and zero-point Energies||0.133493
|-
| Sum of electronic and thermal Energies ||0.143682
|-
| Sum of electronic and thermal Enthalpies ||0.144626
|-
| Sum of electronic and thermal Free Energies ||0.097350
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 9.''' ''Sum of energies of the endo transition structure''</div>

The electronic energy of the endo structure is computed to be -0.05150475 Ha, lower than that of the exo transition structure, indicating a more stable form of the adduct.

Comparing the sum of energies of the exo and endo adduct transition structures list in '''Table 7.''' and '''9.''', it is noticeable that the energies of the endo structure are lower than the exo one.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
| ''' '''
| width="80" align="center" | '''HOMO'''
| width="100" align="center" | '''LUMO'''
| width="200" align="center" | '''Discussion'''
|-
| '''Ethylene-butadiene transition structure'''
| align="center" | [[File:Iii-endo-HOMO.PNG|200px]]
| align="center" | [[File:Iii-endo-LUMO.PNG|200px]]
| align="center" | Both of the HOMO and LUMO are anti-symmetrical
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 10.''' ''The HOMO and LUMO of the endo transition structure''</div>

The existing secondary orbital overlap in the endo transition structure makes its formation more preferred over the exo form. In the endo form, the HOMO of the -(C=O)-O-(C=O)- fragment and the LUMO of the rest of the structure or the LUMO of that fragment and the remaining structure can overlap to form π bond; however the overlapping can not take place in the exo structure (The corresponding MOs are shown in '''Figure ?.''' in blue). This result in lower energy of the endo transition structure, thus more favorable.<ref>J. Org. Chem., '''1987''', 52 (8), pp 1469–1474 {{DOI|10.1021/jo00384a016}}</ref>

[[File:SXC-SECONDARY ORBITAL INTERACTION.PNG|center]]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure ?.''' ''Secondary orbital overlap effect of the endo and exo structure''</div>

=Reference=

<references/>

Rep:Mod:SXC13

2013-12-06T15:30:28Z

Xs1711: /* Endo transition structure */

=Transition states and reactivity=
In this experiment, the transition structures on potential surfaces for the Cope rearrangement and Diels-Alder cycloaddition reactions are generated.

==The cope rearrangement of 1,5-hexadiene==

In this part, the favored reaction mechanism is determined by finding out the low-energy minima and transition structures on the 1,5-hexadiene potential energy surface.

The structure of 1,5-hexadiene with an "anti" linkage for the central four carbon atoms was optimized at the '''HF/3-21G''' level of the theory. As shown in '''Figure.1''', the molecule has a symmetry of C<sub>i</sub> and the energy of this structure is calculated to be -231.69253528 Ha. Therefore the result structure is determined to be ''anti2'' by comparing this energy with the structures in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>1,5-hexadiene (Anti)</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>1,5 hexadienegauche.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 1.''' ''1,5-hexadiene (anti2)''</div>

The structure of 1,5-hexadiene with a "gauche" linkage for the central four carbon atoms was then optimized also at the '''HF/3-21G''' level of theory. The calculated energy of this structure is -231.69266120 Ha, which is slightly higher than the anti2 structure. Comparing with [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1], the structure is equivalent to ''gauche3'', for which the point group is C<sub>1</sub>.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>1,5-hexadiene (Anti)</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>1-reoptimized.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 2.''' ''1,5-hexadiene (gauche3)''</div>

The lowest energy conformation of a reactant is used as a reference in the calculations of lowest activation energies and enthalpies. In this case, ''gauche3'' structure, shown in '''Figure. 2''', is the lowest energy conformation, with zero relative energy.

The structure of ''anti2'' was reoptimized at the '''B3LYP/3-21G''' level of theory. The energy is calculated to be -234.55970873 Ha.
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>1,5-hexadiene (Anti)</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>1-reoptimized.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 3.''' ''1,5-hexadiene (anti2 re-optimized)''</div>

The computed energies are shown in '''Table 1.'''

{| class="wikitable" border="1"
| align="center" style="background:#f0f0f0;"|'''Energy'''
| align="center" style="background:#f0f0f0;"|'''Description'''
| align="center" style="background:#f0f0f0;"|'''Relative Energy/Hartrees B3LYP/6-31G'''
|-
| Sum of electronic and zero-point Energies||Potential energy at 0 K including the zero-point vibrational energy (E = E<sub>elec</sub> + ZPE)||-234.416224
|-
| Sum of electronic and thermal Energies||Energy contributions from the translational, rotational, and vibrational energy modes (E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>) at 298.15 K and 1 atm||-234.408935
|-
| Sum of electronic and thermal Enthalpies||As above but contains an additional correction for RT (H = E + RT)||-234.407991
|-
| Sum of electronic and thermal Free Energies||As above but includes entropic contribution to the free energy (G = H - TS)||-234.44783
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 1.''' ''Sum of energies of ''anti2'' 1,5-hexadiene optimized at '''B3LYP/6-31G'''''</div>

==Optimization of "Chair" and "Boat" structures==

Computing the force constants at the start of the calculation, using the redundant coordinate editor and using the '''QST2''' are the methods for optimizing the transition structure.

==="Chair" conformation of cope rearrangement===

In this part, to work out the "Chair" rearrangement, Hartree Fock and the default set 3-21G were used.
'''Figure 10.''' shown below is the infrared spectrum simulated by the optimization to a Berny TS. The force constant was calculated once and the frequency calculation carried out spontaneously. The result for this simulation is summarized in '''Table 2.'''.

{| class="wikitable" border="1"
| align="center" style="background:#f0f0f0;"|'''Energy'''
| align="center" style="background:#f0f0f0;"|'''Bond Length'''
| align="center" style="background:#f0f0f0;"|'''Energy/Hartrees'''
| align="center" style="background:#f0f0f0;"|'''Frequency'''
| align="center" style="background:#f0f0f0;"|'''Animation'''
|-
| '''"Chair" TS'''|| [[File:B-bond lenghth.PNG|200px]]|| -231.61932242||Imaginary frequency of -818.07||[[File:B2.gif]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 2.''' ''The result for the simulation of chair structure at '''HF/3-21G'''''</div>

[[File:B-IR.svg|thumb|right|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 4.''' ''Simulated IR Spectrum of the Chair Transition State at '''HF/3-21G'''''' ''</div>]]

The transition structure was then re-optimized by using the frozen coordinate method. The distance between the two terminal ends of the allyl fragments were frozen to 2.2 Å. This gives a similar structure as the '''HF/3-21G''' one (in '''Table 2.''') and the computed energy is -231.61502682 Ha.

{| class="wikitable" border="1"
| align="center" style="background:#f0f0f0;"|''' '''
| align="center" style="background:#f0f0f0;"|'''Bond Length'''
| align="center" style="background:#f0f0f0;"|'''Energy/Hartrees'''
|-
| '''"Chair" TS''' || [[File:C-bond length.PNG|200px]] || -231.61502682
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 3.''' ''The result for the re-optimized chair structure with freezing coordinates''</div>

The structure was then optimized again without the distance between the two fragment ends fixed. This time the energy is calculated to be -231.69166702 Ha and the bond lengths of the two ends are quite different, one 1.55 Å while the other 4.39 Å. Therefore the structure is not a "chair" shape, unlike the optimized structure gives by freezing the coordinates shown above.

{| class="wikitable" border="1"
| align="center" style="background:#f0f0f0;"|''' '''
| align="center" style="background:#f0f0f0;"|'''Bond Length'''
| align="center" style="background:#f0f0f0;"|'''Energy/Hartrees'''
|-
| '''"Chair" TS''' || [[File:D-bond length.PNG|200px]] || -231.69166702
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 4.''' ''The result for the re-optimized chair structure without freezing coordinates''</div>

==="Boat" conformation of cope rearrangement===

'''QST2''', a different method, was used in this part to compute the "boat" rearrangement.

Firstly, the ''anti2'' structure optimized above was used as a template. The two copies of ''anti2'' gives the reactant and the product molecules. The molecules were re-numbered as '''Figure 5.''' shown below.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">[[File:Numbering atoms.JPG|650px|thumb|center|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 5.''' ''The atoms of the reactant and the product were renumbered''</div>]]</div>

Then the optimization was carried out at '''HF/3-21G''' using '''QST2''' method.

The first optimization failed, as shown in '''Figure 6.''' and '''7''', produced a C<sub>2h</sub> symmetry point group. The transition state is dissociated. This leads to an unsuccessful optimization. The frequency calculation gives an imaginary frequency of magnitude of 817.94 cm<sup>-1</sup>. And '''Figure 8.''' displays the simulated IR spectrum of this optimization.

[[File:E-failure.svg|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 8.''' ''Simulated IR spectrum of the failed optimization''</div>]]

[[File:E-failure.PNG|200px|center]]

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 6.''' ''Failure in optimization''</div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>E</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>E.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 7.''' ''Failure in optimization''</div>

Then the bond angles of the central four carbon atoms was set to 0° and the angles of each inside C-C-C bond to 100° therefore the geometry of the reactant and the product were more like the "boat" structure.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">[[File:Numbering atoms2.JPG|500px|thumb|center|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 9.''' ''The atoms of the reactant and the product were renumbered''</div>]]</div>

After the modification of their geometries (modified structure shown in '''Figure 9.'''), the optimization was carried out by using '''QST2''' method again.

[[File:E-success.svg|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 10.''' ''Simulated IR spectrum of the success optimization''</div>]]

[[File:E-success.PNG|200px|center]]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 11.''' ''Success Optimization using QST2''</div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>SXC-E2-QST2</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>SXC-E2-QST2.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 12.''' ''Success Optimization using QST2''</div>

'''Figure 11.''' and '''12''' displays the result "boat" structure. This optimization gives a C<sub>2v</sub> transition structure with an electronic energy of -231.602802 Ha. The IR spectrum of the boat transition structure was also optimized ('''Figure 10.'''). And a magnitude of imaginary frequency was calculated to be 839.34 cm<sup>-1</sup>.

Intrinsic Reaction Coordinate or IRC method, makes the tracking of the minimum energy path from a transition structure down to its local minimum on a potential energy surface possible. Small geometry steps are taken in the direction at the largest gradient of the energy surface so that a series of points are generated. The "chair" transition structure was then optimized by using IRC method, with 50 points specified.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>E IRC</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>E IRC.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 13.''' ''Structure of the last point on the simulated IRC ''</div>

[[File:E IRC.PNG|thumb|right|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 14.''' ''Simulated IRC Spectrum of the Chair Transition State''' ''</div>]]

From the last point (with electronic energy = -231.68298131 Ha) on the IRC spectrum simulated, shown in '''Figure 14.''', it is clear that the structure has not reached the energy minimum yet. Therefore further simulations were carried out in the following three different ways.

<u>Method 1</u> The last point on the IRC was taken and a normal optimization was ran. The resultant electronic energy is -231.68302549 Ha, which slightly lower than the first simulation. The optimized structure is shown in '''Figure.15'''.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>E IRC i</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>E IRC i.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 15.''' ''Simulated structure via method 1''</div>

<u>Method 2</u> The simulation was restarted with specified 100 points. This gives energy of -231.68298131 Ha, which is the same as the first simulation done with 50 points specified. Therefore the re-simulated geometry has not reach a minimum as well. The IRC path, displays in '''Figure. 16''', is quite similar to the first simulation.

[[File:E IRC ii.PNG|thumb|right|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 16.''' ''Re-simulated IRC Spectrum of the Chair Transition State via method 2''' ''</div>]]

<u>Method 3</u> The IRC was carried out again with force constants calculated at every step and no specified number of points. This time the calculated energy is -231.65069991 Ha, the lowest simulated energy among these three methods. '''Figure 17.''' shown below is the IRC spectrum for this effective simulation.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>E IRC iii</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>E IRC iii.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 18.''' ''Simulated structure via method 3''</div>

[[File:E IRC iii.PNG|thumb|right|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 17.''' ''Re-simulated IRC Spectrum of the Chair Transition State via method 3''' ''</div>]]

The chair and boat transition structures are re-optimized by using the '''B3LYP/6-31G''' level of theory and the frequency calculations was carried out. Therefore the energies calculated by using different methods can be compared.

===Comparison of the "Chair" and the "Boat" transition structures===

''' Summary of energies (in hartree) '''

{| class="wikitable" style="margin: 1em auto 1em auto;"
| align="center" style="background:#f0f0f0;"|'''Energy'''
| align="center" style="background:#f0f0f0;"|'''Relative Energy/Hartrees, HF/3-21G'''
| align="center" style="background:#f0f0f0;"|'''Relative Energy/Hartrees, B3LYP/6-31G'''
|-
| Sum of electronic and zero-point Energies||-231.4667||-234.4149
|-
| Sum of electronic and thermal Energies||-231.4613||-234.4090
|-
| Sum of electronic and thermal Enthalpies||-231.4604||-234.4081
|-
| Sum of electronic and thermal Free Energies||-231.4952||-234.4438
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 5.''' ''Sum of energies of the chair transition state optimized at '''HF/3-21G''' and '''B3LYP/6-31G''' ''</div>

{| class="wikitable" style="margin: 1em auto 1em auto;"
| align="center" style="background:#f0f0f0;"|'''Energy'''
| align="center" style="background:#f0f0f0;"|'''Relative Energy/Hartrees, HF/3-21G'''
| align="center" style="background:#f0f0f0;"|'''Relative Energy/Hartrees, B3LYP/6-31G'''
|-
| Sum of electronic and zero-point Energies||-231.4510||-234.4023
|-
| Sum of electronic and thermal Energies||-231.4453||-234.3960
|-
| Sum of electronic and thermal Enthalpies||-231.4444||-234.3951
|-
| Sum of electronic and thermal Free Energies||-231.4798||-234.4318
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 6.''' ''Sum of energies of the boat transition state optimized at '''HF/3-21G''' and '''B3LYP/6-31G''' ''</div>

<br /> *1 hartree = 627.509 kcal/mol <br /><br />

The sum of energies of the chair and the boat transition structures are summarized in Table '''5.''' and '''6.'''. It is noticeable that the energies calculated at the theory of '''B3LYP/6-31G''' are lower than those at '''HF/3-21G'''.

''' Summary of activation energies (in kcal/mol) '''

{| class="wikitable" style="margin: 1em auto 1em auto;"
| align="center" style="background:#f0f0f0;"|'''Transition Structure'''
| align="center" style="background:#f0f0f0;"|'''HF/3-21G at 0 K'''
| align="center" style="background:#f0f0f0;"|'''HF/3-21G at 298.15 K'''
| align="center" style="background:#f0f0f0;"|'''B3LYP/6-31G at 0 K'''
| align="center" style="background:#f0f0f0;"|'''B3LYP/6-31G at 298.15 K'''
|-
| ΔE (Chair)||45.68||44.73||34.07||33.19
|-
| ΔE (Boat)||55.61||54.76||41.96||41.34
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 7.''' ''Activateion energies of the chair and the boat transition state optimized at '''HF/3-21G''' and '''B3LYP/6-31G''' under 0 K and 298.15 K''</div>

The experimental activation energies are given to be 33.5 ± 0.5 kcal/mol and 44.7 ± 0.5 kcal/mol for chair and boat transition structures respectively. Therefore simulations at '''B3LYP/6-31G''' give closer value to the experimental.

[[File:Ii-IR.svg|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 19.''' ''Simulated IR spectrum of the exo transition structure''</div>]]

==The Diels Alder Cycloaddition==
The AM1 semi-empirical molecular orbital method was used for the following calculations.

===Ethylene and cis-butadiene reaction===

The HOMO and LUMO of cis-butadiene were plotted and the symmetry was determined.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
| ''' '''
| width="80" align="center" | '''HOMO'''
| width="100" align="center" | '''LUMO'''
| width="100" align="center" | '''Discussion'''
|-
| '''cis-Butadiene'''
| align="center" | [[File:SXC-Cis butadiene HOMO.PNG|200px]]
| align="center" | [[File:SXC-Cis butadiene LUMO.PNG|200px]]
| align="center" | The HOMO is anti-symmetrical and the LUMO is symmetrical
|-
| '''Ethylene'''
| align="center" | [[File:SXC-Ethylene HOMO.PNG|200px]]
| align="center" | [[File:SXC-Ethylene LUMO.PNG|200px]]
| align="center" | The HOMO is symmetrical and the LUMO is anti-symmetrical
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 8.''' ''The HOMO and LUMO of cis-butadiene and ethylene''</div>

'''Calculation of ethylene + cis-butadiene transition structure'''

As the reaction scheme shown below in '''Figure ?.''', the reaction between ethylene and cis-butadiene gives cyclohexene.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">[[File:Reaction between ethylene and butadiene.PNG|650px|thumb|center|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure ?.''' ''The reaction scheme of reaction between ethylene and cis-butadiene''</div>]]</div>

The MOs of the transition structure were simulated and result displays in '''Table 9.'''. The HOMO at the transition state is anti-symmetrical, thus, it is should be formed by the anti-symmetrical HOMO of the cis-butadiene fragment and the anti-symmetrical LUMO of the ethylene fragment.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
| ''' '''
| width="80" align="center" | '''HOMO'''
| width="100" align="center" | '''LUMO'''
| width="100" align="center" | '''Discussion'''
|-
| '''Ethylene-butadiene transition structure'''
| align="center" | [[File:Ii-HOMO.PNG|200px]]
| align="center" | [[File:Ii-LUMO.PNG|200px]]
| align="center" | The HOMO is anti-symmetrical and the LUMO is symmetrical
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 9.''' ''The HOMO and LUMO of ethylene-butadiene transition structure''</div>

As shown in '''Figure 16.''', The bond length of the partly bonded σ C-C bond of the ethylene-butadiene transition structure are calculated to be 2.12 Å. sp<sup>2</sup> and sp<sup>3</sup> C-C bondlengths are typically 1.476 Å and 1.537 Å respectively.<ref>J M Baranowski 1986 J. Phys. C: Solid State Phys. '''19''' 4613 {{DOI|10.1088/0022-3719/19/24/006}}</ref> And the van der Waals radius of the carbon atom is 1.70 Å.<ref>J. Phys. Chem., '''1996''', 100 (18), pp 7384–7391 {{DOI|10.1021/jp953141+}}</ref> Therefore bond forming interaction should takes place as the simulated bondlength of 2.12 Å is within this literature value.

[[File:Ii bond length.PNG|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 16.''' ''Calculated bond length of partly formed σ C-C bond in ethylene-dibutadiene transition structure''</div>]]

The vibration was animated ('''Figure 17.'''), which indicates that the formation of the two bonds are synchronous. The calculation gave an imaginary frequency of 955.67 cm<sup>-1</sup>.

[[File:Ii ethylene-dibutadiene.gif|center]]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 17.''' ''The animation of ethylene-dibutadiene transition structure''</div>

At 147.28 cm<sup>-1</sup>, the lowest positive frequency, the vibration is quite different ('''Figure. ?'''). It is not vibrating along the bond forming between the two fragments but simply vibrating themselves.

[[File:Ii-lowest frequency.gif|center]]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure ?.''' ''The animation of ethylene-dibutadiene transition structure at the lowest positive frequency''</div>

===Cyclohexa-1,3-diene and maleic anhydride reaction===

Cyclohexa-1,3-diene reacts with maleic anhydride to give two adducts ('''Figure ?.'''), in which the endo one is believed to be the major product. In this part, to study the regiochemistry of the Diels Alder cycloaddition, the transition structures of this reaction was investigated.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">[[File:Reaction scheme cyclohexa-1,3-diene and maleic anhydride.PNG|500px|thumb|center|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure ?.''' ''The reaction scheme of Cyclohexa-1,3-diene and maleic anhydride reaction''</div>]]</div>

====Exo transition structure====

The animation of the vibrating exo transition structure displays in '''Figure 17.''', accounts for the synchronous forming of the two bonds. The magnitude of the imaginary frequency given by the calculation is 812.19 cm<sup>-1</sup>. And the partly formed σ C-C bond length is calculated to be 2.17 Å, as shown in '''Figure 19.'''.

[[File:Iii-exo.svg|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 18.''' ''Simulated IR spectrum of the exo transition structure''</div>]]

[[File:Iii-exo.gif|center]]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 17.''' ''The animation of the exo transition structure''</div>

[[File:Iii-exo-bond length.PNG|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 19.''' ''Calculated bond length of partly formed σ C-C bond in the exo transition structure''</div>]]

The sum of energies are list in '''Table 7.''' below and the electronic energy of this exo structure is -0.05041983 Ha by the simulation.

{| class="wikitable" style="margin: 1em auto 1em auto;"
| align="center" style="background:#f0f0f0;"|'''Energy'''
| align="center" style="background:#f0f0f0;"|'''Energy/Hatrees ''Exo'''''
|-
| Sum of electronic and zero-point Energies||0.134882
|-
| Sum of electronic and thermal Energies ||0.144882
|-
| Sum of electronic and thermal Enthalpies ||0.145826
|-
| Sum of electronic and thermal Free Energies ||0.099118
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 7.''' ''Sum of energies of the exo transition structure''</div>

{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
| ''' '''
| width="80" align="center" | '''HOMO'''
| width="100" align="center" | '''LUMO'''
| width="200" align="center" | '''Discussion'''
|-
| '''Ethylene-butadiene transition structure'''
| align="center" | [[File:Iii-exo-HOMO.PNG|200px]]
| align="center" | [[File:Iii-exo-LUMO.PNG|200px]]
| align="center" | Both of the HOMO and LUMO are anti-symmetrical
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 8.''' ''The HOMO and LUMO of the exo transition structure''</div>

====Endo transition structure====
The animated vibration of the endo transition structure ('''Figure 18.'''), shows the synchronous formation of the bonds. The simulation gave an imaginary frequency of magnitude 806.22 cm<sup>-1</sup>. According to the simulation, the partially formed σ C-C bond of the endo transition structure is 2.16 Å apart ('''Figure 21.'''), which is slightly shorter than the exo one. A infrared spectrum was generated ('''Figure 20.''').

[[File:Iii-endo-final.svg|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 20.''' ''Simulated IR spectrum of the endo transition structure''</div>]]

[[File:Iii-endo-final.gif|center]]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 19.''' ''The animation of the endo transition structure''</div>

[[File:Iii-endo-bond length.PNG|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 21.''' ''Calculated bond length of partly formed σ C-C bond in the endo transition structure''</div>]]

{| class="wikitable" style="margin: 1em auto 1em auto;"
| align="center" style="background:#f0f0f0;"|'''Energy'''
| align="center" style="background:#f0f0f0;"|'''Energy/Hatrees ''Endo'''''
|-
| Sum of electronic and zero-point Energies||0.133493
|-
| Sum of electronic and thermal Energies ||0.143682
|-
| Sum of electronic and thermal Enthalpies ||0.144626
|-
| Sum of electronic and thermal Free Energies ||0.097350
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 9.''' ''Sum of energies of the endo transition structure''</div>

The electronic energy of the endo structure is computed to be -0.05150475 Ha, lower than that of the exo transition structure, indicating a more stable form of the adduct.

Comparing the sum of energies of the exo and endo adduct transition structures list in '''Table 7.''' and '''9.''', it is noticeable that the energies of the endo structure are lower than the exo one.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
| ''' '''
| width="80" align="center" | '''HOMO'''
| width="100" align="center" | '''LUMO'''
| width="200" align="center" | '''Discussion'''
|-
| '''Ethylene-butadiene transition structure'''
| align="center" | [[File:Iii-endo-HOMO.PNG|200px]]
| align="center" | [[File:Iii-endo-LUMO.PNG|200px]]
| align="center" | Both of the HOMO and LUMO are anti-symmetrical
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 10.''' ''The HOMO and LUMO of the endo transition structure''</div>

The existing secondary orbital overlap in the endo transition structure makes it more preferred over the exo form. In the endo form, the HOMO of the -(C=O)-O-(C=O)- fragment and the LUMO of the rest of the structure or the LUMO of that fragment and the remaining structure can overlap to form π bond; however the overlapping can not take place in the exo structure (The corresponding MOs are shown in '''Figure ?.''' in blue.). This result in lower energy of the endo transition structure, thus more favorable.

[[File:SXC-SECONDARY ORBITAL INTERACTION.PNG|center]]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure ?.''' ''Secondary orbital overlap effect of the endo and exo structure''</div>

=Reference=

<references/>

Rep:Mod:SXC13

2013-12-06T15:28:20Z

Xs1711: /* Endo transition structure */

=Transition states and reactivity=
In this experiment, the transition structures on potential surfaces for the Cope rearrangement and Diels-Alder cycloaddition reactions are generated.

==The cope rearrangement of 1,5-hexadiene==

In this part, the favored reaction mechanism is determined by finding out the low-energy minima and transition structures on the 1,5-hexadiene potential energy surface.

The structure of 1,5-hexadiene with an "anti" linkage for the central four carbon atoms was optimized at the '''HF/3-21G''' level of the theory. As shown in '''Figure.1''', the molecule has a symmetry of C<sub>i</sub> and the energy of this structure is calculated to be -231.69253528 Ha. Therefore the result structure is determined to be ''anti2'' by comparing this energy with the structures in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>1,5-hexadiene (Anti)</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>1,5 hexadienegauche.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 1.''' ''1,5-hexadiene (anti2)''</div>

The structure of 1,5-hexadiene with a "gauche" linkage for the central four carbon atoms was then optimized also at the '''HF/3-21G''' level of theory. The calculated energy of this structure is -231.69266120 Ha, which is slightly higher than the anti2 structure. Comparing with [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1], the structure is equivalent to ''gauche3'', for which the point group is C<sub>1</sub>.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>1,5-hexadiene (Anti)</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>1-reoptimized.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 2.''' ''1,5-hexadiene (gauche3)''</div>

The lowest energy conformation of a reactant is used as a reference in the calculations of lowest activation energies and enthalpies. In this case, ''gauche3'' structure, shown in '''Figure. 2''', is the lowest energy conformation, with zero relative energy.

The structure of ''anti2'' was reoptimized at the '''B3LYP/3-21G''' level of theory. The energy is calculated to be -234.55970873 Ha.
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>1,5-hexadiene (Anti)</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>1-reoptimized.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 3.''' ''1,5-hexadiene (anti2 re-optimized)''</div>

The computed energies are shown in '''Table 1.'''

{| class="wikitable" border="1"
| align="center" style="background:#f0f0f0;"|'''Energy'''
| align="center" style="background:#f0f0f0;"|'''Description'''
| align="center" style="background:#f0f0f0;"|'''Relative Energy/Hartrees B3LYP/6-31G'''
|-
| Sum of electronic and zero-point Energies||Potential energy at 0 K including the zero-point vibrational energy (E = E<sub>elec</sub> + ZPE)||-234.416224
|-
| Sum of electronic and thermal Energies||Energy contributions from the translational, rotational, and vibrational energy modes (E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>) at 298.15 K and 1 atm||-234.408935
|-
| Sum of electronic and thermal Enthalpies||As above but contains an additional correction for RT (H = E + RT)||-234.407991
|-
| Sum of electronic and thermal Free Energies||As above but includes entropic contribution to the free energy (G = H - TS)||-234.44783
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 1.''' ''Sum of energies of ''anti2'' 1,5-hexadiene optimized at '''B3LYP/6-31G'''''</div>

==Optimization of "Chair" and "Boat" structures==

Computing the force constants at the start of the calculation, using the redundant coordinate editor and using the '''QST2''' are the methods for optimizing the transition structure.

==="Chair" conformation of cope rearrangement===

In this part, to work out the "Chair" rearrangement, Hartree Fock and the default set 3-21G were used.
'''Figure 10.''' shown below is the infrared spectrum simulated by the optimization to a Berny TS. The force constant was calculated once and the frequency calculation carried out spontaneously. The result for this simulation is summarized in '''Table 2.'''.

{| class="wikitable" border="1"
| align="center" style="background:#f0f0f0;"|'''Energy'''
| align="center" style="background:#f0f0f0;"|'''Bond Length'''
| align="center" style="background:#f0f0f0;"|'''Energy/Hartrees'''
| align="center" style="background:#f0f0f0;"|'''Frequency'''
| align="center" style="background:#f0f0f0;"|'''Animation'''
|-
| '''"Chair" TS'''|| [[File:B-bond lenghth.PNG|200px]]|| -231.61932242||Imaginary frequency of -818.07||[[File:B2.gif]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 2.''' ''The result for the simulation of chair structure at '''HF/3-21G'''''</div>

[[File:B-IR.svg|thumb|right|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 4.''' ''Simulated IR Spectrum of the Chair Transition State at '''HF/3-21G'''''' ''</div>]]

The transition structure was then re-optimized by using the frozen coordinate method. The distance between the two terminal ends of the allyl fragments were frozen to 2.2 Å. This gives a similar structure as the '''HF/3-21G''' one (in '''Table 2.''') and the computed energy is -231.61502682 Ha.

{| class="wikitable" border="1"
| align="center" style="background:#f0f0f0;"|''' '''
| align="center" style="background:#f0f0f0;"|'''Bond Length'''
| align="center" style="background:#f0f0f0;"|'''Energy/Hartrees'''
|-
| '''"Chair" TS''' || [[File:C-bond length.PNG|200px]] || -231.61502682
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 3.''' ''The result for the re-optimized chair structure with freezing coordinates''</div>

The structure was then optimized again without the distance between the two fragment ends fixed. This time the energy is calculated to be -231.69166702 Ha and the bond lengths of the two ends are quite different, one 1.55 Å while the other 4.39 Å. Therefore the structure is not a "chair" shape, unlike the optimized structure gives by freezing the coordinates shown above.

{| class="wikitable" border="1"
| align="center" style="background:#f0f0f0;"|''' '''
| align="center" style="background:#f0f0f0;"|'''Bond Length'''
| align="center" style="background:#f0f0f0;"|'''Energy/Hartrees'''
|-
| '''"Chair" TS''' || [[File:D-bond length.PNG|200px]] || -231.69166702
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 4.''' ''The result for the re-optimized chair structure without freezing coordinates''</div>

==="Boat" conformation of cope rearrangement===

'''QST2''', a different method, was used in this part to compute the "boat" rearrangement.

Firstly, the ''anti2'' structure optimized above was used as a template. The two copies of ''anti2'' gives the reactant and the product molecules. The molecules were re-numbered as '''Figure 5.''' shown below.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">[[File:Numbering atoms.JPG|650px|thumb|center|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 5.''' ''The atoms of the reactant and the product were renumbered''</div>]]</div>

Then the optimization was carried out at '''HF/3-21G''' using '''QST2''' method.

The first optimization failed, as shown in '''Figure 6.''' and '''7''', produced a C<sub>2h</sub> symmetry point group. The transition state is dissociated. This leads to an unsuccessful optimization. The frequency calculation gives an imaginary frequency of magnitude of 817.94 cm<sup>-1</sup>. And '''Figure 8.''' displays the simulated IR spectrum of this optimization.

[[File:E-failure.svg|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 8.''' ''Simulated IR spectrum of the failed optimization''</div>]]

[[File:E-failure.PNG|200px|center]]

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 6.''' ''Failure in optimization''</div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>E</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>E.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 7.''' ''Failure in optimization''</div>

Then the bond angles of the central four carbon atoms was set to 0° and the angles of each inside C-C-C bond to 100° therefore the geometry of the reactant and the product were more like the "boat" structure.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">[[File:Numbering atoms2.JPG|500px|thumb|center|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 9.''' ''The atoms of the reactant and the product were renumbered''</div>]]</div>

After the modification of their geometries (modified structure shown in '''Figure 9.'''), the optimization was carried out by using '''QST2''' method again.

[[File:E-success.svg|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 10.''' ''Simulated IR spectrum of the success optimization''</div>]]

[[File:E-success.PNG|200px|center]]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 11.''' ''Success Optimization using QST2''</div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>SXC-E2-QST2</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>SXC-E2-QST2.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 12.''' ''Success Optimization using QST2''</div>

'''Figure 11.''' and '''12''' displays the result "boat" structure. This optimization gives a C<sub>2v</sub> transition structure with an electronic energy of -231.602802 Ha. The IR spectrum of the boat transition structure was also optimized ('''Figure 10.'''). And a magnitude of imaginary frequency was calculated to be 839.34 cm<sup>-1</sup>.

Intrinsic Reaction Coordinate or IRC method, makes the tracking of the minimum energy path from a transition structure down to its local minimum on a potential energy surface possible. Small geometry steps are taken in the direction at the largest gradient of the energy surface so that a series of points are generated. The "chair" transition structure was then optimized by using IRC method, with 50 points specified.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>E IRC</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>E IRC.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 13.''' ''Structure of the last point on the simulated IRC ''</div>

[[File:E IRC.PNG|thumb|right|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 14.''' ''Simulated IRC Spectrum of the Chair Transition State''' ''</div>]]

From the last point (with electronic energy = -231.68298131 Ha) on the IRC spectrum simulated, shown in '''Figure 14.''', it is clear that the structure has not reached the energy minimum yet. Therefore further simulations were carried out in the following three different ways.

<u>Method 1</u> The last point on the IRC was taken and a normal optimization was ran. The resultant electronic energy is -231.68302549 Ha, which slightly lower than the first simulation. The optimized structure is shown in '''Figure.15'''.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>E IRC i</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>E IRC i.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 15.''' ''Simulated structure via method 1''</div>

<u>Method 2</u> The simulation was restarted with specified 100 points. This gives energy of -231.68298131 Ha, which is the same as the first simulation done with 50 points specified. Therefore the re-simulated geometry has not reach a minimum as well. The IRC path, displays in '''Figure. 16''', is quite similar to the first simulation.

[[File:E IRC ii.PNG|thumb|right|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 16.''' ''Re-simulated IRC Spectrum of the Chair Transition State via method 2''' ''</div>]]

<u>Method 3</u> The IRC was carried out again with force constants calculated at every step and no specified number of points. This time the calculated energy is -231.65069991 Ha, the lowest simulated energy among these three methods. '''Figure 17.''' shown below is the IRC spectrum for this effective simulation.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>E IRC iii</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>E IRC iii.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 18.''' ''Simulated structure via method 3''</div>

[[File:E IRC iii.PNG|thumb|right|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 17.''' ''Re-simulated IRC Spectrum of the Chair Transition State via method 3''' ''</div>]]

The chair and boat transition structures are re-optimized by using the '''B3LYP/6-31G''' level of theory and the frequency calculations was carried out. Therefore the energies calculated by using different methods can be compared.

===Comparison of the "Chair" and the "Boat" transition structures===

''' Summary of energies (in hartree) '''

{| class="wikitable" style="margin: 1em auto 1em auto;"
| align="center" style="background:#f0f0f0;"|'''Energy'''
| align="center" style="background:#f0f0f0;"|'''Relative Energy/Hartrees, HF/3-21G'''
| align="center" style="background:#f0f0f0;"|'''Relative Energy/Hartrees, B3LYP/6-31G'''
|-
| Sum of electronic and zero-point Energies||-231.4667||-234.4149
|-
| Sum of electronic and thermal Energies||-231.4613||-234.4090
|-
| Sum of electronic and thermal Enthalpies||-231.4604||-234.4081
|-
| Sum of electronic and thermal Free Energies||-231.4952||-234.4438
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 5.''' ''Sum of energies of the chair transition state optimized at '''HF/3-21G''' and '''B3LYP/6-31G''' ''</div>

{| class="wikitable" style="margin: 1em auto 1em auto;"
| align="center" style="background:#f0f0f0;"|'''Energy'''
| align="center" style="background:#f0f0f0;"|'''Relative Energy/Hartrees, HF/3-21G'''
| align="center" style="background:#f0f0f0;"|'''Relative Energy/Hartrees, B3LYP/6-31G'''
|-
| Sum of electronic and zero-point Energies||-231.4510||-234.4023
|-
| Sum of electronic and thermal Energies||-231.4453||-234.3960
|-
| Sum of electronic and thermal Enthalpies||-231.4444||-234.3951
|-
| Sum of electronic and thermal Free Energies||-231.4798||-234.4318
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 6.''' ''Sum of energies of the boat transition state optimized at '''HF/3-21G''' and '''B3LYP/6-31G''' ''</div>

<br /> *1 hartree = 627.509 kcal/mol <br /><br />

The sum of energies of the chair and the boat transition structures are summarized in Table '''5.''' and '''6.'''. It is noticeable that the energies calculated at the theory of '''B3LYP/6-31G''' are lower than those at '''HF/3-21G'''.

''' Summary of activation energies (in kcal/mol) '''

{| class="wikitable" style="margin: 1em auto 1em auto;"
| align="center" style="background:#f0f0f0;"|'''Transition Structure'''
| align="center" style="background:#f0f0f0;"|'''HF/3-21G at 0 K'''
| align="center" style="background:#f0f0f0;"|'''HF/3-21G at 298.15 K'''
| align="center" style="background:#f0f0f0;"|'''B3LYP/6-31G at 0 K'''
| align="center" style="background:#f0f0f0;"|'''B3LYP/6-31G at 298.15 K'''
|-
| ΔE (Chair)||45.68||44.73||34.07||33.19
|-
| ΔE (Boat)||55.61||54.76||41.96||41.34
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 7.''' ''Activateion energies of the chair and the boat transition state optimized at '''HF/3-21G''' and '''B3LYP/6-31G''' under 0 K and 298.15 K''</div>

The experimental activation energies are given to be 33.5 ± 0.5 kcal/mol and 44.7 ± 0.5 kcal/mol for chair and boat transition structures respectively. Therefore simulations at '''B3LYP/6-31G''' give closer value to the experimental.

[[File:Ii-IR.svg|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 19.''' ''Simulated IR spectrum of the exo transition structure''</div>]]

==The Diels Alder Cycloaddition==
The AM1 semi-empirical molecular orbital method was used for the following calculations.

===Ethylene and cis-butadiene reaction===

The HOMO and LUMO of cis-butadiene were plotted and the symmetry was determined.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
| ''' '''
| width="80" align="center" | '''HOMO'''
| width="100" align="center" | '''LUMO'''
| width="100" align="center" | '''Discussion'''
|-
| '''cis-Butadiene'''
| align="center" | [[File:SXC-Cis butadiene HOMO.PNG|200px]]
| align="center" | [[File:SXC-Cis butadiene LUMO.PNG|200px]]
| align="center" | The HOMO is anti-symmetrical and the LUMO is symmetrical
|-
| '''Ethylene'''
| align="center" | [[File:SXC-Ethylene HOMO.PNG|200px]]
| align="center" | [[File:SXC-Ethylene LUMO.PNG|200px]]
| align="center" | The HOMO is symmetrical and the LUMO is anti-symmetrical
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 8.''' ''The HOMO and LUMO of cis-butadiene and ethylene''</div>

'''Calculation of ethylene + cis-butadiene transition structure'''

As the reaction scheme shown below in '''Figure ?.''', the reaction between ethylene and cis-butadiene gives cyclohexene.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">[[File:Reaction between ethylene and butadiene.PNG|650px|thumb|center|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure ?.''' ''The reaction scheme of reaction between ethylene and cis-butadiene''</div>]]</div>

The MOs of the transition structure were simulated and result displays in '''Table 9.'''. The HOMO at the transition state is anti-symmetrical, thus, it is should be formed by the anti-symmetrical HOMO of the cis-butadiene fragment and the anti-symmetrical LUMO of the ethylene fragment.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
| ''' '''
| width="80" align="center" | '''HOMO'''
| width="100" align="center" | '''LUMO'''
| width="100" align="center" | '''Discussion'''
|-
| '''Ethylene-butadiene transition structure'''
| align="center" | [[File:Ii-HOMO.PNG|200px]]
| align="center" | [[File:Ii-LUMO.PNG|200px]]
| align="center" | The HOMO is anti-symmetrical and the LUMO is symmetrical
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 9.''' ''The HOMO and LUMO of ethylene-butadiene transition structure''</div>

As shown in '''Figure 16.''', The bond length of the partly bonded σ C-C bond of the ethylene-butadiene transition structure are calculated to be 2.12 Å. sp<sup>2</sup> and sp<sup>3</sup> C-C bondlengths are typically 1.476 Å and 1.537 Å respectively.<ref>J M Baranowski 1986 J. Phys. C: Solid State Phys. '''19''' 4613 {{DOI|10.1088/0022-3719/19/24/006}}</ref> And the van der Waals radius of the carbon atom is 1.70 Å.<ref>J. Phys. Chem., '''1996''', 100 (18), pp 7384–7391 {{DOI|10.1021/jp953141+}}</ref> Therefore bond forming interaction should takes place as the simulated bondlength of 2.12 Å is within this literature value.

[[File:Ii bond length.PNG|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 16.''' ''Calculated bond length of partly formed σ C-C bond in ethylene-dibutadiene transition structure''</div>]]

The vibration was animated ('''Figure 17.'''), which indicates that the formation of the two bonds are synchronous. The calculation gave an imaginary frequency of 955.67 cm<sup>-1</sup>.

[[File:Ii ethylene-dibutadiene.gif|center]]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 17.''' ''The animation of ethylene-dibutadiene transition structure''</div>

At 147.28 cm<sup>-1</sup>, the lowest positive frequency, the vibration is quite different ('''Figure. ?'''). It is not vibrating along the bond forming between the two fragments but simply vibrating themselves.

[[File:Ii-lowest frequency.gif|center]]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure ?.''' ''The animation of ethylene-dibutadiene transition structure at the lowest positive frequency''</div>

===Cyclohexa-1,3-diene and maleic anhydride reaction===

Cyclohexa-1,3-diene reacts with maleic anhydride to give two adducts ('''Figure ?.'''), in which the endo one is believed to be the major product. In this part, to study the regiochemistry of the Diels Alder cycloaddition, the transition structures of this reaction was investigated.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">[[File:Reaction scheme cyclohexa-1,3-diene and maleic anhydride.PNG|500px|thumb|center|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure ?.''' ''The reaction scheme of Cyclohexa-1,3-diene and maleic anhydride reaction''</div>]]</div>

====Exo transition structure====

The animation of the vibrating exo transition structure displays in '''Figure 17.''', accounts for the synchronous forming of the two bonds. The magnitude of the imaginary frequency given by the calculation is 812.19 cm<sup>-1</sup>. And the partly formed σ C-C bond length is calculated to be 2.17 Å, as shown in '''Figure 19.'''.

[[File:Iii-exo.svg|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 18.''' ''Simulated IR spectrum of the exo transition structure''</div>]]

[[File:Iii-exo.gif|center]]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 17.''' ''The animation of the exo transition structure''</div>

[[File:Iii-exo-bond length.PNG|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 19.''' ''Calculated bond length of partly formed σ C-C bond in the exo transition structure''</div>]]

The sum of energies are list in '''Table 7.''' below and the electronic energy of this exo structure is -0.05041983 Ha by the simulation.

{| class="wikitable" style="margin: 1em auto 1em auto;"
| align="center" style="background:#f0f0f0;"|'''Energy'''
| align="center" style="background:#f0f0f0;"|'''Energy/Hatrees ''Exo'''''
|-
| Sum of electronic and zero-point Energies||0.134882
|-
| Sum of electronic and thermal Energies ||0.144882
|-
| Sum of electronic and thermal Enthalpies ||0.145826
|-
| Sum of electronic and thermal Free Energies ||0.099118
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 7.''' ''Sum of energies of the exo transition structure''</div>

{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
| ''' '''
| width="80" align="center" | '''HOMO'''
| width="100" align="center" | '''LUMO'''
| width="200" align="center" | '''Discussion'''
|-
| '''Ethylene-butadiene transition structure'''
| align="center" | [[File:Iii-exo-HOMO.PNG|200px]]
| align="center" | [[File:Iii-exo-LUMO.PNG|200px]]
| align="center" | Both of the HOMO and LUMO are anti-symmetrical
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 8.''' ''The HOMO and LUMO of the exo transition structure''</div>

====Endo transition structure====
The animated vibration of the endo transition structure ('''Figure 18.'''), shows the synchronous formation of the bonds. The simulation gave an imaginary frequency of magnitude 806.22 cm<sup>-1</sup>. According to the simulation, the partially formed σ C-C bond of the endo transition structure is 2.16 Å apart ('''Figure 21.'''), which is slightly shorter than the exo one. A infrared spectrum was generated ('''Figure 20.''').

[[File:Iii-endo-final.svg|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 20.''' ''Simulated IR spectrum of the endo transition structure''</div>]]

[[File:Iii-endo-final.gif|center]]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 19.''' ''The animation of the endo transition structure''</div>

[[File:Iii-endo-bond length.PNG|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 21.''' ''Calculated bond length of partly formed σ C-C bond in the endo transition structure''</div>]]

{| class="wikitable" style="margin: 1em auto 1em auto;"
| align="center" style="background:#f0f0f0;"|'''Energy'''
| align="center" style="background:#f0f0f0;"|'''Energy/Hatrees ''Endo'''''
|-
| Sum of electronic and zero-point Energies||0.133493
|-
| Sum of electronic and thermal Energies ||0.143682
|-
| Sum of electronic and thermal Enthalpies ||0.144626
|-
| Sum of electronic and thermal Free Energies ||0.097350
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 9.''' ''Sum of energies of the endo transition structure''</div>

The electronic energy of the endo structure is computed to be -0.05150475 Ha, lower than that of the exo transition structure, indicating a more stable form of the adduct.

Comparing the sum of energies of the exo and endo adduct transition structures list in '''Table 7.''' and '''9.''', it is noticeable that the energies of the endo structure are lower than the exo one.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
| ''' '''
| width="80" align="center" | '''HOMO'''
| width="100" align="center" | '''LUMO'''
| width="200" align="center" | '''Discussion'''
|-
| '''Ethylene-butadiene transition structure'''
| align="center" | [[File:Iii-endo-HOMO.PNG|200px]]
| align="center" | [[File:Iii-endo-LUMO.PNG|200px]]
| align="center" | Both of the HOMO and LUMO are anti-symmetrical
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 10.''' ''The HOMO and LUMO of the endo transition structure''</div>

The existing secondary orbital overlap in the endo transition structure makes it more preferred over the exo form. In the endo form, the HOMO of the -(C=O)-O-(C=O)- fragment and the LUMO of the rest of the structure or the LUMO of that fragment and the remaining structure can overlap to form π bond. However the overlapping can not take place in the exo structure, as shown in '''Figure ?.'''. This result in lower energy of the endo transition structure, thus more favorable.

[[File:SXC-SECONDARY ORBITAL INTERACTION.PNG|center]]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure ?.''' ''Secondary orbital overlap effect of the endo and exo structure''</div>

=Reference=

<references/>

File:SXC-SECONDARY ORBITAL INTERACTION.PNG

2013-12-06T15:27:13Z

Xs1711:

Rep:Mod:SXC13

2013-12-06T15:10:18Z

2013-12-06T13:16:44Z

Xs1711: /* Ethylene and cis-butadiene reaction */

=Transition states and reactivity=
In this experiment, the transition structures on potential surfaces for the Cope rearrangement and Diels-Alder cycloaddition reactions are generated.

==The cope rearrangement of 1,5-hexadiene==

In this part, the favored reaction mechanism is determined by finding out the low-energy minima and transition structures on the 1,5-hexadiene potential energy surface.

The structure of 1,5-hexadiene with an "anti" linkage for the central four carbon atoms was optimized at the '''HF/3-21G''' level of the theory. As shown in '''Figure.1''', the molecule has a symmetry of C<sub>i</sub> and the energy of this structure is calculated to be -231.69253528 Ha. Therefore the result structure is determined to be ''anti2'' by comparing this energy with the structures in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>1,5-hexadiene (Anti)</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>1,5 hexadienegauche.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 1.''' ''1,5-hexadiene (anti2)''</div>

The structure of 1,5-hexadiene with a "gauche" linkage for the central four carbon atoms was then optimized also at the '''HF/3-21G''' level of theory. The calculated energy of this structure is -231.69266120 Ha, which is slightly higher than the anti2 structure. Comparing with [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1], the structure is equivalent to ''gauche3'', for which the point group is C<sub>1</sub>.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>1,5-hexadiene (Anti)</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>1-reoptimized.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 2.''' ''1,5-hexadiene (gauche3)''</div>

The lowest energy conformation of a reactant is used as a reference in the calculations of lowest activation energies and enthalpies. In this case, ''gauche3'' structure, shown in '''Figure. 2''', is the lowest energy conformation, with zero relative energy.

The structure of ''anti2'' was reoptimized at the '''B3LYP/3-21G''' level of theory. The energy is calculated to be -234.55970873 Ha.
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>1,5-hexadiene (Anti)</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>1-reoptimized.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 3.''' ''1,5-hexadiene (anti2 re-optimized)''</div>

The computed energies are shown in '''Table 1.'''

{| class="wikitable" border="1"
| align="center" style="background:#f0f0f0;"|'''Energy'''
| align="center" style="background:#f0f0f0;"|'''Description'''
| align="center" style="background:#f0f0f0;"|'''Relative Energy/Hartrees B3LYP/6-31G'''
|-
| Sum of electronic and zero-point Energies||Potential energy at 0 K including the zero-point vibrational energy (E = E<sub>elec</sub> + ZPE)||-234.416224
|-
| Sum of electronic and thermal Energies||Energy contributions from the translational, rotational, and vibrational energy modes (E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>) at 298.15 K and 1 atm||-234.408935
|-
| Sum of electronic and thermal Enthalpies||As above but contains an additional correction for RT (H = E + RT)||-234.407991
|-
| Sum of electronic and thermal Free Energies||As above but includes entropic contribution to the free energy (G = H - TS)||-234.44783
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 1.''' ''Sum of energies of ''anti2'' 1,5-hexadiene optimized at '''B3LYP/6-31G'''''</div>

==Optimization of "Chair" and "Boat" structures==

Computing the force constants at the start of the calculation, using the redundant coordinate editor and using the '''QST2''' are the methods for optimizing the transition structure.

==="Chair" conformation of cope rearrangement===

In this part, to work out the "Chair" rearrangement, Hartree Fock and the default set 3-21G were used.
'''Figure 10.''' shown below is the infrared spectrum simulated by the optimization to a Berny TS. The force constant was calculated once and the frequency calculation carried out spontaneously. The result for this simulation is summarized in '''Table 2.'''.

{| class="wikitable" border="1"
| align="center" style="background:#f0f0f0;"|'''Energy'''
| align="center" style="background:#f0f0f0;"|'''Bond Length'''
| align="center" style="background:#f0f0f0;"|'''Energy/Hartrees'''
| align="center" style="background:#f0f0f0;"|'''Frequency'''
| align="center" style="background:#f0f0f0;"|'''Animation'''
|-
| '''"Chair" TS'''|| [[File:B-bond lenghth.PNG|200px]]|| -231.61932242||Imaginary frequency of -818.07||[[File:B2.gif]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 2.''' ''The result for the simulation of chair structure at '''HF/3-21G'''''</div>

[[File:B-IR.svg|thumb|right|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 4.''' ''Simulated IR Spectrum of the Chair Transition State at '''HF/3-21G'''''' ''</div>]]

The transition structure was then re-optimized by using the frozen coordinate method. The distance between the two terminal ends of the allyl fragments were frozen to 2.2 Å. This gives a similar structure as the '''HF/3-21G''' one (in '''Table 2.''') and the computed energy is -231.61502682 Ha.

{| class="wikitable" border="1"
| align="center" style="background:#f0f0f0;"|''' '''
| align="center" style="background:#f0f0f0;"|'''Bond Length'''
| align="center" style="background:#f0f0f0;"|'''Energy/Hartrees'''
|-
| '''"Chair" TS''' || [[File:C-bond length.PNG|200px]] || -231.61502682
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 3.''' ''The result for the re-optimized chair structure with freezing coordinates''</div>

The structure was then optimized again without the distance between the two fragment ends fixed. This time the energy is calculated to be -231.69166702 Ha and the bond lengths of the two ends are quite different, one 1.55 Å while the other 4.39 Å. Therefore the structure is not a "chair" shape, unlike the optimized structure gives by freezing the coordinates shown above.

{| class="wikitable" border="1"
| align="center" style="background:#f0f0f0;"|''' '''
| align="center" style="background:#f0f0f0;"|'''Bond Length'''
| align="center" style="background:#f0f0f0;"|'''Energy/Hartrees'''
|-
| '''"Chair" TS''' || [[File:D-bond length.PNG|200px]] || -231.69166702
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 4.''' ''The result for the re-optimized chair structure without freezing coordinates''</div>

==="Boat" conformation of cope rearrangement===

'''QST2''', a different method, was used in this part to compute the "boat" rearrangement.

Firstly, the ''anti2'' structure optimized above was used as a template. The two copies of ''anti2'' gives the reactant and the product molecules. The molecules were re-numbered as '''Figure 5.''' shown below.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">[[File:Numbering atoms.JPG|650px|thumb|center|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 5.''' ''The atoms of the reactant and the product were renumbered''</div>]]</div>

Then the optimization was carried out at '''HF/3-21G''' using '''QST2''' method.

The first optimization failed, as shown in '''Figure 6.''' and '''7''', produced a C<sub>2h</sub> symmetry point group. The transition state is dissociated. This leads to an unsuccessful optimization. The frequency calculation gives an imaginary frequency of magnitude of 817.94 cm<sup>-1</sup>. And '''Figure 8.''' displays the simulated IR spectrum of this optimization.

[[File:E-failure.svg|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 8.''' ''Simulated IR spectrum of the failed optimization''</div>]]

[[File:E-failure.PNG|200px|center]]

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 6.''' ''Failure in optimization''</div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>E</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>E.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 7.''' ''Failure in optimization''</div>

Then the bond angles of the central four carbon atoms was set to 0° and the angles of each inside C-C-C bond to 100° therefore the geometry of the reactant and the product were more like the "boat" structure.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">[[File:Numbering atoms2.JPG|500px|thumb|center|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 9.''' ''The atoms of the reactant and the product were renumbered''</div>]]</div>

After the modification of their geometries (modified structure shown in '''Figure 9.'''), the optimization was carried out by using '''QST2''' method again.

[[File:E-success.svg|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 10.''' ''Simulated IR spectrum of the success optimization''</div>]]

[[File:E-success.PNG|200px|center]]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 11.''' ''Success Optimization using QST2''</div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>SXC-E2-QST2</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>SXC-E2-QST2.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 12.''' ''Success Optimization using QST2''</div>

'''Figure 11.''' and '''12''' displays the result "boat" structure. This optimization gives a C<sub>2v</sub> transition structure with an electronic energy of -231.602802 Ha. The IR spectrum of the boat transition structure was also optimized ('''Figure 10.'''). And a magnitude of imaginary frequency was calculated to be 839.34 cm<sup>-1</sup>.

Intrinsic Reaction Coordinate or IRC method, makes the tracking of the minimum energy path from a transition structure down to its local minimum on a potential energy surface possible. Small geometry steps are taken in the direction at the largest gradient of the energy surface so that a series of points are generated. The "chair" transition structure was then optimized by using IRC method, with 50 points specified.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>E IRC</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>E IRC.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 13.''' ''Structure of the last point on the simulated IRC ''</div>

[[File:E IRC.PNG|thumb|right|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 14.''' ''Simulated IRC Spectrum of the Chair Transition State''' ''</div>]]

From the last point (with electronic energy = -231.68298131 Ha) on the IRC spectrum simulated, shown in '''Figure 14.''', it is clear that the structure has not reached the energy minimum yet. Therefore further simulations were carried out in the following three different ways.

<u>Method 1</u> The last point on the IRC was taken and a normal optimization was ran. The resultant electronic energy is -231.68302549 Ha, which slightly lower than the first simulation. The optimized structure is shown in '''Figure.15'''.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>E IRC i</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>E IRC i.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 15.''' ''Simulated structure via method 1''</div>

<u>Method 2</u> The simulation was restarted with specified 100 points. This gives energy of -231.68298131 Ha, which is the same as the first simulation done with 50 points specified. Therefore the re-simulated geometry has not reach a minimum as well. The IRC path, displays in '''Figure. 16''', is quite similar to the first simulation.

[[File:E IRC ii.PNG|thumb|right|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 16.''' ''Re-simulated IRC Spectrum of the Chair Transition State via method 2''' ''</div>]]

<u>Method 3</u> The IRC was carried out again with force constants calculated at every step and no specified number of points. This time the calculated energy is -231.65069991 Ha, the lowest simulated energy among these three methods. '''Figure 17.''' shown below is the IRC spectrum for this effective simulation.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>E IRC iii</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>E IRC iii.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 18.''' ''Simulated structure via method 3''</div>

[[File:E IRC iii.PNG|thumb|right|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 17.''' ''Re-simulated IRC Spectrum of the Chair Transition State via method 3''' ''</div>]]

The chair and boat transition structures are re-optimized by using the '''B3LYP/6-31G''' level of theory and the frequency calculations was carried out. Therefore the energies calculated by using different methods can be compared.

===Comparison of the "Chair" and the "Boat" transition structures===

''' Summary of energies (in hartree) '''

{| class="wikitable" style="margin: 1em auto 1em auto;"
| align="center" style="background:#f0f0f0;"|'''Energy'''
| align="center" style="background:#f0f0f0;"|'''Relative Energy/Hartrees, HF/3-21G'''
| align="center" style="background:#f0f0f0;"|'''Relative Energy/Hartrees, B3LYP/6-31G'''
|-
| Sum of electronic and zero-point Energies||-231.4667||-234.4149
|-
| Sum of electronic and thermal Energies||-231.4613||-234.4090
|-
| Sum of electronic and thermal Enthalpies||-231.4604||-234.4081
|-
| Sum of electronic and thermal Free Energies||-231.4952||-234.4438
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 5.''' ''Sum of energies of the chair transition state optimized at '''HF/3-21G''' and '''B3LYP/6-31G''' ''</div>

{| class="wikitable" style="margin: 1em auto 1em auto;"
| align="center" style="background:#f0f0f0;"|'''Energy'''
| align="center" style="background:#f0f0f0;"|'''Relative Energy/Hartrees, HF/3-21G'''
| align="center" style="background:#f0f0f0;"|'''Relative Energy/Hartrees, B3LYP/6-31G'''
|-
| Sum of electronic and zero-point Energies||-231.4510||-234.4023
|-
| Sum of electronic and thermal Energies||-231.4453||-234.3960
|-
| Sum of electronic and thermal Enthalpies||-231.4444||-234.3951
|-
| Sum of electronic and thermal Free Energies||-231.4798||-234.4318
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 6.''' ''Sum of energies of the boat transition state optimized at '''HF/3-21G''' and '''B3LYP/6-31G''' ''</div>

<br /> *1 hartree = 627.509 kcal/mol <br /><br />

The sum of energies of the chair and the boat transition structures are summarized in Table '''5.''' and '''6.'''. It is noticeable that the energies calculated at the theory of '''B3LYP/6-31G''' are lower than those at '''HF/3-21G'''.

''' Summary of activation energies (in kcal/mol) '''

{| class="wikitable" style="margin: 1em auto 1em auto;"
| align="center" style="background:#f0f0f0;"|'''Transition Structure'''
| align="center" style="background:#f0f0f0;"|'''HF/3-21G at 0 K'''
| align="center" style="background:#f0f0f0;"|'''HF/3-21G at 298.15 K'''
| align="center" style="background:#f0f0f0;"|'''B3LYP/6-31G at 0 K'''
| align="center" style="background:#f0f0f0;"|'''B3LYP/6-31G at 298.15 K'''
|-
| ΔE (Chair)||45.68||44.73||34.07||33.19
|-
| ΔE (Boat)||55.61||54.76||41.96||41.34
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 7.''' ''Activateion energies of the chair and the boat transition state optimized at '''HF/3-21G''' and '''B3LYP/6-31G''' under 0 K and 298.15 K''</div>

The experimental activation energies are given to be 33.5 ± 0.5 kcal/mol and 44.7 ± 0.5 kcal/mol for chair and boat transition structures respectively. Therefore simulations at '''B3LYP/6-31G''' give closer value to the experimental.

[[File:Ii-IR.svg|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 19.''' ''Simulated IR spectrum of the exo transition structure''</div>]]

==The Diels Alder Cycloaddition==
The AM1 semi-empirical molecular orbital method was used for the following calculations.

===Ethylene and cis-butadiene reaction===

The HOMO and LUMO of cis-butadiene were plotted and the symmetry was determined.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
| ''' '''
| width="80" align="center" | '''HOMO'''
| width="100" align="center" | '''LUMO'''
| width="100" align="center" | '''Discussion'''
|-
| '''cis-Butadiene'''
| align="center" | [[File:SXC-Cis butadiene HOMO.PNG|200px]]
| align="center" | [[File:SXC-Cis butadiene LUMO.PNG|200px]]
| align="center" | The HOMO is anti-symmetrical and the LUMO is symmetrical
|-
| '''Ethylene'''
| align="center" | [[File:SXC-Ethylene HOMO.PNG|200px]]
| align="center" | [[File:SXC-Ethylene LUMO.PNG|200px]]
| align="center" | The HOMO is symmetrical and the LUMO is anti-symmetrical
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 8.''' ''The HOMO and LUMO of cis-butadiene and ethylene''</div>

'''Calculation of ethylene + cis-butadiene transition structure'''

As the reaction scheme shown below in '''Figure ?.''', the reaction between ethylene and cis-butadiene gives cyclohexene.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">[[File:Reaction between ethylene and butadiene.PNG|650px|thumb|center|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure ?.''' ''The reaction scheme of reaction between ethylene and cis-butadiene''</div>]]</div>

The MOs of the transition structure were simulated and result displays in '''Table 9.'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
| ''' '''
| width="80" align="center" | '''HOMO'''
| width="100" align="center" | '''LUMO'''
| width="100" align="center" | '''Discussion'''
|-
| '''Ethylene-butadiene transition structure'''
| align="center" | [[File:Ii-HOMO.PNG|200px]]
| align="center" | [[File:Ii-LUMO.PNG|200px]]
| align="center" | The HOMO is anti-symmetrical and the LUMO is symmetrical
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 9.''' ''The HOMO and LUMO of ethylene-butadiene transition structure''</div>

As shown in '''Figure 16.''', The bond length of the partly bonded σ C-C bond of the ethylene-butadiene transition structure are calculated to be 2.12 Å. sp<sup>2</sup> and sp<sup>3</sup> C-C bondlengths are typically 1.476 Å and 1.537 Å respectively.<ref>J M Baranowski 1986 J. Phys. C: Solid State Phys. '''19''' 4613 {{DOI|10.1088/0022-3719/19/24/006}}</ref> And the van der Waals radius of the carbon atom is 1.70 Å.<ref>J. Phys. Chem., '''1996''', 100 (18), pp 7384–7391 {{DOI|10.1021/jp953141+}}</ref> Therefore bond forming interaction should takes place as the simulated bondlength of 2.12 Å is within this literature value.

[[File:Ii bond length.PNG|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 16.''' ''Calculated bond length of partly formed σ C-C bond in ethylene-dibutadiene transition structure''</div>]]

The vibration was animated ('''Figure 17.'''), which indicates that the formation of the two bonds are synchronous. The calculation gave an imaginary frequency of 955.67 cm<sup>-1</sup>.

[[File:Ii ethylene-dibutadiene.gif|center]]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 17.''' ''The animation of ethylene-dibutadiene transition structure''</div>

At 147.28 cm<sup>-1</sup>, the lowest positive frequency, the vibration is quite different ('''Figure. ?'''). It is not vibrating along the bond forming between the two fragments but simply vibrating themselves.

[[File:Ii-lowest frequency.gif]]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure ?.''' ''The animation of ethylene-dibutadiene transition structure at the lowest positive frequency''</div>

===Cyclohexa-1,3-diene and maleic anhydride reaction===

Cyclohexa-1,3-diene reacts with maleic anhydride to give two adducts ('''Figure ?.'''), in which the endo one is believed to be the major product. In this part, to study the regiochemistry of the Diels Alder cycloaddition, the transition structures of this reaction was investigated.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">[[File:Reaction scheme cyclohexa-1,3-diene and maleic anhydride.PNG|500px|thumb|center|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure ?.''' ''The reaction scheme of Cyclohexa-1,3-diene and maleic anhydride reaction''</div>]]</div>

====Exo transition structure====

The animation of the vibrating exo transition structure is displayed in '''Figure 17.''', and the magnitude of the imaginary frequency given by the calculation is 812.19 cm<sup>-1</sup>. And the partly formed σ C-C bond length is calculated to be 2.17 Å, as shown in '''Figure 19.'''.

[[File:Iii-exo.svg|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 18.''' ''Simulated IR spectrum of the exo transition structure''</div>]]

[[File:Iii-exo.gif|center]]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 17.''' ''The animation of the exo transition structure''</div>

[[File:Iii-exo-bond length.PNG|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 19.''' ''Calculated bond length of partly formed σ C-C bond in the exo transition structure''</div>]]

{| class="wikitable" style="margin: 1em auto 1em auto;"
| align="center" style="background:#f0f0f0;"|'''Energy'''
| align="center" style="background:#f0f0f0;"|'''Energy/Hatrees ''Exo'''''
|-
| Sum of electronic and zero-point Energies||0.134882
|-
| Sum of electronic and thermal Energies ||0.144882
|-
| Sum of electronic and thermal Enthalpies ||0.145826
|-
| Sum of electronic and thermal Free Energies ||0.099118
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 7.''' ''Sum of energies of the exo transition structure''</div>

{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
| ''' '''
| width="80" align="center" | '''HOMO'''
| width="100" align="center" | '''LUMO'''
| width="200" align="center" | '''Discussion'''
|-
| '''Ethylene-butadiene transition structure'''
| align="center" | [[File:Iii-exo-HOMO.PNG|200px]]
| align="center" | [[File:Iii-exo-LUMO.PNG|200px]]
| align="center" | Both of the HOMO and LUMO are anti-symmetrical
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 8.''' ''The HOMO and LUMO of the exo transition structure''</div>

====Endo transition structure====
Figure '''18.''' shown is the animated vibration of the endo transition structure. The simulation gave an imaginary frequency of magnitude 806.22 cm<sup>-1</sup>. According to the simulation, the partially formed σ C-C bond of the endo transition structure is 2.16 Å apart ('''Figure 21.'''), which is slightly shorter than the exo one.

[[File:Iii-endo.svg|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 20.''' ''Simulated IR spectrum of the endo transition structure''</div>]]

[[File:Iii-endo.gif|center]]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 19.''' ''The animation of the endo transition structure''</div>

[[File:Iii-endo-bond length.PNG|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 21.''' ''Calculated bond length of partly formed σ C-C bond in the endo transition structure''</div>]]

{| class="wikitable" style="margin: 1em auto 1em auto;"
| align="center" style="background:#f0f0f0;"|'''Energy'''
| align="center" style="background:#f0f0f0;"|'''Energy/Hatrees ''Endo'''''
|-
| Sum of electronic and zero-point Energies||0.133493
|-
| Sum of electronic and thermal Energies ||0.143682
|-
| Sum of electronic and thermal Enthalpies ||0.144626
|-
| Sum of electronic and thermal Free Energies ||0.097350
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 9.''' ''Sum of energies of the endo transition structure''</div>

{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
| ''' '''
| width="80" align="center" | '''HOMO'''
| width="100" align="center" | '''LUMO'''
| width="200" align="center" | '''Discussion'''
|-
| '''Ethylene-butadiene transition structure'''
| align="center" | [[File:Iii-endo-HOMO.PNG|200px]]
| align="center" | [[File:Iii-endo-LUMO.PNG|200px]]
| align="center" | Both of the HOMO and LUMO are anti-symmetrical
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 10.''' ''The HOMO and LUMO of the endo transition structure''</div>

=Reference=

<references/>

2013-12-06T11:21:45Z

Xs1711: /* Exo transition structure */

=Transition states and reactivity=
In this experiment, the transition structures on potential surfaces for the Cope rearrangement and Diels-Alder cycloaddition reactions are generated.

==The cope rearrangement of 1,5-hexadiene==

In this part, the favored reaction mechanism is determined by finding out the low-energy minima and transition structures on the 1,5-hexadiene potential energy surface.

The structure of 1,5-hexadiene with an "anti" linkage for the central four carbon atoms was optimized at the '''HF/3-21G''' level of the theory. As shown in '''Figure.1''', the molecule has a symmetry of C<sub>i</sub> and the energy of this structure is calculated to be -231.69253528 Ha. Therefore the result structure is determined to be ''anti2'' by comparing this energy with the structures in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>1,5-hexadiene (Anti)</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>1,5 hexadienegauche.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 1.''' ''1,5-hexadiene (anti2)''</div>

The structure of 1,5-hexadiene with a "gauche" linkage for the central four carbon atoms was then optimized also at the '''HF/3-21G''' level of theory. The calculated energy of this structure is -231.69266120 Ha, which is slightly higher than the anti2 structure. Comparing with [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1], the structure is equivalent to ''gauche3'', for which the point group is C<sub>1</sub>.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>1,5-hexadiene (Anti)</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>1-reoptimized.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 2.''' ''1,5-hexadiene (gauche3)''</div>

The lowest energy conformation of a reactant is used as a reference in the calculations of lowest activation energies and enthalpies. In this case, ''gauche3'' structure, shown in '''Figure. 2''', is the lowest energy conformation, with zero relative energy.

The structure of ''anti2'' was reoptimized at the '''B3LYP/3-21G''' level of theory. The energy is calculated to be -234.55970873 Ha.
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>1,5-hexadiene (Anti)</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>1-reoptimized.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 3.''' ''1,5-hexadiene (anti2 re-optimized)''</div>

The computed energies are shown in '''Table 1.'''

{| class="wikitable" border="1"
| align="center" style="background:#f0f0f0;"|'''Energy'''
| align="center" style="background:#f0f0f0;"|'''Description'''
| align="center" style="background:#f0f0f0;"|'''Relative Energy/Hartrees B3LYP/6-31G'''
|-
| Sum of electronic and zero-point Energies||Potential energy at 0 K including the zero-point vibrational energy (E = E<sub>elec</sub> + ZPE)||-234.416224
|-
| Sum of electronic and thermal Energies||Energy contributions from the translational, rotational, and vibrational energy modes (E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>) at 298.15 K and 1 atm||-234.408935
|-
| Sum of electronic and thermal Enthalpies||As above but contains an additional correction for RT (H = E + RT)||-234.407991
|-
| Sum of electronic and thermal Free Energies||As above but includes entropic contribution to the free energy (G = H - TS)||-234.44783
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 1.''' ''Sum of energies of ''anti2'' 1,5-hexadiene optimized at '''B3LYP/6-31G'''''</div>

==Optimization of "Chair" and "Boat" structures==

Computing the force constants at the start of the calculation, using the redundant coordinate editor and using the '''QST2''' are the methods for optimizing the transition structure.

==="Chair" conformation of cope rearrangement===

In this part, to work out the "Chair" rearrangement, Hartree Fock and the default set 3-21G were used.
'''Figure 10.''' shown below is the infrared spectrum simulated by the optimization to a Berny TS. The force constant was calculated once and the frequency calculation carried out spontaneously. The result for this simulation is summarized in '''Table 2.'''.

{| class="wikitable" border="1"
| align="center" style="background:#f0f0f0;"|'''Energy'''
| align="center" style="background:#f0f0f0;"|'''Bond Length'''
| align="center" style="background:#f0f0f0;"|'''Energy/Hartrees'''
| align="center" style="background:#f0f0f0;"|'''Frequency'''
| align="center" style="background:#f0f0f0;"|'''Animation'''
|-
| '''"Chair" TS'''|| [[File:B-bond lenghth.PNG|200px]]|| -231.61932242||Imaginary frequency of -818.07||[[File:B2.gif]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 2.''' ''The result for the simulation of chair structure at '''HF/3-21G'''''</div>

[[File:B-IR.svg|thumb|right|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 4.''' ''Simulated IR Spectrum of the Chair Transition State at '''HF/3-21G'''''' ''</div>]]

The transition structure was then re-optimized by using the frozen coordinate method. The distance between the two terminal ends of the allyl fragments were frozen to 2.2 Å. This gives a similar structure as the '''HF/3-21G''' one (in '''Table 2.''') and the computed energy is -231.61502682 Ha.

{| class="wikitable" border="1"
| align="center" style="background:#f0f0f0;"|''' '''
| align="center" style="background:#f0f0f0;"|'''Bond Length'''
| align="center" style="background:#f0f0f0;"|'''Energy/Hartrees'''
|-
| '''"Chair" TS''' || [[File:C-bond length.PNG|200px]] || -231.61502682
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 3.''' ''The result for the re-optimized chair structure with freezing coordinates''</div>

The structure was then optimized again without the distance between the two fragment ends fixed. This time the energy is calculated to be -231.69166702 Ha and the bond lengths of the two ends are quite different, one 1.55 Å while the other 4.39 Å. Therefore the structure is not a "chair" shape, unlike the optimized structure gives by freezing the coordinates shown above.

{| class="wikitable" border="1"
| align="center" style="background:#f0f0f0;"|''' '''
| align="center" style="background:#f0f0f0;"|'''Bond Length'''
| align="center" style="background:#f0f0f0;"|'''Energy/Hartrees'''
|-
| '''"Chair" TS''' || [[File:D-bond length.PNG|200px]] || -231.69166702
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 4.''' ''The result for the re-optimized chair structure without freezing coordinates''</div>

==="Boat" conformation of cope rearrangement===

'''QST2''', a different method, was used in this part to compute the "boat" rearrangement.

Firstly, the ''anti2'' structure optimized above was used as a template. The two copies of ''anti2'' gives the reactant and the product molecules. The molecules were re-numbered as '''Figure 5.''' shown below.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">[[File:Numbering atoms.JPG|650px|thumb|center|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 5.''' ''The atoms of the reactant and the product were renumbered''</div>]]</div>

Then the optimization was carried out at '''HF/3-21G''' using '''QST2''' method.

The first optimization failed, as shown in '''Figure 6.''' and '''7''', produced a C<sub>2h</sub> symmetry point group. The transition state is dissociated. This leads to an unsuccessful optimization. The frequency calculation gives an imaginary frequency of magnitude of 817.94 cm<sup>-1</sup>. And '''Figure 8.''' displays the simulated IR spectrum of this optimization.

[[File:E-failure.svg|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 8.''' ''Simulated IR spectrum of the failed optimization''</div>]]

[[File:E-failure.PNG|200px|center]]

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 6.''' ''Failure in optimization''</div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>E</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>E.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 7.''' ''Failure in optimization''</div>

Then the bond angles of the central four carbon atoms was set to 0° and the angles of each inside C-C-C bond to 100° therefore the geometry of the reactant and the product were more like the "boat" structure.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">[[File:Numbering atoms2.JPG|500px|thumb|center|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 9.''' ''The atoms of the reactant and the product were renumbered''</div>]]</div>

After the modification of their geometries (modified structure shown in '''Figure 9.'''), the optimization was carried out by using '''QST2''' method again.

[[File:E-success.svg|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 10.''' ''Simulated IR spectrum of the success optimization''</div>]]

[[File:E-success.PNG|200px|center]]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 11.''' ''Success Optimization using QST2''</div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>SXC-E2-QST2</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>SXC-E2-QST2.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 12.''' ''Success Optimization using QST2''</div>

'''Figure 11.''' and '''12''' displays the result "boat" structure. This optimization gives a C<sub>2v</sub> transition structure with an electronic energy of -231.602802 Ha. The IR spectrum of the boat transition structure was also optimized ('''Figure 10.'''). And a magnitude of imaginary frequency was calculated to be 839.34 cm<sup>-1</sup>.

Intrinsic Reaction Coordinate or IRC method, makes the tracking of the minimum energy path from a transition structure down to its local minimum on a potential energy surface possible. Small geometry steps are taken in the direction at the largest gradient of the energy surface so that a series of points are generated. The "chair" transition structure was then optimized by using IRC method, with 50 points specified.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>E IRC</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>E IRC.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 13.''' ''Structure of the last point on the simulated IRC ''</div>

[[File:E IRC.PNG|thumb|right|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 14.''' ''Simulated IRC Spectrum of the Chair Transition State''' ''</div>]]

From the last point (with electronic energy = -231.68298131 Ha) on the IRC spectrum simulated, shown in '''Figure 14.''', it is clear that the structure has not reached the energy minimum yet. Therefore further simulations were carried out in the following three different ways.

<u>Method 1</u> The last point on the IRC was taken and a normal optimization was ran. The resultant electronic energy is -231.68302549 Ha, which slightly lower than the first simulation. The optimized structure is shown in '''Figure.15'''.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>E IRC i</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>E IRC i.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 15.''' ''Simulated structure via method 1''</div>

<u>Method 2</u> The simulation was restarted with specified 100 points. This gives energy of -231.68298131 Ha, which is the same as the first simulation done with 50 points specified. Therefore the re-simulated geometry has not reach a minimum as well. The IRC path, displays in '''Figure. 16''', is quite similar to the first simulation.

[[File:E IRC ii.PNG|thumb|right|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 16.''' ''Re-simulated IRC Spectrum of the Chair Transition State via method 2''' ''</div>]]

<u>Method 3</u> The IRC was carried out again with force constants calculated at every step and no specified number of points. This time the calculated energy is -231.65069991 Ha, the lowest simulated energy among these three methods. '''Figure 17.''' shown below is the IRC spectrum for this effective simulation.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>E IRC iii</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>E IRC iii.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 18.''' ''Simulated structure via method 3''</div>

[[File:E IRC iii.PNG|thumb|right|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 17.''' ''Re-simulated IRC Spectrum of the Chair Transition State via method 3''' ''</div>]]

The chair and boat transition structures are re-optimized by using the '''B3LYP/6-31G''' level of theory and the frequency calculations was carried out. Therefore the energies calculated by using different methods can be compared.

===Comparison of the "Chair" and the "Boat" transition structures===

''' Summary of energies (in hartree) '''

{| class="wikitable" style="margin: 1em auto 1em auto;"
| align="center" style="background:#f0f0f0;"|'''Energy'''
| align="center" style="background:#f0f0f0;"|'''Relative Energy/Hartrees, HF/3-21G'''
| align="center" style="background:#f0f0f0;"|'''Relative Energy/Hartrees, B3LYP/6-31G'''
|-
| Sum of electronic and zero-point Energies||-231.4667||-234.4149
|-
| Sum of electronic and thermal Energies||-231.4613||-234.4090
|-
| Sum of electronic and thermal Enthalpies||-231.4604||-234.4081
|-
| Sum of electronic and thermal Free Energies||-231.4952||-234.4438
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 5.''' ''Sum of energies of the chair transition state optimized at '''HF/3-21G''' and '''B3LYP/6-31G''' ''</div>

{| class="wikitable" style="margin: 1em auto 1em auto;"
| align="center" style="background:#f0f0f0;"|'''Energy'''
| align="center" style="background:#f0f0f0;"|'''Relative Energy/Hartrees, HF/3-21G'''
| align="center" style="background:#f0f0f0;"|'''Relative Energy/Hartrees, B3LYP/6-31G'''
|-
| Sum of electronic and zero-point Energies||-231.4510||-234.4023
|-
| Sum of electronic and thermal Energies||-231.4453||-234.3960
|-
| Sum of electronic and thermal Enthalpies||-231.4444||-234.3951
|-
| Sum of electronic and thermal Free Energies||-231.4798||-234.4318
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 6.''' ''Sum of energies of the boat transition state optimized at '''HF/3-21G''' and '''B3LYP/6-31G''' ''</div>

<br /> *1 hartree = 627.509 kcal/mol <br /><br />

The sum of energies of the chair and the boat transition structures are summarized in Table '''5.''' and '''6.'''. It is noticeable that the energies calculated at the theory of '''B3LYP/6-31G''' are lower than those at '''HF/3-21G'''.

''' Summary of activation energies (in kcal/mol) '''

{| class="wikitable" style="margin: 1em auto 1em auto;"
| align="center" style="background:#f0f0f0;"|'''Transition Structure'''
| align="center" style="background:#f0f0f0;"|'''HF/3-21G at 0 K'''
| align="center" style="background:#f0f0f0;"|'''HF/3-21G at 298.15 K'''
| align="center" style="background:#f0f0f0;"|'''B3LYP/6-31G at 0 K'''
| align="center" style="background:#f0f0f0;"|'''B3LYP/6-31G at 298.15 K'''
|-
| ΔE (Chair)||45.68||44.73||34.07||33.19
|-
| ΔE (Boat)||55.61||54.76||41.96||41.34
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 7.''' ''Activateion energies of the chair and the boat transition state optimized at '''HF/3-21G''' and '''B3LYP/6-31G''' under 0 K and 298.15 K''</div>

The experimental activation energies are given to be 33.5 ± 0.5 kcal/mol and 44.7 ± 0.5 kcal/mol for chair and boat transition structures respectively. Therefore simulations at '''B3LYP/6-31G''' give closer value to the experimental.

[[File:Ii-IR.svg|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 19.''' ''Simulated IR spectrum of the exo transition structure''</div>]]

==The Diels Alder Cycloaddition==
The AM1 semi-empirical molecular orbital method was used for the following calculations.

===Ethylene and cis-butadiene reaction===

The HOMO and LUMO of cis-butadiene were plotted and the symmetry was determined.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
| ''' '''
| width="80" align="center" | '''HOMO'''
| width="100" align="center" | '''LUMO'''
| width="100" align="center" | '''Discussion'''
|-
| '''cis-Butadiene'''
| align="center" | [[File:SXC-Cis butadiene HOMO.PNG|200px]]
| align="center" | [[File:SXC-Cis butadiene LUMO.PNG|200px]]
| align="center" | The HOMO is anti-symmetrical and the LUMO is symmetrical
|-
| '''Ethylene'''
| align="center" | [[File:SXC-Ethylene HOMO.PNG|200px]]
| align="center" | [[File:SXC-Ethylene LUMO.PNG|200px]]
| align="center" | The HOMO is symmetrical and the LUMO is anti-symmetrical
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 8.''' ''The HOMO and LUMO of cis-butadiene and ethylene''</div>

'''Calculation of ethylene + cis-butadiene transition structure'''

As the reaction scheme shown below in '''Figure ?.''', the reaction between ethylene and cis-butadiene gives cyclohexene.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">[[File:Reaction between ethylene and butadiene.PNG|650px|thumb|center|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure ?.''' ''The reaction scheme of reaction between ethylene and cis-butadiene''</div>]]</div>

The MOs of the transition structure were simulated and result displays in '''Table 9.'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
| ''' '''
| width="80" align="center" | '''HOMO'''
| width="100" align="center" | '''LUMO'''
| width="100" align="center" | '''Discussion'''
|-
| '''Ethylene-butadiene transition structure'''
| align="center" | [[File:Ii-HOMO.PNG|200px]]
| align="center" | [[File:Ii-LUMO.PNG|200px]]
| align="center" | The HOMO is anti-symmetrical and the LUMO is symmetrical
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 9.''' ''The HOMO and LUMO of ethylene-butadiene transition structure''</div>

As shown in '''Figure 16.''', The bond length of the partly bonded σ C-C bond of the ethylene-butadiene transition structure are calculated to be 2.12 Å. sp<sup>2</sup> and sp<sup>3</sup> C-C bondlengths are typically 1.47 Å and 1.54 Å respectively. And the van der Waals radius of the carbon atom is 1.7 Å.

[[File:Ii bond length.PNG|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 16.''' ''Calculated bond length of partly formed σ C-C bond in ethylene-dibutadiene transition structure''</div>]]

The vibration was animated ('''Figure 17.''') and the calculation gave an imaginary frequency of magnitude 955.67 cm<sup>-1</sup>.
[[File:Ii ethylene-dibutadiene.gif|center]]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 17.''' ''The animation of ethylene-dibutadiene transition structure''</div>

===Cyclohexa-1,3-diene and maleic anhydride reaction===

====Exo transition structure====

The animation of the vibrating exo transition structure is displayed in '''Figure 17.''', and the magnitude of the imaginary frequency given by the calculation is 812.19 cm<sup>-1</sup>. And the partly formed σ C-C bond length is calculated to be 2.17 Å, as shown in '''Figure 19.'''.

[[File:Iii-exo.svg|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 18.''' ''Simulated IR spectrum of the exo transition structure''</div>]]

[[File:Iii-exo.gif|center]]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 17.''' ''The animation of the exo transition structure''</div>

[[File:Iii-exo-bond length.PNG|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 19.''' ''Calculated bond length of partly formed σ C-C bond in the exo transition structure''</div>]]

{| class="wikitable" style="margin: 1em auto 1em auto;"
| align="center" style="background:#f0f0f0;"|'''Energy'''
| align="center" style="background:#f0f0f0;"|'''Energy/Hatrees ''Exo'''''
|-
| Sum of electronic and zero-point Energies||0.134882
|-
| Sum of electronic and thermal Energies ||0.144882
|-
| Sum of electronic and thermal Enthalpies ||0.145826
|-
| Sum of electronic and thermal Free Energies ||0.099118
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 7.''' ''Sum of energies of the exo transition structure''</div>

{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
| ''' '''
| width="80" align="center" | '''HOMO'''
| width="100" align="center" | '''LUMO'''
| width="200" align="center" | '''Discussion'''
|-
| '''Ethylene-butadiene transition structure'''
| align="center" | [[File:Iii-exo-HOMO.PNG|200px]]
| align="center" | [[File:Iii-exo-LUMO.PNG|200px]]
| align="center" | Both of the HOMO and LUMO are anti-symmetrical
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 8.''' ''The HOMO and LUMO of the exo transition structure''</div>

====Endo transition structure====
Figure '''18.''' shown is the animated vibration of the endo transition structure. The simulation gave an imaginary frequency of magnitude 806.22 cm<sup>-1</sup>. According to the simulation, the partially formed σ C-C bond of the endo transition structure is 2.16 Å apart ('''Figure 21.'''), which is slightly shorter than the exo one.

[[File:Iii-endo.svg|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 20.''' ''Simulated IR spectrum of the endo transition structure''</div>]]

[[File:Iii-endo.gif|center]]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 19.''' ''The animation of the endo transition structure''</div>

[[File:Iii-endo-bond length.PNG|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 21.''' ''Calculated bond length of partly formed σ C-C bond in the endo transition structure''</div>]]

{| class="wikitable" style="margin: 1em auto 1em auto;"
| align="center" style="background:#f0f0f0;"|'''Energy'''
| align="center" style="background:#f0f0f0;"|'''Energy/Hatrees ''Endo'''''
|-
| Sum of electronic and zero-point Energies||0.133493
|-
| Sum of electronic and thermal Energies ||0.143682
|-
| Sum of electronic and thermal Enthalpies ||0.144626
|-
| Sum of electronic and thermal Free Energies ||0.097350
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 9.''' ''Sum of energies of the endo transition structure''</div>

{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
| ''' '''
| width="80" align="center" | '''HOMO'''
| width="100" align="center" | '''LUMO'''
| width="200" align="center" | '''Discussion'''
|-
| '''Ethylene-butadiene transition structure'''
| align="center" | [[File:Iii-endo-HOMO.PNG|200px]]
| align="center" | [[File:Iii-endo-LUMO.PNG|200px]]
| align="center" | Both of the HOMO and LUMO are anti-symmetrical
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 10.''' ''The HOMO and LUMO of the endo transition structure''</div>

=Reference=

File:Reaction scheme cyclohexa-1,3-diene and maleic anhydride.PNG

2013-12-06T11:16:28Z

Xs1711:

Rep:Mod:SXC13

2013-12-06T11:05:08Z

Xs1711: /* Endo transition structure */

=Transition states and reactivity=
In this experiment, the transition structures on potential surfaces for the Cope rearrangement and Diels-Alder cycloaddition reactions are generated.

==The cope rearrangement of 1,5-hexadiene==

In this part, the favored reaction mechanism is determined by finding out the low-energy minima and transition structures on the 1,5-hexadiene potential energy surface.

The structure of 1,5-hexadiene with an "anti" linkage for the central four carbon atoms was optimized at the '''HF/3-21G''' level of the theory. As shown in '''Figure.1''', the molecule has a symmetry of C<sub>i</sub> and the energy of this structure is calculated to be -231.69253528 Ha. Therefore the result structure is determined to be ''anti2'' by comparing this energy with the structures in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>1,5-hexadiene (Anti)</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>1,5 hexadienegauche.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 1.''' ''1,5-hexadiene (anti2)''</div>

The structure of 1,5-hexadiene with a "gauche" linkage for the central four carbon atoms was then optimized also at the '''HF/3-21G''' level of theory. The calculated energy of this structure is -231.69266120 Ha, which is slightly higher than the anti2 structure. Comparing with [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1], the structure is equivalent to ''gauche3'', for which the point group is C<sub>1</sub>.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>1,5-hexadiene (Anti)</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>1-reoptimized.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 2.''' ''1,5-hexadiene (gauche3)''</div>

The lowest energy conformation of a reactant is used as a reference in the calculations of lowest activation energies and enthalpies. In this case, ''gauche3'' structure, shown in '''Figure. 2''', is the lowest energy conformation, with zero relative energy.

The structure of ''anti2'' was reoptimized at the '''B3LYP/3-21G''' level of theory. The energy is calculated to be -234.55970873 Ha.
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>1,5-hexadiene (Anti)</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>1-reoptimized.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 3.''' ''1,5-hexadiene (anti2 re-optimized)''</div>

The computed energies are shown in '''Table 1.'''

{| class="wikitable" border="1"
| align="center" style="background:#f0f0f0;"|'''Energy'''
| align="center" style="background:#f0f0f0;"|'''Description'''
| align="center" style="background:#f0f0f0;"|'''Relative Energy/Hartrees B3LYP/6-31G'''
|-
| Sum of electronic and zero-point Energies||Potential energy at 0 K including the zero-point vibrational energy (E = E<sub>elec</sub> + ZPE)||-234.416224
|-
| Sum of electronic and thermal Energies||Energy contributions from the translational, rotational, and vibrational energy modes (E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>) at 298.15 K and 1 atm||-234.408935
|-
| Sum of electronic and thermal Enthalpies||As above but contains an additional correction for RT (H = E + RT)||-234.407991
|-
| Sum of electronic and thermal Free Energies||As above but includes entropic contribution to the free energy (G = H - TS)||-234.44783
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 1.''' ''Sum of energies of ''anti2'' 1,5-hexadiene optimized at '''B3LYP/6-31G'''''</div>

==Optimization of "Chair" and "Boat" structures==

Computing the force constants at the start of the calculation, using the redundant coordinate editor and using the '''QST2''' are the methods for optimizing the transition structure.

==="Chair" conformation of cope rearrangement===

In this part, to work out the "Chair" rearrangement, Hartree Fock and the default set 3-21G were used.
'''Figure 10.''' shown below is the infrared spectrum simulated by the optimization to a Berny TS. The force constant was calculated once and the frequency calculation carried out spontaneously. The result for this simulation is summarized in '''Table 2.'''.

{| class="wikitable" border="1"
| align="center" style="background:#f0f0f0;"|'''Energy'''
| align="center" style="background:#f0f0f0;"|'''Bond Length'''
| align="center" style="background:#f0f0f0;"|'''Energy/Hartrees'''
| align="center" style="background:#f0f0f0;"|'''Frequency'''
| align="center" style="background:#f0f0f0;"|'''Animation'''
|-
| '''"Chair" TS'''|| [[File:B-bond lenghth.PNG|200px]]|| -231.61932242||Imaginary frequency of -818.07||[[File:B2.gif]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 2.''' ''The result for the simulation of chair structure at '''HF/3-21G'''''</div>

[[File:B-IR.svg|thumb|right|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 4.''' ''Simulated IR Spectrum of the Chair Transition State at '''HF/3-21G'''''' ''</div>]]

The transition structure was then re-optimized by using the frozen coordinate method. The distance between the two terminal ends of the allyl fragments were frozen to 2.2 Å. This gives a similar structure as the '''HF/3-21G''' one (in '''Table 2.''') and the computed energy is -231.61502682 Ha.

{| class="wikitable" border="1"
| align="center" style="background:#f0f0f0;"|''' '''
| align="center" style="background:#f0f0f0;"|'''Bond Length'''
| align="center" style="background:#f0f0f0;"|'''Energy/Hartrees'''
|-
| '''"Chair" TS''' || [[File:C-bond length.PNG|200px]] || -231.61502682
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 3.''' ''The result for the re-optimized chair structure with freezing coordinates''</div>

The structure was then optimized again without the distance between the two fragment ends fixed. This time the energy is calculated to be -231.69166702 Ha and the bond lengths of the two ends are quite different, one 1.55 Å while the other 4.39 Å. Therefore the structure is not a "chair" shape, unlike the optimized structure gives by freezing the coordinates shown above.

{| class="wikitable" border="1"
| align="center" style="background:#f0f0f0;"|''' '''
| align="center" style="background:#f0f0f0;"|'''Bond Length'''
| align="center" style="background:#f0f0f0;"|'''Energy/Hartrees'''
|-
| '''"Chair" TS''' || [[File:D-bond length.PNG|200px]] || -231.69166702
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 4.''' ''The result for the re-optimized chair structure without freezing coordinates''</div>

==="Boat" conformation of cope rearrangement===

'''QST2''', a different method, was used in this part to compute the "boat" rearrangement.

Firstly, the ''anti2'' structure optimized above was used as a template. The two copies of ''anti2'' gives the reactant and the product molecules. The molecules were re-numbered as '''Figure 5.''' shown below.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">[[File:Numbering atoms.JPG|650px|thumb|center|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 5.''' ''The atoms of the reactant and the product were renumbered''</div>]]</div>

Then the optimization was carried out at '''HF/3-21G''' using '''QST2''' method.

The first optimization failed, as shown in '''Figure 6.''' and '''7''', produced a C<sub>2h</sub> symmetry point group. The transition state is dissociated. This leads to an unsuccessful optimization. The frequency calculation gives an imaginary frequency of magnitude of 817.94 cm<sup>-1</sup>. And '''Figure 8.''' displays the simulated IR spectrum of this optimization.

[[File:E-failure.svg|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 8.''' ''Simulated IR spectrum of the failed optimization''</div>]]

[[File:E-failure.PNG|200px|center]]

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 6.''' ''Failure in optimization''</div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>E</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>E.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 7.''' ''Failure in optimization''</div>

Then the bond angles of the central four carbon atoms was set to 0° and the angles of each inside C-C-C bond to 100° therefore the geometry of the reactant and the product were more like the "boat" structure.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">[[File:Numbering atoms2.JPG|500px|thumb|center|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 9.''' ''The atoms of the reactant and the product were renumbered''</div>]]</div>

After the modification of their geometries (modified structure shown in '''Figure 9.'''), the optimization was carried out by using '''QST2''' method again.

[[File:E-success.svg|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 10.''' ''Simulated IR spectrum of the success optimization''</div>]]

[[File:E-success.PNG|200px|center]]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 11.''' ''Success Optimization using QST2''</div>

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>SXC-E2-QST2</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>SXC-E2-QST2.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 12.''' ''Success Optimization using QST2''</div>

'''Figure 11.''' and '''12''' displays the result "boat" structure. This optimization gives a C<sub>2v</sub> transition structure with an electronic energy of -231.602802 Ha. The IR spectrum of the boat transition structure was also optimized ('''Figure 10.'''). And a magnitude of imaginary frequency was calculated to be 839.34 cm<sup>-1</sup>.

Intrinsic Reaction Coordinate or IRC method, makes the tracking of the minimum energy path from a transition structure down to its local minimum on a potential energy surface possible. Small geometry steps are taken in the direction at the largest gradient of the energy surface so that a series of points are generated. The "chair" transition structure was then optimized by using IRC method, with 50 points specified.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>E IRC</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>E IRC.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 13.''' ''Structure of the last point on the simulated IRC ''</div>

[[File:E IRC.PNG|thumb|right|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 14.''' ''Simulated IRC Spectrum of the Chair Transition State''' ''</div>]]

From the last point (with electronic energy = -231.68298131 Ha) on the IRC spectrum simulated, shown in '''Figure 14.''', it is clear that the structure has not reached the energy minimum yet. Therefore further simulations were carried out in the following three different ways.

<u>Method 1</u> The last point on the IRC was taken and a normal optimization was ran. The resultant electronic energy is -231.68302549 Ha, which slightly lower than the first simulation. The optimized structure is shown in '''Figure.15'''.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>E IRC i</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>E IRC i.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 15.''' ''Simulated structure via method 1''</div>

<u>Method 2</u> The simulation was restarted with specified 100 points. This gives energy of -231.68298131 Ha, which is the same as the first simulation done with 50 points specified. Therefore the re-simulated geometry has not reach a minimum as well. The IRC path, displays in '''Figure. 16''', is quite similar to the first simulation.

[[File:E IRC ii.PNG|thumb|right|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 16.''' ''Re-simulated IRC Spectrum of the Chair Transition State via method 2''' ''</div>]]

<u>Method 3</u> The IRC was carried out again with force constants calculated at every step and no specified number of points. This time the calculated energy is -231.65069991 Ha, the lowest simulated energy among these three methods. '''Figure 17.''' shown below is the IRC spectrum for this effective simulation.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><jmol><jmolApplet><title>E IRC iii</title><color>white</color>
<size>200</size><script>zoom 5;moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>E IRC iii.mol</uploadedFileContents>
</jmolApplet></jmol></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 18.''' ''Simulated structure via method 3''</div>

[[File:E IRC iii.PNG|thumb|right|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 17.''' ''Re-simulated IRC Spectrum of the Chair Transition State via method 3''' ''</div>]]

The chair and boat transition structures are re-optimized by using the '''B3LYP/6-31G''' level of theory and the frequency calculations was carried out. Therefore the energies calculated by using different methods can be compared.

===Comparison of the "Chair" and the "Boat" transition structures===

''' Summary of energies (in hartree) '''

{| class="wikitable" style="margin: 1em auto 1em auto;"
| align="center" style="background:#f0f0f0;"|'''Energy'''
| align="center" style="background:#f0f0f0;"|'''Relative Energy/Hartrees, HF/3-21G'''
| align="center" style="background:#f0f0f0;"|'''Relative Energy/Hartrees, B3LYP/6-31G'''
|-
| Sum of electronic and zero-point Energies||-231.4667||-234.4149
|-
| Sum of electronic and thermal Energies||-231.4613||-234.4090
|-
| Sum of electronic and thermal Enthalpies||-231.4604||-234.4081
|-
| Sum of electronic and thermal Free Energies||-231.4952||-234.4438
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 5.''' ''Sum of energies of the chair transition state optimized at '''HF/3-21G''' and '''B3LYP/6-31G''' ''</div>

{| class="wikitable" style="margin: 1em auto 1em auto;"
| align="center" style="background:#f0f0f0;"|'''Energy'''
| align="center" style="background:#f0f0f0;"|'''Relative Energy/Hartrees, HF/3-21G'''
| align="center" style="background:#f0f0f0;"|'''Relative Energy/Hartrees, B3LYP/6-31G'''
|-
| Sum of electronic and zero-point Energies||-231.4510||-234.4023
|-
| Sum of electronic and thermal Energies||-231.4453||-234.3960
|-
| Sum of electronic and thermal Enthalpies||-231.4444||-234.3951
|-
| Sum of electronic and thermal Free Energies||-231.4798||-234.4318
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 6.''' ''Sum of energies of the boat transition state optimized at '''HF/3-21G''' and '''B3LYP/6-31G''' ''</div>

<br /> *1 hartree = 627.509 kcal/mol <br /><br />

The sum of energies of the chair and the boat transition structures are summarized in Table '''5.''' and '''6.'''. It is noticeable that the energies calculated at the theory of '''B3LYP/6-31G''' are lower than those at '''HF/3-21G'''.

''' Summary of activation energies (in kcal/mol) '''

{| class="wikitable" style="margin: 1em auto 1em auto;"
| align="center" style="background:#f0f0f0;"|'''Transition Structure'''
| align="center" style="background:#f0f0f0;"|'''HF/3-21G at 0 K'''
| align="center" style="background:#f0f0f0;"|'''HF/3-21G at 298.15 K'''
| align="center" style="background:#f0f0f0;"|'''B3LYP/6-31G at 0 K'''
| align="center" style="background:#f0f0f0;"|'''B3LYP/6-31G at 298.15 K'''
|-
| ΔE (Chair)||45.68||44.73||34.07||33.19
|-
| ΔE (Boat)||55.61||54.76||41.96||41.34
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 7.''' ''Activateion energies of the chair and the boat transition state optimized at '''HF/3-21G''' and '''B3LYP/6-31G''' under 0 K and 298.15 K''</div>

The experimental activation energies are given to be 33.5 ± 0.5 kcal/mol and 44.7 ± 0.5 kcal/mol for chair and boat transition structures respectively. Therefore simulations at '''B3LYP/6-31G''' give closer value to the experimental.

[[File:Ii-IR.svg|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 19.''' ''Simulated IR spectrum of the exo transition structure''</div>]]

==The Diels Alder Cycloaddition==
The AM1 semi-empirical molecular orbital method was used for the following calculations.

===Ethylene and cis-butadiene reaction===

The HOMO and LUMO of cis-butadiene were plotted and the symmetry was determined.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
| ''' '''
| width="80" align="center" | '''HOMO'''
| width="100" align="center" | '''LUMO'''
| width="100" align="center" | '''Discussion'''
|-
| '''cis-Butadiene'''
| align="center" | [[File:SXC-Cis butadiene HOMO.PNG|200px]]
| align="center" | [[File:SXC-Cis butadiene LUMO.PNG|200px]]
| align="center" | The HOMO is anti-symmetrical and the LUMO is symmetrical
|-
| '''Ethylene'''
| align="center" | [[File:SXC-Ethylene HOMO.PNG|200px]]
| align="center" | [[File:SXC-Ethylene LUMO.PNG|200px]]
| align="center" | The HOMO is symmetrical and the LUMO is anti-symmetrical
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 8.''' ''The HOMO and LUMO of cis-butadiene and ethylene''</div>

'''Calculation of ethylene + cis-butadiene transition structure'''

As the reaction scheme shown below in '''Figure ?.''', the reaction between ethylene and cis-butadiene gives cyclohexene.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">[[File:Reaction between ethylene and butadiene.PNG|650px|thumb|center|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure ?.''' ''The reaction scheme of reaction between ethylene and cis-butadiene''</div>]]</div>

The MOs of the transition structure were simulated and result displays in '''Table 9.'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
| ''' '''
| width="80" align="center" | '''HOMO'''
| width="100" align="center" | '''LUMO'''
| width="100" align="center" | '''Discussion'''
|-
| '''Ethylene-butadiene transition structure'''
| align="center" | [[File:Ii-HOMO.PNG|200px]]
| align="center" | [[File:Ii-LUMO.PNG|200px]]
| align="center" | The HOMO is anti-symmetrical and the LUMO is symmetrical
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 9.''' ''The HOMO and LUMO of ethylene-butadiene transition structure''</div>

As shown in '''Figure 16.''', The bond length of the partly bonded σ C-C bond of the ethylene-butadiene transition structure are calculated to be 2.12 Å. sp<sup>2</sup> and sp<sup>3</sup> C-C bondlengths are typically 1.47 Å and 1.54 Å respectively. And the van der Waals radius of the carbon atom is 1.7 Å.

[[File:Ii bond length.PNG|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 16.''' ''Calculated bond length of partly formed σ C-C bond in ethylene-dibutadiene transition structure''</div>]]

The vibration was animated ('''Figure 17.''') and the calculation gave an imaginary frequency of magnitude 955.67 cm<sup>-1</sup>.
[[File:Ii ethylene-dibutadiene.gif|center]]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 17.''' ''The animation of ethylene-dibutadiene transition structure''</div>

===Cyclohexa-1,3-diene and maleic anhydride reaction===

====Exo transition structure====
The animation of the vibrating exo transition structure is displayed in '''Figure 17.''', and the magnitude of the imaginary frequency given by the calculation is 812.19 cm<sup>-1</sup>. And the partly formed σ C-C bond length is calculated to be 2.17 Å, as shown in '''Figure 19.'''.

[[File:Iii-exo.svg|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 18.''' ''Simulated IR spectrum of the exo transition structure''</div>]]

[[File:Iii-exo.gif|center]]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 17.''' ''The animation of the exo transition structure''</div>

[[File:Iii-exo-bond length.PNG|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 19.''' ''Calculated bond length of partly formed σ C-C bond in the exo transition structure''</div>]]

{| class="wikitable" style="margin: 1em auto 1em auto;"
| align="center" style="background:#f0f0f0;"|'''Energy'''
| align="center" style="background:#f0f0f0;"|'''Energy/Hatrees ''Exo'''''
|-
| Sum of electronic and zero-point Energies||0.134882
|-
| Sum of electronic and thermal Energies ||0.144882
|-
| Sum of electronic and thermal Enthalpies ||0.145826
|-
| Sum of electronic and thermal Free Energies ||0.099118
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 7.''' ''Sum of energies of the exo transition structure''</div>

{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
| ''' '''
| width="80" align="center" | '''HOMO'''
| width="100" align="center" | '''LUMO'''
| width="200" align="center" | '''Discussion'''
|-
| '''Ethylene-butadiene transition structure'''
| align="center" | [[File:Iii-exo-HOMO.PNG|200px]]
| align="center" | [[File:Iii-exo-LUMO.PNG|200px]]
| align="center" | Both of the HOMO and LUMO are anti-symmetrical
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 8.''' ''The HOMO and LUMO of the exo transition structure''</div>

====Endo transition structure====
Figure '''18.''' shown is the animated vibration of the endo transition structure. The simulation gave an imaginary frequency of magnitude 806.22 cm<sup>-1</sup>. According to the simulation, the partially formed σ C-C bond of the endo transition structure is 2.16 Å apart ('''Figure 21.'''), which is slightly shorter than the exo one.

[[File:Iii-endo.svg|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 20.''' ''Simulated IR spectrum of the endo transition structure''</div>]]

[[File:Iii-endo.gif|center]]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 19.''' ''The animation of the endo transition structure''</div>

[[File:Iii-endo-bond length.PNG|thumb|<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Figure 21.''' ''Calculated bond length of partly formed σ C-C bond in the endo transition structure''</div>]]

{| class="wikitable" style="margin: 1em auto 1em auto;"
| align="center" style="background:#f0f0f0;"|'''Energy'''
| align="center" style="background:#f0f0f0;"|'''Energy/Hatrees ''Endo'''''
|-
| Sum of electronic and zero-point Energies||0.133493
|-
| Sum of electronic and thermal Energies ||0.143682
|-
| Sum of electronic and thermal Enthalpies ||0.144626
|-
| Sum of electronic and thermal Free Energies ||0.097350
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 9.''' ''Sum of energies of the endo transition structure''</div>

{| class="wikitable" style="margin: 1em auto 1em auto;"
|-
| ''' '''
| width="80" align="center" | '''HOMO'''
| width="100" align="center" | '''LUMO'''
| width="200" align="center" | '''Discussion'''
|-
| '''Ethylene-butadiene transition structure'''
| align="center" | [[File:Iii-endo-HOMO.PNG|200px]]
| align="center" | [[File:Iii-endo-LUMO.PNG|200px]]
| align="center" | Both of the HOMO and LUMO are anti-symmetrical
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 10.''' ''The HOMO and LUMO of the endo transition structure''</div>

=Reference=