ChemWiki - User contributions [en]

File:XJB exots2.jpg

2013-12-06T16:15:19Z

Jx1011:

File:XJB exo ts reopt.log

2013-12-06T16:15:01Z

Jx1011:

Rep:Mod:XJB1130

2013-12-06T16:04:52Z

Jx1011: /* Further work */

==The Cope Rearrangement of 1,5-hexadiene==

The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement reaction ('''Figure 1'''). Its mechanism has been studied by a large number of experimental and computational researches, which is recently accepted that the reaction occurs via either a "chair" or a "boat" transition state structure in a concerted manner. The aim of this exercise is to locate the low-energy minimum and the transition structures on the potential energy surface by Gaussian calculation in order to study its mechanism. The reaction pathway and the activation energy can also be calculated by this calculation.

<center>[[Image:XJB_cope rearrangement.jpg|500px|center|thumb|'''Figure 1'''. ''The Cope rearrangement'']]</center>

===Optimizing the Reactants and Products===

Different conformations of 1,5-hexadiene can be optimized using '''Gaussview'''. The optimized structure of four conformers involved in this discussion and their corresponding energies optimized at '''HF/3-21G''' level of theory are concluded in the following table ('''Table 1''').

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | anti4
! scope="col" | anti2
! scope="col" | gauche
! scope="col" | gauche3
|-
| <jmol><jmolApplet><title>XJB_anti4.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti4.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche3.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| -231.69097Ha || -231.69254Ha || -231.68772Ha ||-231.69266Ha
|}

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 1.''' ''Four conformers of 1,5-hexadiene''</div>

This exercise first involved optimizing a molecule of 1,5-hexadiene with "anti" linkage for the cetral four C atoms at the '''HF/3-21G''' level of theory. The final optimization structure had the total electronic energy of '''-231.69097Ha''' with a '''C<sub>1</sub>''' symmetry. By comparing the energy and point group to that in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1], this structure is confirmed to be ''anti4''. The compositions of energies are also listed in '''Table 2''' as i) the sum of electronic and zero-point energies (potential energy at 0K including the zero-point vibrational energy), ii) the sum of electronic and thermal energies (energy including translational, rotational and vibrational at 298.15K and 1 atm), iii) the sum of electronic and thermal enthalpies (containning an additional correction for RT), and iv) the sum of electronic and thermal free energies (including the entropic contribution to the free energy). For what we are considering, the first two terms are the most useful for further comparisons.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Formula Description'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||E = E<sub>elec</sub> + ZPE||-231.53785
|-
| Sum of electronic and thermal Energies||E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>||-231.53096
|-
| Sum of electronic and thermal Enthalpie||H = E + RT||-231.53002
|-
| Sum of electronic and thermal Free Energie||G = H - TS||-231.56891
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 2.''' ''Frequency calculations of anti4 optimized at '''HF/3-21G''' ([[File:XJB_anti4.log]])''</div>

Next another 1,5-hexadiene molecule with "gauche" linkage for the central four atoms was drawn and optimized at the '''HF/3-21G''' level of theory as before. Energy for this conformation is expected to be higher than that for the ''anti4'' conformation due to the closer position of two alkene ends thus the steric interaction. The final structure had the electronic energy of '''-231.68772Ha''' which was indeed higher than that of the ''anti4'', indicating less favour towards this gauche structure. It had a '''C<sub>2</sub>''' symmetry and was confirmed to be the ''gauche'' conformation.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53439
|-
| Sum of electronic and thermal Energies||-231.52770
|-
| Sum of electronic and thermal Enthalpie||-231.52676
|-
| Sum of electronic and thermal Free Energie||-231.56483
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 3.''' ''Frequency calculations of gauche optimized at '''HF/3-21G''' ([[File:XJB_gauche.log]])''</div>

A lowest energy conformation ''gauche3'' of the reactant molecule was drawn based on and confirmed by [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. The optimization gave a final structure with an energy of '''-231.69266Ha''' with a symmetry of '''C<sub>1</sub>'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53949
|-
| Sum of electronic and thermal Energies||-231.53265
|-
| Sum of electronic and thermal Enthalpie||-231.5317
|-
| Sum of electronic and thermal Free Energie||-231.57064
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 4.''' ''Frequency calculations of gauche3 optimized at '''HF/3-21G''' ([[File:XJB_gauche3.log]])''</div>

The next step required us to draw the C<sub>i</sub> symmetry ''anti2'' conformation of 1,5-hexadiene and optimize it at the '''HF/3-21G''' level of theory. The optimized energy was calculated to be '''-231.69254Ha''' with point group of '''C<sub>i</sub>''', which was consistent with the one given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53954
|-
| Sum of electronic and thermal Energies||-231.53257
|-
| Sum of electronic and thermal Enthalpie||-231.53162
|-
| Sum of electronic and thermal Free Energie||-231.57092
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 5.''' ''Frequency calculations of anti2 optimized at '''HF/3-21G''' ([[File:XJB_anti2.log]])''</div>

The ''anti2'' structure was then reoptimized at '''B3LYP/6-31G*''' which was a improvement over the '''HF/3-21G''' basis set. It adds polarization to atoms and improves the modeling of core electrons thus producing more accurate description of orbitals<ref name="soo">''Nigerian Journal of Chemical Research'', 2007, '''12'''. {{DOI|10.4314/njcr.v12i1.}}</ref>. The reoptimized ''anti2'' structure had the total electronic energy of '''-234.61171Ha''' and the same symmetry C<sub>i</sub>. The comparison of two structures optimized using different levels of theory is summarized in '''Table 6'''. We can see that they have exactly the same dihedral angle of the central four C atoms at 180.00<sup>o</sup> and angles between three of them are very similar as 111.3<sup>o</sup> and 112.7<sup>o</sup> respectively. Thus the the geometry change using '''HF/3-21G''' or '''B3LYP/6-31G*''' is trivial.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Before re-optimization
! scope="col" | After re-optimization
|-
| <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2reoptimized</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2reoptimized.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| ''anti2 optimized at '''HF/3-21G''''' || ''anti2 optimized at '''B3LYP/6-31G*'''''
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 6.''' ''Two optimization of anti2 at '''HF/3-21G''' and '''B3LYP/6-31G*'''([[File:XJB_anti2.log]])''</div>

The frequency calculation was run as well and a list of energies is summarized in '''Table 7'''. A list of all the vibrational frequencies was also generated and no imaginary (negative) frequency was observed and these vibrations are visualized by simulating the infrared spectrum in '''Figure 2'''.

{| align="center"
|- align="center"
|
{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-234.46920
|-
| Sum of electronic and thermal Energies||-234.46186
|-
| Sum of electronic and thermal Enthalpie||-234.46091
|-
| Sum of electronic and thermal Free Energie||-234.50078
|}
|
[[Image:XJB_IR_anti2.jpg|300px]]
|- align="center"
| align="center" | '''Table 7.''' ''Frequency calculations of anti2 optimized at '''B3LYP/6-31G*'''
([[File:XJB_anti2_reoptimized.log]])''
| align="center" | '''Figure 2.''' ''The generated IR spectrum of optimized '''B3LYP/6-31G*''' structure''
|}

===Optimizing the "Chair" and "Boat" Transition Structures===

The study of the chair and boat transition states was carried out using three different approaches. The first one was called the Hessian which involved computing the force constant matrix; the second one was to freeze the reaction coordinate and then optimized the structure once the molecule was fully relaxed; the last one was to use QST2 method to specify the reactants and products then finding the transition structure.

<center>[[Image:XJB_2 ts.jpg|500px|center|thumb|'''Figure 3'''. ''The two transition states of Cope rearrangement'']]</center>

====The Chair Transition State====

In order to set up a transition state optimization, first an allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimized at the '''HF/3-21G''' level of theory. Then this half of the transition structure was duplicated with rotation to make the approximate resembling of the chair transition state with terminal ends of the fragments 2.2 Å apart. The chair transition structure optimization was then done by the first two different approaches.

<center>[[Image:XJB_guess ts.jpg|400px|center|thumb|'''Figure 4'''. ''The guessed chair transition structure of two allyl fragments'']]</center>

For the first optimization approach, the force constants were computed (the Hessian) and the chair transition state was optimized to a TS (Berny) at the '''HF/3-21G''' level of theory. The frequency calculation was carried out at the same time and gave an imaginary frequency at -818 cm<sup>-1</sup>. The electronic energy was calculated to be '''-231.61932Ha''' with C<sub>2h</sub> symmetry point group. The distance between the terminal carbons was 2.0203 Å. An animation of this transition state vibrations was generated to confirm the corresponding Cope rearrangement.

<center>[[File:XJB_chair ts animation.GIF|400px|center|thumb|'''Figure 5.''' ''Animation of the Cope rearrangement via the chair transition state'']]</center>

The chair transition state was then reoptimized using the '''B3LYP/6-31G*''' level of theory and carried out frequency calculations as well. The electronic energy calculated this time was '''-234.55698Ha''' with an imaginary frequency at -566 cm<sup>-1</sup>. We can see that with the geometries quite similar, the energies differ markedly calculated using two levels of theory. The summary and comparison of the calculations optimized at two levels of theory are summarized in '''Table 8'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46670||-234.41493
|-
| Sum of electronic and thermal Energies||-231.46134||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40807
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir1.jpg|200px]]||[[Image:XJB_chairir2.jpg|200px]]
|-
| ||[[File:XJB_chairts1.1.log]]||[[File:XJB_chairts1.2.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 8.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the first approach''</div>

The second approach is the frozen coordinate method using the Redundant Coord Editor to freeze the bond lengths of the terminal ends to 2.2 Å. First the calculation failed because of the recent update of GaussView. After editting the coordinate distance (B 5 1 2.2 F) and the frozen distance (B 5 1 F) the optimization went well this time. The structure was then optimized again using a normal guess Hessian modified including the two differentiating coordinates. This chair transition state was then reoptimized again using the '''B3LYP/6-31G*''' level of theory. The electronic energies were calculated to be the same as in the first approach and frequency calculations were summarized in '''Table 9'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46671||-234.41490
|-
| Sum of electronic and thermal Energies||-231.46135||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40806
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir3.jpg|200px]]||[[Image:XJB_chairir4.jpg|200px]]
|-
| ||[[File:XJB_chairts2.1.log]]||[[File:XJB_chairts2.2.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 9.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the second approach''</div>

As the same energy results given, the bond forming/breaking bond lengths in the chair transition structures in the above two different approaches were compared. The first approach gave the bond lengths of 2.0203 Å while the second approach gave 2.0207 Å. Considering the accuracy limit of bond length estimation in Gaussview, these two bond lengths can be seen as exactly the same. Two methods worked reasonably well because of the good assumption of transition structures. However for more complicated and larger molecule, the Hessian method may fail due to the difficulty to predict the transition geometry (different surface curvature). Thus better way to generate such transition structures is to freeze reaction coordinate.

{| align="center"
|- align="center"
|
[[Image:XJB_chairts bondlength1.jpg|350px|thumb|'''Figure 6.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by Hession'']]
|
[[Image:XJB_chairts bondlength2.jpg|350px|thumb|'''Figure 7.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by frozen coordinate'']]
|- align="center"
|}

====The Boat Transition State====

The boat transition structure was then optimized using the '''QST2''' method. The same numbering for the reactant and product molecules was important in order to specify for this reaction. The renumbering of the molecules is shown as in '''Figure 8''', but the following optimization failed and gave a twisted structure. The reactant and product geometries were then modified further by changing the central C-C-C-C dihedral angle to 0<sup>o</sup> and the inside two C-C-C (i.e. C2-C3-C4 and C3-C4-C5) angles to 100<sup>o</sup>. The job was run successfully and frequency calculations were done with the boat transition state generated. Only one imaginary frequency was detected at -840 cm<sup>-1</sup>. The electronic energy was '''-231.60280Ha''' with C<sub>2v</sub> symmetry. The distances between the terminal carbons of the allyl fragments were examined to be both 2.140 Å which were comparably longer than that of the chair transition structure.

{| align="center"
|- align="center"
|
[[Image:XJB_QST1.jpg|400px|thumb|'''Figure 8.''' ''Numbering of reactant and product'']]
|
[[Image:XJB_QST2.jpg|410px|thumb|'''Figure 9.''' ''Re-numbering of reactant and product'']]
|- align="center"
|}

<center>[[File:XJB_boat ts animation.GIF|400px|center|thumb|'''Figure 10.''' ''Animation of the Cope rearrangement via the boat transition state'']]</center>

The structure was then reoptimized at the higher '''B3LYP/6-31G*''' theory of level and the electronic energy was calculated to be '''-234.54309Ha''' with very similar geometry. The frequency calculation generated an imaginary frequency at -530 cm<sup>-1</sup>. The summary and comparison of two levels' calculations are carried out in the following table.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.45093||-234.40234
|-
| Sum of electronic and thermal Energies||-231.44530||-234.39601
|-
| Sum of electronic and thermal Enthalpie|| -231.44436||-234.39506
|-
| Sum of electronic and thermal Free Energie||-231.47977||-234.43175
|-
| Generated IR spectrum||[[Image:XJB_boatir1.jpg|200px]]||[[Image:XJB_boatir2.jpg|200px]]
|-
| ||[[File:XJB_boat ts1.log]]||[[File:XJB_boat ts2.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 10.''' ''Frequency calculations of the boat transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by QST2''</div>

====Intrinsic Reaction Coordinate====

Next a useful method called '''Intrinct Reaction Coordinate (IRC)''' was applied to the Hessian optimized chair transition structure to follow the mininmum energy path (MEP) from a transition structure down to its local minimum on a potential energy surface. IRC can be used to verify that the transition state we found actually connects the reactants and products, or to identify any reaction intermediates, or to generate an animation of the reaction<ref name="IRC">''The Journal of Chemical Physics.'', 2005, '''122''', pp 1-17. {{DOI|10.1063/1.1927521}}</ref>. The symmetrical reaction coordinate was computed only in the forward direction with force constant always calculated and the number of points along IRC 50. The last structure generated along IRC is given in '''Figure 11''' with the energy calculated as '''-231.69158Ha''' and the dihedral angle of the central four C atoms and the internal angle between three C atoms measured as ''D'' = -67.188<sup>o</sup> ''A'' = 111.78<sup>o</sup>. This energy does not match to any of the conformations discussed in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] thus a minimum geometry has not reached yet.

{| align="center"
|- align="center"
|
<jmol><jmolApplet><title>XJB_IRC.mol</title><color>white</color>
<size>200</size><script>zoom=5</script>
<uploadedFileContents>XJB_IRC.mol</uploadedFileContents>
</jmolApplet></jmol>
|
[[Image:XJB_irc.jpg|500px]]
|- align="center"
| align="center" | '''Figure 11.''' ''The energy minimum structure along IRC ([[File:XJB_IRC.log]])''
| align="center" | '''Figure 12.''' ''Total energy along IRC''
|}

Three improving options can be used to reach the minimum energy. The first one is to run a normal Hessian optimization at the last point on IRC; the second one involves using a larger number of points (in this case 100 was used); the last one is to redo the IRC computation specifying to compute force constant at every step. Three methods were all used and the results were summarized in '''Table 11'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Improved IRC method 1
! scope="col" | Improved IRC method 2
! scope="col" | Improved IRC method 3
|-
| [[Image:XJB_IRC1.jpg|300px|center|IRC1]]||[[Image:XJB_IRC2.jpg|300px|center|IRC3]] || [[Image:XJB_IRC3.jpg|300px|center|IRC3]]
|-
| ''D'' = -64.17<sup>o</sup> ''A'' = 112.04<sup>o</sup> || ''D'' = -66.83<sup>o</sup> ''A'' = 111.82<sup>o</sup> || ''D'' = -64.17<sup>o</sup> ''A'' = 112.04<sup>o</sup>
|-
| -231.69167Ha || -231.69150Ha || -231.69167Ha
|-
| [[File:XJB_IRC1.log]]||[[File:XJB_IRC2.log]]||[[File:XJB_IRC3.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 11.''' ''Comparison of three improvement methods of IRC computation upon the chair transition structure''</div>

The first method was very quick but maybe generated relatively low-accuracy result; the second approach involved the controlling of number of points which might result in ending up at the wrong direction and wrong structure; the third method was the most suitable option for this small system but it took longer time than the previous two. From '''Table 11''' it is concluded that method 1 and 3 gave the same lowest energy which was consistent to the energy of the ''gauche2'' conformation given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

====Activation Energies====

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | '''HF/3-21G''' at 0K
! scope="col" | '''HF/3-21G''' at 298.15K
! scope="col" | '''B3LYP/6-31G*''' at 0K
! scope="col" | '''B3LYP/6-31G*''' at 298.15K
! scope="col" | Expt. at 0K
|-
| '''ΔE (chair)''' in kcal/mol||45.71||44.70||34.05||33.16||33.5 ± 0.5
|-
| '''ΔE (boat)''' in kcal/mol||55.60||54.76||41.96||41.32||44.7 ± 0.5
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 12.''' ''Summary of activation energies for the Cope rearrangement via both transition states in kcal/mol''</div>

The activation energy calculation was done by calculating the difference in energies between the optimized reactant ''anti2'' and the optimized chair and boat transition states. It is observed that chair conformation gives generally lower activation energy than the boat structure thus more favoured transition state. The structure optimized at '''B3LYP/6-31G*''' level of theory at 0K has closer activation energy to the experimental result. Thus '''B3LYP/6-31G*''' gives a better estimation of this transition structure than '''HF/3-21G'''.

==The Diels Alder Cycloaddition==

In this exercise, AM1 semi-empirical method was used to compute the Diels Alder cycloaddition which is a concerted pericyclic reaction. The driving force of this reaction is the formation of a new σ bond between the π orbitals of the dieneophile and the diene which is stronger than the π bond. The reaction is not allowed without the HOMO-LUMO interation and a symmetry overlap.

===Cis Butadiene===

<center>[[Image:prototype Diels Alder.jpg|500px|center|thumb|'''Figure 13'''. ''The prototype reaction between butadiene and ethylene'']]</center>

To study a prototype Diels Alder cycloaddition reaction between butadiene and ethylene, first a cis butadiene molecule was built using Gaussview and optimized using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the butadiene were plotted. As seen the HOMO of cis-butadiene is anti-symmetric ('''a''') and the LUMO is symmetric ('''s''') with respect to plane.

{| align="center"
|- align="center"
|
[[Image:XJB_HOMO.jpg|280px|thumb|'''Figure 14.''' ''Anti-symmetric HOMO of cis-butadiene'']]
|
[[Image:XJB_LUMO.jpg|300px|thumb|'''Figure 15.''' ''Symmetric HOMO of cis-butadiene'']]
|- align="center"
|}

===The Transition State of prototype reaction between ethylene and butadiene===

The Hessian method was used to estimate transition state for the reaction between ethylene and butadiene because these two reactants are relatively small and simple molecules. The transition state was drawn by first building a structure as (a) in '''Figure 16''' then removing the -CH<sub>2</sub>-CH<sub>2</sub>- group to form the envelop structure as in (b). The '''clean''' option should not be used and the dashed bond length was guessed to be larger than 2.2 Å otherwise the geometry change would cause a failure in optimization.

<center>[[Image:XJB_dielsguessts.jpg|300px|center|thumb|'''Figure 16'''. ''The guessing transition state for the reaction between butadiene and ethylene'']]</center>

<center>[[File:XJB_butadiene and ethylene ts.GIF|400px|center|thumb|'''Figure 17.''' ''Animation of the butadiene and ethylene cycloaddition transition state'' [[File:XJB_prototype DA.log]]]]</center>

After the Hessian optimization, the transition structure has been successfully determined and an animation was generated in '''Figure 17'''. It has a C<sub>1</sub> symmetry point group and the bond lengths of two partly formed C-C bonds were measured to be 2.12Å. The typical sp<sup>3</sup> and sp<sup>2</sup> C-C bond lengths are 1.54 Å and 1.47 Å respectively<ref name="bondlength">''J. Phys. C: Solid State Phys.'', 1986, '''19''', pp 4613-4621.</ref>. The van der Waals radius of the C atom is 1.7 Å<ref name="bondlength2">''Z. Naturforsch.'', 2007, '''62b''', pp 235-243.</ref>. The measured bond distance of the transition structure is larger than typical bond lengths but still within the van der Waals radius, suggesting the bond forming interation between the carbon atoms.

From the animation ('''Figure 17''') the vibration directions of two molecules are opposite and they are approaching and leaving each other at the same time. This suggests the formation of the two bonds synchronous. The vibrational frequencies were also calculated and an infrared spectrum was simulated ('''Figure 20'''). An imaginary frequency of magnitude -955 cm<sup>-1</sup> was seen and the lowest positive frequency was observed at 147 cm<sup>-1</sup> which corresponded to the reactant vibration but not the bond formation.

{| align="center"
|
[[Image:XJB_HOMOts.jpg|300px|thumb|'''Figure 18'''. HOMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_LUMOts.jpg|300px|thumb|'''Figure 19'''. LUMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_IRts.jpg|300px|thumb|'''Figure 20'''. Generated IR of transition state of reaction between cis-butadiene and ethylene]]
|}

The HOMO and LUMO molecular orbitals of this transition structure were visualized in '''Figure 18''' and '''Figure 19'''. From these two figures it can be observed that the HOMO at the transition structure is anti-symmetric ('''a''') while the LUMO is symmetric ('''s'''). Thus this anti-symmetric transition state HOMO is formed by the HOMO of cis-butadiene and the LUMO of ethylene with the knowledge that they are both anti-symmetrical. This conclusion agrees with the previous talked conditions for allowed reaction.

===The cyclohexa-1,3-diene reaction with maleic anhydride===

<center>[[Image:XJB_mechanism3.jpg|500px|center|thumb|'''Figure 21'''. ''The endo and exo products of the Diels Alder reaction between cyclohexa-1,3-diene and maleic anhydride'']]</center>

The exo and endo transition states were drawn according to '''Figure 21''' with the new C-C bond formation line dashed and bond length estimation as 2.5 Å. Then the same Hessian method was applied to optimize the structures. The electronic energy of the endo structure was calculated to be '''-0.05150Ha''' and '''-0.05042Ha''' for the exo. This reaction is supposed to be kinetically controlled thus the lower energy endo transition state is more favoured.

The bond lengths of the partly formed C-C bond are 2.16 Å and 2.17 Å for endo and exo transition states respectively. Another measured C-C bond length is shown in '''Table 13''' as 2.89 Å for endo (C3-C8 and C2-C7) and 2.95 Å for endo (C7-C10 and C8-C9). The longer atom distance indicates the larger steric repulsion in the exo transition state because of the extra existence of hydrogen atoms on C7 and C8.

Woodward and Hoffmann also proposed a secondary orbital overlap statement as an explanation of the endo stereo-preference in Diels Alder reactions<ref name="soot">''J. Org. Chem.'', 1987, '''52(8)''', pp 1469–1474. {{DOI|10.1021/jo00384a016}}</ref>. This secondary orbital overlap does not involve bond changes but π orbital interactions as the highlighted carbon atoms of the endo transition state in '''Table 13'''. Such overlap stablises the structure and lowers the energy of the endo transition state while there is no such effect existed in the exo transition state.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | endo
! scope="col" | exo
|-
| '''Animation'''||[[File:XJB_endots.GIF|300px]]||[[File:XJB_exots.GIF|300px]]
|-
| '''Sum of electronic and zero-point Energies'''||0.133493||0.134880
|-
| '''Sum of electronic and thermal Energies'''||0.143683||0.144881
|-
| '''Sum of electronic and thermal Enthalpies'''||0.144627||0.145825
|-
| '''Sum of electronic and thermal Free Energies'''||0.097349||0.099117
|-
| '''Bond lengths'''||[[Image:XJB_endomeasure.jpg|300px]]||[[Image:XJB_exomeasure.jpg|300px]]
|-
| '''HOMO'''||[[Image:XJB_endoHOMO.jpg|300px]]||[[Image:XJB_exoHOMO.jpg|285px]]
|-
| '''Secondary orbital overlap effect'''||[[Image:XJB_endosoo.jpg|300px]]||[[Image:XJB_exosoo.jpg|300px]]
|-
| ||[[File:XJB_endo ts.log]]||[[File:XJB_exo ts.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 13.''' ''Summary of comparison of properties of endo and exo transition structures for reaction between cyclohexa-1,3-diene and maleic anhydride''</div>

===Further work===

The semiempirical/AM1 calculation is based on the "Neglect of Differential Diatomic Overlap (NDDO)" integral approximation, which neglects all two-electron integrals involving two-center charge distributions are neglected<ref>http://www.cup.uni-muenchen.de/ch/compchem/energy/semi1.html</ref>. The better optimization towards the endo and exo transition states should be done first by AM1 optimization followed by '''B3LYP/6-31G*''' reoptimization<ref>''J. Org. Chem.'', 2003, '''68 (19)''', pp 7158–7166. {{DOI|10.1021/jo0348827}}</ref>. The results are summarized in '''Table 14'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | endo
! scope="col" | exo
|-
| '''Structure'''||[[Image:XJB_endots2.jpg|300px]]||[[Image:XJB_exots2.jpg|300px]]
|-
| '''Electronic energy/Hartrees'''||-612.68340||-612.67931
|-
| ||[[File:XJB_endo ts reopt.log]]||[[File:XJB_exo ts reopt.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 14.''' ''Reoptimization using '''B3LYP/6-31G*''' of endo and exo transition structures for reaction between cyclohexa-1,3-diene and maleic anhydride''</div>

==Reference==

<references/>

File:XJB endo ts reopt.log

2013-12-06T16:04:18Z

Jx1011:

Rep:Mod:XJB1130

2013-12-06T16:03:59Z

Jx1011: /* Further work */

==The Cope Rearrangement of 1,5-hexadiene==

The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement reaction ('''Figure 1'''). Its mechanism has been studied by a large number of experimental and computational researches, which is recently accepted that the reaction occurs via either a "chair" or a "boat" transition state structure in a concerted manner. The aim of this exercise is to locate the low-energy minimum and the transition structures on the potential energy surface by Gaussian calculation in order to study its mechanism. The reaction pathway and the activation energy can also be calculated by this calculation.

<center>[[Image:XJB_cope rearrangement.jpg|500px|center|thumb|'''Figure 1'''. ''The Cope rearrangement'']]</center>

===Optimizing the Reactants and Products===

Different conformations of 1,5-hexadiene can be optimized using '''Gaussview'''. The optimized structure of four conformers involved in this discussion and their corresponding energies optimized at '''HF/3-21G''' level of theory are concluded in the following table ('''Table 1''').

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | anti4
! scope="col" | anti2
! scope="col" | gauche
! scope="col" | gauche3
|-
| <jmol><jmolApplet><title>XJB_anti4.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti4.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche3.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| -231.69097Ha || -231.69254Ha || -231.68772Ha ||-231.69266Ha
|}

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 1.''' ''Four conformers of 1,5-hexadiene''</div>

This exercise first involved optimizing a molecule of 1,5-hexadiene with "anti" linkage for the cetral four C atoms at the '''HF/3-21G''' level of theory. The final optimization structure had the total electronic energy of '''-231.69097Ha''' with a '''C<sub>1</sub>''' symmetry. By comparing the energy and point group to that in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1], this structure is confirmed to be ''anti4''. The compositions of energies are also listed in '''Table 2''' as i) the sum of electronic and zero-point energies (potential energy at 0K including the zero-point vibrational energy), ii) the sum of electronic and thermal energies (energy including translational, rotational and vibrational at 298.15K and 1 atm), iii) the sum of electronic and thermal enthalpies (containning an additional correction for RT), and iv) the sum of electronic and thermal free energies (including the entropic contribution to the free energy). For what we are considering, the first two terms are the most useful for further comparisons.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Formula Description'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||E = E<sub>elec</sub> + ZPE||-231.53785
|-
| Sum of electronic and thermal Energies||E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>||-231.53096
|-
| Sum of electronic and thermal Enthalpie||H = E + RT||-231.53002
|-
| Sum of electronic and thermal Free Energie||G = H - TS||-231.56891
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 2.''' ''Frequency calculations of anti4 optimized at '''HF/3-21G''' ([[File:XJB_anti4.log]])''</div>

Next another 1,5-hexadiene molecule with "gauche" linkage for the central four atoms was drawn and optimized at the '''HF/3-21G''' level of theory as before. Energy for this conformation is expected to be higher than that for the ''anti4'' conformation due to the closer position of two alkene ends thus the steric interaction. The final structure had the electronic energy of '''-231.68772Ha''' which was indeed higher than that of the ''anti4'', indicating less favour towards this gauche structure. It had a '''C<sub>2</sub>''' symmetry and was confirmed to be the ''gauche'' conformation.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53439
|-
| Sum of electronic and thermal Energies||-231.52770
|-
| Sum of electronic and thermal Enthalpie||-231.52676
|-
| Sum of electronic and thermal Free Energie||-231.56483
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 3.''' ''Frequency calculations of gauche optimized at '''HF/3-21G''' ([[File:XJB_gauche.log]])''</div>

A lowest energy conformation ''gauche3'' of the reactant molecule was drawn based on and confirmed by [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. The optimization gave a final structure with an energy of '''-231.69266Ha''' with a symmetry of '''C<sub>1</sub>'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53949
|-
| Sum of electronic and thermal Energies||-231.53265
|-
| Sum of electronic and thermal Enthalpie||-231.5317
|-
| Sum of electronic and thermal Free Energie||-231.57064
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 4.''' ''Frequency calculations of gauche3 optimized at '''HF/3-21G''' ([[File:XJB_gauche3.log]])''</div>

The next step required us to draw the C<sub>i</sub> symmetry ''anti2'' conformation of 1,5-hexadiene and optimize it at the '''HF/3-21G''' level of theory. The optimized energy was calculated to be '''-231.69254Ha''' with point group of '''C<sub>i</sub>''', which was consistent with the one given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53954
|-
| Sum of electronic and thermal Energies||-231.53257
|-
| Sum of electronic and thermal Enthalpie||-231.53162
|-
| Sum of electronic and thermal Free Energie||-231.57092
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 5.''' ''Frequency calculations of anti2 optimized at '''HF/3-21G''' ([[File:XJB_anti2.log]])''</div>

The ''anti2'' structure was then reoptimized at '''B3LYP/6-31G*''' which was a improvement over the '''HF/3-21G''' basis set. It adds polarization to atoms and improves the modeling of core electrons thus producing more accurate description of orbitals<ref name="soo">''Nigerian Journal of Chemical Research'', 2007, '''12'''. {{DOI|10.4314/njcr.v12i1.}}</ref>. The reoptimized ''anti2'' structure had the total electronic energy of '''-234.61171Ha''' and the same symmetry C<sub>i</sub>. The comparison of two structures optimized using different levels of theory is summarized in '''Table 6'''. We can see that they have exactly the same dihedral angle of the central four C atoms at 180.00<sup>o</sup> and angles between three of them are very similar as 111.3<sup>o</sup> and 112.7<sup>o</sup> respectively. Thus the the geometry change using '''HF/3-21G''' or '''B3LYP/6-31G*''' is trivial.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Before re-optimization
! scope="col" | After re-optimization
|-
| <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2reoptimized</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2reoptimized.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| ''anti2 optimized at '''HF/3-21G''''' || ''anti2 optimized at '''B3LYP/6-31G*'''''
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 6.''' ''Two optimization of anti2 at '''HF/3-21G''' and '''B3LYP/6-31G*'''([[File:XJB_anti2.log]])''</div>

The frequency calculation was run as well and a list of energies is summarized in '''Table 7'''. A list of all the vibrational frequencies was also generated and no imaginary (negative) frequency was observed and these vibrations are visualized by simulating the infrared spectrum in '''Figure 2'''.

{| align="center"
|- align="center"
|
{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-234.46920
|-
| Sum of electronic and thermal Energies||-234.46186
|-
| Sum of electronic and thermal Enthalpie||-234.46091
|-
| Sum of electronic and thermal Free Energie||-234.50078
|}
|
[[Image:XJB_IR_anti2.jpg|300px]]
|- align="center"
| align="center" | '''Table 7.''' ''Frequency calculations of anti2 optimized at '''B3LYP/6-31G*'''
([[File:XJB_anti2_reoptimized.log]])''
| align="center" | '''Figure 2.''' ''The generated IR spectrum of optimized '''B3LYP/6-31G*''' structure''
|}

===Optimizing the "Chair" and "Boat" Transition Structures===

The study of the chair and boat transition states was carried out using three different approaches. The first one was called the Hessian which involved computing the force constant matrix; the second one was to freeze the reaction coordinate and then optimized the structure once the molecule was fully relaxed; the last one was to use QST2 method to specify the reactants and products then finding the transition structure.

<center>[[Image:XJB_2 ts.jpg|500px|center|thumb|'''Figure 3'''. ''The two transition states of Cope rearrangement'']]</center>

====The Chair Transition State====

In order to set up a transition state optimization, first an allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimized at the '''HF/3-21G''' level of theory. Then this half of the transition structure was duplicated with rotation to make the approximate resembling of the chair transition state with terminal ends of the fragments 2.2 Å apart. The chair transition structure optimization was then done by the first two different approaches.

<center>[[Image:XJB_guess ts.jpg|400px|center|thumb|'''Figure 4'''. ''The guessed chair transition structure of two allyl fragments'']]</center>

For the first optimization approach, the force constants were computed (the Hessian) and the chair transition state was optimized to a TS (Berny) at the '''HF/3-21G''' level of theory. The frequency calculation was carried out at the same time and gave an imaginary frequency at -818 cm<sup>-1</sup>. The electronic energy was calculated to be '''-231.61932Ha''' with C<sub>2h</sub> symmetry point group. The distance between the terminal carbons was 2.0203 Å. An animation of this transition state vibrations was generated to confirm the corresponding Cope rearrangement.

<center>[[File:XJB_chair ts animation.GIF|400px|center|thumb|'''Figure 5.''' ''Animation of the Cope rearrangement via the chair transition state'']]</center>

The chair transition state was then reoptimized using the '''B3LYP/6-31G*''' level of theory and carried out frequency calculations as well. The electronic energy calculated this time was '''-234.55698Ha''' with an imaginary frequency at -566 cm<sup>-1</sup>. We can see that with the geometries quite similar, the energies differ markedly calculated using two levels of theory. The summary and comparison of the calculations optimized at two levels of theory are summarized in '''Table 8'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46670||-234.41493
|-
| Sum of electronic and thermal Energies||-231.46134||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40807
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir1.jpg|200px]]||[[Image:XJB_chairir2.jpg|200px]]
|-
| ||[[File:XJB_chairts1.1.log]]||[[File:XJB_chairts1.2.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 8.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the first approach''</div>

The second approach is the frozen coordinate method using the Redundant Coord Editor to freeze the bond lengths of the terminal ends to 2.2 Å. First the calculation failed because of the recent update of GaussView. After editting the coordinate distance (B 5 1 2.2 F) and the frozen distance (B 5 1 F) the optimization went well this time. The structure was then optimized again using a normal guess Hessian modified including the two differentiating coordinates. This chair transition state was then reoptimized again using the '''B3LYP/6-31G*''' level of theory. The electronic energies were calculated to be the same as in the first approach and frequency calculations were summarized in '''Table 9'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46671||-234.41490
|-
| Sum of electronic and thermal Energies||-231.46135||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40806
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir3.jpg|200px]]||[[Image:XJB_chairir4.jpg|200px]]
|-
| ||[[File:XJB_chairts2.1.log]]||[[File:XJB_chairts2.2.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 9.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the second approach''</div>

As the same energy results given, the bond forming/breaking bond lengths in the chair transition structures in the above two different approaches were compared. The first approach gave the bond lengths of 2.0203 Å while the second approach gave 2.0207 Å. Considering the accuracy limit of bond length estimation in Gaussview, these two bond lengths can be seen as exactly the same. Two methods worked reasonably well because of the good assumption of transition structures. However for more complicated and larger molecule, the Hessian method may fail due to the difficulty to predict the transition geometry (different surface curvature). Thus better way to generate such transition structures is to freeze reaction coordinate.

{| align="center"
|- align="center"
|
[[Image:XJB_chairts bondlength1.jpg|350px|thumb|'''Figure 6.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by Hession'']]
|
[[Image:XJB_chairts bondlength2.jpg|350px|thumb|'''Figure 7.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by frozen coordinate'']]
|- align="center"
|}

====The Boat Transition State====

The boat transition structure was then optimized using the '''QST2''' method. The same numbering for the reactant and product molecules was important in order to specify for this reaction. The renumbering of the molecules is shown as in '''Figure 8''', but the following optimization failed and gave a twisted structure. The reactant and product geometries were then modified further by changing the central C-C-C-C dihedral angle to 0<sup>o</sup> and the inside two C-C-C (i.e. C2-C3-C4 and C3-C4-C5) angles to 100<sup>o</sup>. The job was run successfully and frequency calculations were done with the boat transition state generated. Only one imaginary frequency was detected at -840 cm<sup>-1</sup>. The electronic energy was '''-231.60280Ha''' with C<sub>2v</sub> symmetry. The distances between the terminal carbons of the allyl fragments were examined to be both 2.140 Å which were comparably longer than that of the chair transition structure.

{| align="center"
|- align="center"
|
[[Image:XJB_QST1.jpg|400px|thumb|'''Figure 8.''' ''Numbering of reactant and product'']]
|
[[Image:XJB_QST2.jpg|410px|thumb|'''Figure 9.''' ''Re-numbering of reactant and product'']]
|- align="center"
|}

<center>[[File:XJB_boat ts animation.GIF|400px|center|thumb|'''Figure 10.''' ''Animation of the Cope rearrangement via the boat transition state'']]</center>

The structure was then reoptimized at the higher '''B3LYP/6-31G*''' theory of level and the electronic energy was calculated to be '''-234.54309Ha''' with very similar geometry. The frequency calculation generated an imaginary frequency at -530 cm<sup>-1</sup>. The summary and comparison of two levels' calculations are carried out in the following table.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.45093||-234.40234
|-
| Sum of electronic and thermal Energies||-231.44530||-234.39601
|-
| Sum of electronic and thermal Enthalpie|| -231.44436||-234.39506
|-
| Sum of electronic and thermal Free Energie||-231.47977||-234.43175
|-
| Generated IR spectrum||[[Image:XJB_boatir1.jpg|200px]]||[[Image:XJB_boatir2.jpg|200px]]
|-
| ||[[File:XJB_boat ts1.log]]||[[File:XJB_boat ts2.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 10.''' ''Frequency calculations of the boat transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by QST2''</div>

====Intrinsic Reaction Coordinate====

Next a useful method called '''Intrinct Reaction Coordinate (IRC)''' was applied to the Hessian optimized chair transition structure to follow the mininmum energy path (MEP) from a transition structure down to its local minimum on a potential energy surface. IRC can be used to verify that the transition state we found actually connects the reactants and products, or to identify any reaction intermediates, or to generate an animation of the reaction<ref name="IRC">''The Journal of Chemical Physics.'', 2005, '''122''', pp 1-17. {{DOI|10.1063/1.1927521}}</ref>. The symmetrical reaction coordinate was computed only in the forward direction with force constant always calculated and the number of points along IRC 50. The last structure generated along IRC is given in '''Figure 11''' with the energy calculated as '''-231.69158Ha''' and the dihedral angle of the central four C atoms and the internal angle between three C atoms measured as ''D'' = -67.188<sup>o</sup> ''A'' = 111.78<sup>o</sup>. This energy does not match to any of the conformations discussed in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] thus a minimum geometry has not reached yet.

{| align="center"
|- align="center"
|
<jmol><jmolApplet><title>XJB_IRC.mol</title><color>white</color>
<size>200</size><script>zoom=5</script>
<uploadedFileContents>XJB_IRC.mol</uploadedFileContents>
</jmolApplet></jmol>
|
[[Image:XJB_irc.jpg|500px]]
|- align="center"
| align="center" | '''Figure 11.''' ''The energy minimum structure along IRC ([[File:XJB_IRC.log]])''
| align="center" | '''Figure 12.''' ''Total energy along IRC''
|}

Three improving options can be used to reach the minimum energy. The first one is to run a normal Hessian optimization at the last point on IRC; the second one involves using a larger number of points (in this case 100 was used); the last one is to redo the IRC computation specifying to compute force constant at every step. Three methods were all used and the results were summarized in '''Table 11'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Improved IRC method 1
! scope="col" | Improved IRC method 2
! scope="col" | Improved IRC method 3
|-
| [[Image:XJB_IRC1.jpg|300px|center|IRC1]]||[[Image:XJB_IRC2.jpg|300px|center|IRC3]] || [[Image:XJB_IRC3.jpg|300px|center|IRC3]]
|-
| ''D'' = -64.17<sup>o</sup> ''A'' = 112.04<sup>o</sup> || ''D'' = -66.83<sup>o</sup> ''A'' = 111.82<sup>o</sup> || ''D'' = -64.17<sup>o</sup> ''A'' = 112.04<sup>o</sup>
|-
| -231.69167Ha || -231.69150Ha || -231.69167Ha
|-
| [[File:XJB_IRC1.log]]||[[File:XJB_IRC2.log]]||[[File:XJB_IRC3.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 11.''' ''Comparison of three improvement methods of IRC computation upon the chair transition structure''</div>

The first method was very quick but maybe generated relatively low-accuracy result; the second approach involved the controlling of number of points which might result in ending up at the wrong direction and wrong structure; the third method was the most suitable option for this small system but it took longer time than the previous two. From '''Table 11''' it is concluded that method 1 and 3 gave the same lowest energy which was consistent to the energy of the ''gauche2'' conformation given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

====Activation Energies====

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | '''HF/3-21G''' at 0K
! scope="col" | '''HF/3-21G''' at 298.15K
! scope="col" | '''B3LYP/6-31G*''' at 0K
! scope="col" | '''B3LYP/6-31G*''' at 298.15K
! scope="col" | Expt. at 0K
|-
| '''ΔE (chair)''' in kcal/mol||45.71||44.70||34.05||33.16||33.5 ± 0.5
|-
| '''ΔE (boat)''' in kcal/mol||55.60||54.76||41.96||41.32||44.7 ± 0.5
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 12.''' ''Summary of activation energies for the Cope rearrangement via both transition states in kcal/mol''</div>

The activation energy calculation was done by calculating the difference in energies between the optimized reactant ''anti2'' and the optimized chair and boat transition states. It is observed that chair conformation gives generally lower activation energy than the boat structure thus more favoured transition state. The structure optimized at '''B3LYP/6-31G*''' level of theory at 0K has closer activation energy to the experimental result. Thus '''B3LYP/6-31G*''' gives a better estimation of this transition structure than '''HF/3-21G'''.

==The Diels Alder Cycloaddition==

In this exercise, AM1 semi-empirical method was used to compute the Diels Alder cycloaddition which is a concerted pericyclic reaction. The driving force of this reaction is the formation of a new σ bond between the π orbitals of the dieneophile and the diene which is stronger than the π bond. The reaction is not allowed without the HOMO-LUMO interation and a symmetry overlap.

===Cis Butadiene===

<center>[[Image:prototype Diels Alder.jpg|500px|center|thumb|'''Figure 13'''. ''The prototype reaction between butadiene and ethylene'']]</center>

To study a prototype Diels Alder cycloaddition reaction between butadiene and ethylene, first a cis butadiene molecule was built using Gaussview and optimized using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the butadiene were plotted. As seen the HOMO of cis-butadiene is anti-symmetric ('''a''') and the LUMO is symmetric ('''s''') with respect to plane.

{| align="center"
|- align="center"
|
[[Image:XJB_HOMO.jpg|280px|thumb|'''Figure 14.''' ''Anti-symmetric HOMO of cis-butadiene'']]
|
[[Image:XJB_LUMO.jpg|300px|thumb|'''Figure 15.''' ''Symmetric HOMO of cis-butadiene'']]
|- align="center"
|}

===The Transition State of prototype reaction between ethylene and butadiene===

The Hessian method was used to estimate transition state for the reaction between ethylene and butadiene because these two reactants are relatively small and simple molecules. The transition state was drawn by first building a structure as (a) in '''Figure 16''' then removing the -CH<sub>2</sub>-CH<sub>2</sub>- group to form the envelop structure as in (b). The '''clean''' option should not be used and the dashed bond length was guessed to be larger than 2.2 Å otherwise the geometry change would cause a failure in optimization.

<center>[[Image:XJB_dielsguessts.jpg|300px|center|thumb|'''Figure 16'''. ''The guessing transition state for the reaction between butadiene and ethylene'']]</center>

<center>[[File:XJB_butadiene and ethylene ts.GIF|400px|center|thumb|'''Figure 17.''' ''Animation of the butadiene and ethylene cycloaddition transition state'' [[File:XJB_prototype DA.log]]]]</center>

After the Hessian optimization, the transition structure has been successfully determined and an animation was generated in '''Figure 17'''. It has a C<sub>1</sub> symmetry point group and the bond lengths of two partly formed C-C bonds were measured to be 2.12Å. The typical sp<sup>3</sup> and sp<sup>2</sup> C-C bond lengths are 1.54 Å and 1.47 Å respectively<ref name="bondlength">''J. Phys. C: Solid State Phys.'', 1986, '''19''', pp 4613-4621.</ref>. The van der Waals radius of the C atom is 1.7 Å<ref name="bondlength2">''Z. Naturforsch.'', 2007, '''62b''', pp 235-243.</ref>. The measured bond distance of the transition structure is larger than typical bond lengths but still within the van der Waals radius, suggesting the bond forming interation between the carbon atoms.

From the animation ('''Figure 17''') the vibration directions of two molecules are opposite and they are approaching and leaving each other at the same time. This suggests the formation of the two bonds synchronous. The vibrational frequencies were also calculated and an infrared spectrum was simulated ('''Figure 20'''). An imaginary frequency of magnitude -955 cm<sup>-1</sup> was seen and the lowest positive frequency was observed at 147 cm<sup>-1</sup> which corresponded to the reactant vibration but not the bond formation.

{| align="center"
|
[[Image:XJB_HOMOts.jpg|300px|thumb|'''Figure 18'''. HOMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_LUMOts.jpg|300px|thumb|'''Figure 19'''. LUMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_IRts.jpg|300px|thumb|'''Figure 20'''. Generated IR of transition state of reaction between cis-butadiene and ethylene]]
|}

The HOMO and LUMO molecular orbitals of this transition structure were visualized in '''Figure 18''' and '''Figure 19'''. From these two figures it can be observed that the HOMO at the transition structure is anti-symmetric ('''a''') while the LUMO is symmetric ('''s'''). Thus this anti-symmetric transition state HOMO is formed by the HOMO of cis-butadiene and the LUMO of ethylene with the knowledge that they are both anti-symmetrical. This conclusion agrees with the previous talked conditions for allowed reaction.

===The cyclohexa-1,3-diene reaction with maleic anhydride===

<center>[[Image:XJB_mechanism3.jpg|500px|center|thumb|'''Figure 21'''. ''The endo and exo products of the Diels Alder reaction between cyclohexa-1,3-diene and maleic anhydride'']]</center>

The exo and endo transition states were drawn according to '''Figure 21''' with the new C-C bond formation line dashed and bond length estimation as 2.5 Å. Then the same Hessian method was applied to optimize the structures. The electronic energy of the endo structure was calculated to be '''-0.05150Ha''' and '''-0.05042Ha''' for the exo. This reaction is supposed to be kinetically controlled thus the lower energy endo transition state is more favoured.

The bond lengths of the partly formed C-C bond are 2.16 Å and 2.17 Å for endo and exo transition states respectively. Another measured C-C bond length is shown in '''Table 13''' as 2.89 Å for endo (C3-C8 and C2-C7) and 2.95 Å for endo (C7-C10 and C8-C9). The longer atom distance indicates the larger steric repulsion in the exo transition state because of the extra existence of hydrogen atoms on C7 and C8.

Woodward and Hoffmann also proposed a secondary orbital overlap statement as an explanation of the endo stereo-preference in Diels Alder reactions<ref name="soot">''J. Org. Chem.'', 1987, '''52(8)''', pp 1469–1474. {{DOI|10.1021/jo00384a016}}</ref>. This secondary orbital overlap does not involve bond changes but π orbital interactions as the highlighted carbon atoms of the endo transition state in '''Table 13'''. Such overlap stablises the structure and lowers the energy of the endo transition state while there is no such effect existed in the exo transition state.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | endo
! scope="col" | exo
|-
| '''Animation'''||[[File:XJB_endots.GIF|300px]]||[[File:XJB_exots.GIF|300px]]
|-
| '''Sum of electronic and zero-point Energies'''||0.133493||0.134880
|-
| '''Sum of electronic and thermal Energies'''||0.143683||0.144881
|-
| '''Sum of electronic and thermal Enthalpies'''||0.144627||0.145825
|-
| '''Sum of electronic and thermal Free Energies'''||0.097349||0.099117
|-
| '''Bond lengths'''||[[Image:XJB_endomeasure.jpg|300px]]||[[Image:XJB_exomeasure.jpg|300px]]
|-
| '''HOMO'''||[[Image:XJB_endoHOMO.jpg|300px]]||[[Image:XJB_exoHOMO.jpg|285px]]
|-
| '''Secondary orbital overlap effect'''||[[Image:XJB_endosoo.jpg|300px]]||[[Image:XJB_exosoo.jpg|300px]]
|-
| ||[[File:XJB_endo ts.log]]||[[File:XJB_exo ts.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 13.''' ''Summary of comparison of properties of endo and exo transition structures for reaction between cyclohexa-1,3-diene and maleic anhydride''</div>

===Further work===

The semiempirical/AM1 calculation is based on the "Neglect of Differential Diatomic Overlap (NDDO)" integral approximation, which neglects all two-electron integrals involving two-center charge distributions are neglected<ref>http://www.cup.uni-muenchen.de/ch/compchem/energy/semi1.html</ref>. The better optimization towards the endo and exo transition states should be done first by AM1 optimization followed by '''B3LYP/6-31G*''' reoptimization<ref>''J. Org. Chem.'', 2003, '''68 (19)''', pp 7158–7166. {{DOI|10.1021/jo0348827}}</ref>. The results are summarized in '''Table 14'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | endo
! scope="col" | exo
|-
| '''Structure'''||[[Image:XJB_endots2.jpg|300px]]||[[Image:XJB_exots2.jpg|300px]]
|-
| '''Electronic energy'''||-612.68340||-612.67931
|-
| ||[[File:XJB_endo ts reopt.log]]||[[File:XJB_exo ts reopt.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 14.''' ''Reoptimization using '''B3LYP/6-31G*''' of endo and exo transition structures for reaction between cyclohexa-1,3-diene and maleic anhydride''</div>

==Reference==

<references/>

File:XJB endots2.jpg

2013-12-06T16:02:47Z

Jx1011:

Rep:Mod:XJB1130

2013-12-06T16:02:05Z

Jx1011: /* Activation Energies */

==The Cope Rearrangement of 1,5-hexadiene==

The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement reaction ('''Figure 1'''). Its mechanism has been studied by a large number of experimental and computational researches, which is recently accepted that the reaction occurs via either a "chair" or a "boat" transition state structure in a concerted manner. The aim of this exercise is to locate the low-energy minimum and the transition structures on the potential energy surface by Gaussian calculation in order to study its mechanism. The reaction pathway and the activation energy can also be calculated by this calculation.

<center>[[Image:XJB_cope rearrangement.jpg|500px|center|thumb|'''Figure 1'''. ''The Cope rearrangement'']]</center>

===Optimizing the Reactants and Products===

Different conformations of 1,5-hexadiene can be optimized using '''Gaussview'''. The optimized structure of four conformers involved in this discussion and their corresponding energies optimized at '''HF/3-21G''' level of theory are concluded in the following table ('''Table 1''').

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | anti4
! scope="col" | anti2
! scope="col" | gauche
! scope="col" | gauche3
|-
| <jmol><jmolApplet><title>XJB_anti4.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti4.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche3.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| -231.69097Ha || -231.69254Ha || -231.68772Ha ||-231.69266Ha
|}

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 1.''' ''Four conformers of 1,5-hexadiene''</div>

This exercise first involved optimizing a molecule of 1,5-hexadiene with "anti" linkage for the cetral four C atoms at the '''HF/3-21G''' level of theory. The final optimization structure had the total electronic energy of '''-231.69097Ha''' with a '''C<sub>1</sub>''' symmetry. By comparing the energy and point group to that in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1], this structure is confirmed to be ''anti4''. The compositions of energies are also listed in '''Table 2''' as i) the sum of electronic and zero-point energies (potential energy at 0K including the zero-point vibrational energy), ii) the sum of electronic and thermal energies (energy including translational, rotational and vibrational at 298.15K and 1 atm), iii) the sum of electronic and thermal enthalpies (containning an additional correction for RT), and iv) the sum of electronic and thermal free energies (including the entropic contribution to the free energy). For what we are considering, the first two terms are the most useful for further comparisons.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Formula Description'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||E = E<sub>elec</sub> + ZPE||-231.53785
|-
| Sum of electronic and thermal Energies||E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>||-231.53096
|-
| Sum of electronic and thermal Enthalpie||H = E + RT||-231.53002
|-
| Sum of electronic and thermal Free Energie||G = H - TS||-231.56891
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 2.''' ''Frequency calculations of anti4 optimized at '''HF/3-21G''' ([[File:XJB_anti4.log]])''</div>

Next another 1,5-hexadiene molecule with "gauche" linkage for the central four atoms was drawn and optimized at the '''HF/3-21G''' level of theory as before. Energy for this conformation is expected to be higher than that for the ''anti4'' conformation due to the closer position of two alkene ends thus the steric interaction. The final structure had the electronic energy of '''-231.68772Ha''' which was indeed higher than that of the ''anti4'', indicating less favour towards this gauche structure. It had a '''C<sub>2</sub>''' symmetry and was confirmed to be the ''gauche'' conformation.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53439
|-
| Sum of electronic and thermal Energies||-231.52770
|-
| Sum of electronic and thermal Enthalpie||-231.52676
|-
| Sum of electronic and thermal Free Energie||-231.56483
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 3.''' ''Frequency calculations of gauche optimized at '''HF/3-21G''' ([[File:XJB_gauche.log]])''</div>

A lowest energy conformation ''gauche3'' of the reactant molecule was drawn based on and confirmed by [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. The optimization gave a final structure with an energy of '''-231.69266Ha''' with a symmetry of '''C<sub>1</sub>'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53949
|-
| Sum of electronic and thermal Energies||-231.53265
|-
| Sum of electronic and thermal Enthalpie||-231.5317
|-
| Sum of electronic and thermal Free Energie||-231.57064
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 4.''' ''Frequency calculations of gauche3 optimized at '''HF/3-21G''' ([[File:XJB_gauche3.log]])''</div>

The next step required us to draw the C<sub>i</sub> symmetry ''anti2'' conformation of 1,5-hexadiene and optimize it at the '''HF/3-21G''' level of theory. The optimized energy was calculated to be '''-231.69254Ha''' with point group of '''C<sub>i</sub>''', which was consistent with the one given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53954
|-
| Sum of electronic and thermal Energies||-231.53257
|-
| Sum of electronic and thermal Enthalpie||-231.53162
|-
| Sum of electronic and thermal Free Energie||-231.57092
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 5.''' ''Frequency calculations of anti2 optimized at '''HF/3-21G''' ([[File:XJB_anti2.log]])''</div>

The ''anti2'' structure was then reoptimized at '''B3LYP/6-31G*''' which was a improvement over the '''HF/3-21G''' basis set. It adds polarization to atoms and improves the modeling of core electrons thus producing more accurate description of orbitals<ref name="soo">''Nigerian Journal of Chemical Research'', 2007, '''12'''. {{DOI|10.4314/njcr.v12i1.}}</ref>. The reoptimized ''anti2'' structure had the total electronic energy of '''-234.61171Ha''' and the same symmetry C<sub>i</sub>. The comparison of two structures optimized using different levels of theory is summarized in '''Table 6'''. We can see that they have exactly the same dihedral angle of the central four C atoms at 180.00<sup>o</sup> and angles between three of them are very similar as 111.3<sup>o</sup> and 112.7<sup>o</sup> respectively. Thus the the geometry change using '''HF/3-21G''' or '''B3LYP/6-31G*''' is trivial.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Before re-optimization
! scope="col" | After re-optimization
|-
| <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2reoptimized</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2reoptimized.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| ''anti2 optimized at '''HF/3-21G''''' || ''anti2 optimized at '''B3LYP/6-31G*'''''
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 6.''' ''Two optimization of anti2 at '''HF/3-21G''' and '''B3LYP/6-31G*'''([[File:XJB_anti2.log]])''</div>

The frequency calculation was run as well and a list of energies is summarized in '''Table 7'''. A list of all the vibrational frequencies was also generated and no imaginary (negative) frequency was observed and these vibrations are visualized by simulating the infrared spectrum in '''Figure 2'''.

{| align="center"
|- align="center"
|
{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-234.46920
|-
| Sum of electronic and thermal Energies||-234.46186
|-
| Sum of electronic and thermal Enthalpie||-234.46091
|-
| Sum of electronic and thermal Free Energie||-234.50078
|}
|
[[Image:XJB_IR_anti2.jpg|300px]]
|- align="center"
| align="center" | '''Table 7.''' ''Frequency calculations of anti2 optimized at '''B3LYP/6-31G*'''
([[File:XJB_anti2_reoptimized.log]])''
| align="center" | '''Figure 2.''' ''The generated IR spectrum of optimized '''B3LYP/6-31G*''' structure''
|}

===Optimizing the "Chair" and "Boat" Transition Structures===

The study of the chair and boat transition states was carried out using three different approaches. The first one was called the Hessian which involved computing the force constant matrix; the second one was to freeze the reaction coordinate and then optimized the structure once the molecule was fully relaxed; the last one was to use QST2 method to specify the reactants and products then finding the transition structure.

<center>[[Image:XJB_2 ts.jpg|500px|center|thumb|'''Figure 3'''. ''The two transition states of Cope rearrangement'']]</center>

====The Chair Transition State====

In order to set up a transition state optimization, first an allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimized at the '''HF/3-21G''' level of theory. Then this half of the transition structure was duplicated with rotation to make the approximate resembling of the chair transition state with terminal ends of the fragments 2.2 Å apart. The chair transition structure optimization was then done by the first two different approaches.

<center>[[Image:XJB_guess ts.jpg|400px|center|thumb|'''Figure 4'''. ''The guessed chair transition structure of two allyl fragments'']]</center>

For the first optimization approach, the force constants were computed (the Hessian) and the chair transition state was optimized to a TS (Berny) at the '''HF/3-21G''' level of theory. The frequency calculation was carried out at the same time and gave an imaginary frequency at -818 cm<sup>-1</sup>. The electronic energy was calculated to be '''-231.61932Ha''' with C<sub>2h</sub> symmetry point group. The distance between the terminal carbons was 2.0203 Å. An animation of this transition state vibrations was generated to confirm the corresponding Cope rearrangement.

<center>[[File:XJB_chair ts animation.GIF|400px|center|thumb|'''Figure 5.''' ''Animation of the Cope rearrangement via the chair transition state'']]</center>

The chair transition state was then reoptimized using the '''B3LYP/6-31G*''' level of theory and carried out frequency calculations as well. The electronic energy calculated this time was '''-234.55698Ha''' with an imaginary frequency at -566 cm<sup>-1</sup>. We can see that with the geometries quite similar, the energies differ markedly calculated using two levels of theory. The summary and comparison of the calculations optimized at two levels of theory are summarized in '''Table 8'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46670||-234.41493
|-
| Sum of electronic and thermal Energies||-231.46134||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40807
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir1.jpg|200px]]||[[Image:XJB_chairir2.jpg|200px]]
|-
| ||[[File:XJB_chairts1.1.log]]||[[File:XJB_chairts1.2.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 8.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the first approach''</div>

The second approach is the frozen coordinate method using the Redundant Coord Editor to freeze the bond lengths of the terminal ends to 2.2 Å. First the calculation failed because of the recent update of GaussView. After editting the coordinate distance (B 5 1 2.2 F) and the frozen distance (B 5 1 F) the optimization went well this time. The structure was then optimized again using a normal guess Hessian modified including the two differentiating coordinates. This chair transition state was then reoptimized again using the '''B3LYP/6-31G*''' level of theory. The electronic energies were calculated to be the same as in the first approach and frequency calculations were summarized in '''Table 9'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46671||-234.41490
|-
| Sum of electronic and thermal Energies||-231.46135||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40806
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir3.jpg|200px]]||[[Image:XJB_chairir4.jpg|200px]]
|-
| ||[[File:XJB_chairts2.1.log]]||[[File:XJB_chairts2.2.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 9.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the second approach''</div>

As the same energy results given, the bond forming/breaking bond lengths in the chair transition structures in the above two different approaches were compared. The first approach gave the bond lengths of 2.0203 Å while the second approach gave 2.0207 Å. Considering the accuracy limit of bond length estimation in Gaussview, these two bond lengths can be seen as exactly the same. Two methods worked reasonably well because of the good assumption of transition structures. However for more complicated and larger molecule, the Hessian method may fail due to the difficulty to predict the transition geometry (different surface curvature). Thus better way to generate such transition structures is to freeze reaction coordinate.

{| align="center"
|- align="center"
|
[[Image:XJB_chairts bondlength1.jpg|350px|thumb|'''Figure 6.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by Hession'']]
|
[[Image:XJB_chairts bondlength2.jpg|350px|thumb|'''Figure 7.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by frozen coordinate'']]
|- align="center"
|}

====The Boat Transition State====

The boat transition structure was then optimized using the '''QST2''' method. The same numbering for the reactant and product molecules was important in order to specify for this reaction. The renumbering of the molecules is shown as in '''Figure 8''', but the following optimization failed and gave a twisted structure. The reactant and product geometries were then modified further by changing the central C-C-C-C dihedral angle to 0<sup>o</sup> and the inside two C-C-C (i.e. C2-C3-C4 and C3-C4-C5) angles to 100<sup>o</sup>. The job was run successfully and frequency calculations were done with the boat transition state generated. Only one imaginary frequency was detected at -840 cm<sup>-1</sup>. The electronic energy was '''-231.60280Ha''' with C<sub>2v</sub> symmetry. The distances between the terminal carbons of the allyl fragments were examined to be both 2.140 Å which were comparably longer than that of the chair transition structure.

{| align="center"
|- align="center"
|
[[Image:XJB_QST1.jpg|400px|thumb|'''Figure 8.''' ''Numbering of reactant and product'']]
|
[[Image:XJB_QST2.jpg|410px|thumb|'''Figure 9.''' ''Re-numbering of reactant and product'']]
|- align="center"
|}

<center>[[File:XJB_boat ts animation.GIF|400px|center|thumb|'''Figure 10.''' ''Animation of the Cope rearrangement via the boat transition state'']]</center>

The structure was then reoptimized at the higher '''B3LYP/6-31G*''' theory of level and the electronic energy was calculated to be '''-234.54309Ha''' with very similar geometry. The frequency calculation generated an imaginary frequency at -530 cm<sup>-1</sup>. The summary and comparison of two levels' calculations are carried out in the following table.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.45093||-234.40234
|-
| Sum of electronic and thermal Energies||-231.44530||-234.39601
|-
| Sum of electronic and thermal Enthalpie|| -231.44436||-234.39506
|-
| Sum of electronic and thermal Free Energie||-231.47977||-234.43175
|-
| Generated IR spectrum||[[Image:XJB_boatir1.jpg|200px]]||[[Image:XJB_boatir2.jpg|200px]]
|-
| ||[[File:XJB_boat ts1.log]]||[[File:XJB_boat ts2.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 10.''' ''Frequency calculations of the boat transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by QST2''</div>

====Intrinsic Reaction Coordinate====

Next a useful method called '''Intrinct Reaction Coordinate (IRC)''' was applied to the Hessian optimized chair transition structure to follow the mininmum energy path (MEP) from a transition structure down to its local minimum on a potential energy surface. IRC can be used to verify that the transition state we found actually connects the reactants and products, or to identify any reaction intermediates, or to generate an animation of the reaction<ref name="IRC">''The Journal of Chemical Physics.'', 2005, '''122''', pp 1-17. {{DOI|10.1063/1.1927521}}</ref>. The symmetrical reaction coordinate was computed only in the forward direction with force constant always calculated and the number of points along IRC 50. The last structure generated along IRC is given in '''Figure 11''' with the energy calculated as '''-231.69158Ha''' and the dihedral angle of the central four C atoms and the internal angle between three C atoms measured as ''D'' = -67.188<sup>o</sup> ''A'' = 111.78<sup>o</sup>. This energy does not match to any of the conformations discussed in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] thus a minimum geometry has not reached yet.

{| align="center"
|- align="center"
|
<jmol><jmolApplet><title>XJB_IRC.mol</title><color>white</color>
<size>200</size><script>zoom=5</script>
<uploadedFileContents>XJB_IRC.mol</uploadedFileContents>
</jmolApplet></jmol>
|
[[Image:XJB_irc.jpg|500px]]
|- align="center"
| align="center" | '''Figure 11.''' ''The energy minimum structure along IRC ([[File:XJB_IRC.log]])''
| align="center" | '''Figure 12.''' ''Total energy along IRC''
|}

Three improving options can be used to reach the minimum energy. The first one is to run a normal Hessian optimization at the last point on IRC; the second one involves using a larger number of points (in this case 100 was used); the last one is to redo the IRC computation specifying to compute force constant at every step. Three methods were all used and the results were summarized in '''Table 11'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Improved IRC method 1
! scope="col" | Improved IRC method 2
! scope="col" | Improved IRC method 3
|-
| [[Image:XJB_IRC1.jpg|300px|center|IRC1]]||[[Image:XJB_IRC2.jpg|300px|center|IRC3]] || [[Image:XJB_IRC3.jpg|300px|center|IRC3]]
|-
| ''D'' = -64.17<sup>o</sup> ''A'' = 112.04<sup>o</sup> || ''D'' = -66.83<sup>o</sup> ''A'' = 111.82<sup>o</sup> || ''D'' = -64.17<sup>o</sup> ''A'' = 112.04<sup>o</sup>
|-
| -231.69167Ha || -231.69150Ha || -231.69167Ha
|-
| [[File:XJB_IRC1.log]]||[[File:XJB_IRC2.log]]||[[File:XJB_IRC3.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 11.''' ''Comparison of three improvement methods of IRC computation upon the chair transition structure''</div>

The first method was very quick but maybe generated relatively low-accuracy result; the second approach involved the controlling of number of points which might result in ending up at the wrong direction and wrong structure; the third method was the most suitable option for this small system but it took longer time than the previous two. From '''Table 11''' it is concluded that method 1 and 3 gave the same lowest energy which was consistent to the energy of the ''gauche2'' conformation given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

====Activation Energies====

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | '''HF/3-21G''' at 0K
! scope="col" | '''HF/3-21G''' at 298.15K
! scope="col" | '''B3LYP/6-31G*''' at 0K
! scope="col" | '''B3LYP/6-31G*''' at 298.15K
! scope="col" | Expt. at 0K
|-
| '''ΔE (chair)''' in kcal/mol||45.71||44.70||34.05||33.16||33.5 ± 0.5
|-
| '''ΔE (boat)''' in kcal/mol||55.60||54.76||41.96||41.32||44.7 ± 0.5
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 12.''' ''Summary of activation energies for the Cope rearrangement via both transition states in kcal/mol''</div>

The activation energy calculation was done by calculating the difference in energies between the optimized reactant ''anti2'' and the optimized chair and boat transition states. It is observed that chair conformation gives generally lower activation energy than the boat structure thus more favoured transition state. The structure optimized at '''B3LYP/6-31G*''' level of theory at 0K has closer activation energy to the experimental result. Thus '''B3LYP/6-31G*''' gives a better estimation of this transition structure than '''HF/3-21G'''.

==The Diels Alder Cycloaddition==

In this exercise, AM1 semi-empirical method was used to compute the Diels Alder cycloaddition which is a concerted pericyclic reaction. The driving force of this reaction is the formation of a new σ bond between the π orbitals of the dieneophile and the diene which is stronger than the π bond. The reaction is not allowed without the HOMO-LUMO interation and a symmetry overlap.

===Cis Butadiene===

<center>[[Image:prototype Diels Alder.jpg|500px|center|thumb|'''Figure 13'''. ''The prototype reaction between butadiene and ethylene'']]</center>

To study a prototype Diels Alder cycloaddition reaction between butadiene and ethylene, first a cis butadiene molecule was built using Gaussview and optimized using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the butadiene were plotted. As seen the HOMO of cis-butadiene is anti-symmetric ('''a''') and the LUMO is symmetric ('''s''') with respect to plane.

{| align="center"
|- align="center"
|
[[Image:XJB_HOMO.jpg|280px|thumb|'''Figure 14.''' ''Anti-symmetric HOMO of cis-butadiene'']]
|
[[Image:XJB_LUMO.jpg|300px|thumb|'''Figure 15.''' ''Symmetric HOMO of cis-butadiene'']]
|- align="center"
|}

===The Transition State of prototype reaction between ethylene and butadiene===

The Hessian method was used to estimate transition state for the reaction between ethylene and butadiene because these two reactants are relatively small and simple molecules. The transition state was drawn by first building a structure as (a) in '''Figure 16''' then removing the -CH<sub>2</sub>-CH<sub>2</sub>- group to form the envelop structure as in (b). The '''clean''' option should not be used and the dashed bond length was guessed to be larger than 2.2 Å otherwise the geometry change would cause a failure in optimization.

<center>[[Image:XJB_dielsguessts.jpg|300px|center|thumb|'''Figure 16'''. ''The guessing transition state for the reaction between butadiene and ethylene'']]</center>

<center>[[File:XJB_butadiene and ethylene ts.GIF|400px|center|thumb|'''Figure 17.''' ''Animation of the butadiene and ethylene cycloaddition transition state'' [[File:XJB_prototype DA.log]]]]</center>

After the Hessian optimization, the transition structure has been successfully determined and an animation was generated in '''Figure 17'''. It has a C<sub>1</sub> symmetry point group and the bond lengths of two partly formed C-C bonds were measured to be 2.12Å. The typical sp<sup>3</sup> and sp<sup>2</sup> C-C bond lengths are 1.54 Å and 1.47 Å respectively<ref name="bondlength">''J. Phys. C: Solid State Phys.'', 1986, '''19''', pp 4613-4621.</ref>. The van der Waals radius of the C atom is 1.7 Å<ref name="bondlength2">''Z. Naturforsch.'', 2007, '''62b''', pp 235-243.</ref>. The measured bond distance of the transition structure is larger than typical bond lengths but still within the van der Waals radius, suggesting the bond forming interation between the carbon atoms.

From the animation ('''Figure 17''') the vibration directions of two molecules are opposite and they are approaching and leaving each other at the same time. This suggests the formation of the two bonds synchronous. The vibrational frequencies were also calculated and an infrared spectrum was simulated ('''Figure 20'''). An imaginary frequency of magnitude -955 cm<sup>-1</sup> was seen and the lowest positive frequency was observed at 147 cm<sup>-1</sup> which corresponded to the reactant vibration but not the bond formation.

{| align="center"
|
[[Image:XJB_HOMOts.jpg|300px|thumb|'''Figure 18'''. HOMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_LUMOts.jpg|300px|thumb|'''Figure 19'''. LUMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_IRts.jpg|300px|thumb|'''Figure 20'''. Generated IR of transition state of reaction between cis-butadiene and ethylene]]
|}

The HOMO and LUMO molecular orbitals of this transition structure were visualized in '''Figure 18''' and '''Figure 19'''. From these two figures it can be observed that the HOMO at the transition structure is anti-symmetric ('''a''') while the LUMO is symmetric ('''s'''). Thus this anti-symmetric transition state HOMO is formed by the HOMO of cis-butadiene and the LUMO of ethylene with the knowledge that they are both anti-symmetrical. This conclusion agrees with the previous talked conditions for allowed reaction.

===The cyclohexa-1,3-diene reaction with maleic anhydride===

<center>[[Image:XJB_mechanism3.jpg|500px|center|thumb|'''Figure 21'''. ''The endo and exo products of the Diels Alder reaction between cyclohexa-1,3-diene and maleic anhydride'']]</center>

The exo and endo transition states were drawn according to '''Figure 21''' with the new C-C bond formation line dashed and bond length estimation as 2.5 Å. Then the same Hessian method was applied to optimize the structures. The electronic energy of the endo structure was calculated to be '''-0.05150Ha''' and '''-0.05042Ha''' for the exo. This reaction is supposed to be kinetically controlled thus the lower energy endo transition state is more favoured.

The bond lengths of the partly formed C-C bond are 2.16 Å and 2.17 Å for endo and exo transition states respectively. Another measured C-C bond length is shown in '''Table 13''' as 2.89 Å for endo (C3-C8 and C2-C7) and 2.95 Å for endo (C7-C10 and C8-C9). The longer atom distance indicates the larger steric repulsion in the exo transition state because of the extra existence of hydrogen atoms on C7 and C8.

Woodward and Hoffmann also proposed a secondary orbital overlap statement as an explanation of the endo stereo-preference in Diels Alder reactions<ref name="soot">''J. Org. Chem.'', 1987, '''52(8)''', pp 1469–1474. {{DOI|10.1021/jo00384a016}}</ref>. This secondary orbital overlap does not involve bond changes but π orbital interactions as the highlighted carbon atoms of the endo transition state in '''Table 13'''. Such overlap stablises the structure and lowers the energy of the endo transition state while there is no such effect existed in the exo transition state.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | endo
! scope="col" | exo
|-
| '''Animation'''||[[File:XJB_endots.GIF|300px]]||[[File:XJB_exots.GIF|300px]]
|-
| '''Sum of electronic and zero-point Energies'''||0.133493||0.134880
|-
| '''Sum of electronic and thermal Energies'''||0.143683||0.144881
|-
| '''Sum of electronic and thermal Enthalpies'''||0.144627||0.145825
|-
| '''Sum of electronic and thermal Free Energies'''||0.097349||0.099117
|-
| '''Bond lengths'''||[[Image:XJB_endomeasure.jpg|300px]]||[[Image:XJB_exomeasure.jpg|300px]]
|-
| '''HOMO'''||[[Image:XJB_endoHOMO.jpg|300px]]||[[Image:XJB_exoHOMO.jpg|285px]]
|-
| '''Secondary orbital overlap effect'''||[[Image:XJB_endosoo.jpg|300px]]||[[Image:XJB_exosoo.jpg|300px]]
|-
| ||[[File:XJB_endo ts.log]]||[[File:XJB_exo ts.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 13.''' ''Summary of comparison of properties of endo and exo transition structures for reaction between cyclohexa-1,3-diene and maleic anhydride''</div>

===Further work===

The semiempirical/AM1 calculation is based on the "Neglect of Differential Diatomic Overlap (NDDO)" integral approximation, which neglects all two-electron integrals involving two-center charge distributions are neglected<ref>http://www.cup.uni-muenchen.de/ch/compchem/energy/semi1.html</ref>. The better optimization towards the endo and exo transition states should be done first by AM1 optimization followed by '''B3LYP/6-31G*''' reoptimization<ref>''J. Org. Chem.'', 2003, '''68 (19)''', pp 7158–7166. {{DOI|10.1021/jo0348827}}</ref>. The results are summarized in '''Table 14'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | endo
! scope="col" | exo
|-
| '''Structure'''||[[Image:XJB_endots2.jpg|300px]]||[[Image:XJB_exots2.jpg|300px]]
|-
| '''Electronic energy'''||-612.68340||-612.67931
|-
| ||[[File:endots reopt]]||[[File:exots reopt]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 14.''' ''Reoptimization using '''B3LYP/6-31G*''' of endo and exo transition structures for reaction between cyclohexa-1,3-diene and maleic anhydride''</div>

==Reference==

<references/>

Rep:Mod:XJB1130

2013-12-06T16:00:57Z

Jx1011: /* Further work */

==The Cope Rearrangement of 1,5-hexadiene==

The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement reaction ('''Figure 1'''). Its mechanism has been studied by a large number of experimental and computational researches, which is recently accepted that the reaction occurs via either a "chair" or a "boat" transition state structure in a concerted manner. The aim of this exercise is to locate the low-energy minimum and the transition structures on the potential energy surface by Gaussian calculation in order to study its mechanism. The reaction pathway and the activation energy can also be calculated by this calculation.

<center>[[Image:XJB_cope rearrangement.jpg|500px|center|thumb|'''Figure 1'''. ''The Cope rearrangement'']]</center>

===Optimizing the Reactants and Products===

Different conformations of 1,5-hexadiene can be optimized using '''Gaussview'''. The optimized structure of four conformers involved in this discussion and their corresponding energies optimized at '''HF/3-21G''' level of theory are concluded in the following table ('''Table 1''').

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | anti4
! scope="col" | anti2
! scope="col" | gauche
! scope="col" | gauche3
|-
| <jmol><jmolApplet><title>XJB_anti4.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti4.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche3.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| -231.69097Ha || -231.69254Ha || -231.68772Ha ||-231.69266Ha
|}

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 1.''' ''Four conformers of 1,5-hexadiene''</div>

This exercise first involved optimizing a molecule of 1,5-hexadiene with "anti" linkage for the cetral four C atoms at the '''HF/3-21G''' level of theory. The final optimization structure had the total electronic energy of '''-231.69097Ha''' with a '''C<sub>1</sub>''' symmetry. By comparing the energy and point group to that in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1], this structure is confirmed to be ''anti4''. The compositions of energies are also listed in '''Table 2''' as i) the sum of electronic and zero-point energies (potential energy at 0K including the zero-point vibrational energy), ii) the sum of electronic and thermal energies (energy including translational, rotational and vibrational at 298.15K and 1 atm), iii) the sum of electronic and thermal enthalpies (containning an additional correction for RT), and iv) the sum of electronic and thermal free energies (including the entropic contribution to the free energy). For what we are considering, the first two terms are the most useful for further comparisons.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Formula Description'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||E = E<sub>elec</sub> + ZPE||-231.53785
|-
| Sum of electronic and thermal Energies||E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>||-231.53096
|-
| Sum of electronic and thermal Enthalpie||H = E + RT||-231.53002
|-
| Sum of electronic and thermal Free Energie||G = H - TS||-231.56891
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 2.''' ''Frequency calculations of anti4 optimized at '''HF/3-21G''' ([[File:XJB_anti4.log]])''</div>

Next another 1,5-hexadiene molecule with "gauche" linkage for the central four atoms was drawn and optimized at the '''HF/3-21G''' level of theory as before. Energy for this conformation is expected to be higher than that for the ''anti4'' conformation due to the closer position of two alkene ends thus the steric interaction. The final structure had the electronic energy of '''-231.68772Ha''' which was indeed higher than that of the ''anti4'', indicating less favour towards this gauche structure. It had a '''C<sub>2</sub>''' symmetry and was confirmed to be the ''gauche'' conformation.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53439
|-
| Sum of electronic and thermal Energies||-231.52770
|-
| Sum of electronic and thermal Enthalpie||-231.52676
|-
| Sum of electronic and thermal Free Energie||-231.56483
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 3.''' ''Frequency calculations of gauche optimized at '''HF/3-21G''' ([[File:XJB_gauche.log]])''</div>

A lowest energy conformation ''gauche3'' of the reactant molecule was drawn based on and confirmed by [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. The optimization gave a final structure with an energy of '''-231.69266Ha''' with a symmetry of '''C<sub>1</sub>'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53949
|-
| Sum of electronic and thermal Energies||-231.53265
|-
| Sum of electronic and thermal Enthalpie||-231.5317
|-
| Sum of electronic and thermal Free Energie||-231.57064
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 4.''' ''Frequency calculations of gauche3 optimized at '''HF/3-21G''' ([[File:XJB_gauche3.log]])''</div>

The next step required us to draw the C<sub>i</sub> symmetry ''anti2'' conformation of 1,5-hexadiene and optimize it at the '''HF/3-21G''' level of theory. The optimized energy was calculated to be '''-231.69254Ha''' with point group of '''C<sub>i</sub>''', which was consistent with the one given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53954
|-
| Sum of electronic and thermal Energies||-231.53257
|-
| Sum of electronic and thermal Enthalpie||-231.53162
|-
| Sum of electronic and thermal Free Energie||-231.57092
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 5.''' ''Frequency calculations of anti2 optimized at '''HF/3-21G''' ([[File:XJB_anti2.log]])''</div>

The ''anti2'' structure was then reoptimized at '''B3LYP/6-31G*''' which was a improvement over the '''HF/3-21G''' basis set. It adds polarization to atoms and improves the modeling of core electrons thus producing more accurate description of orbitals<ref name="soo">''Nigerian Journal of Chemical Research'', 2007, '''12'''. {{DOI|10.4314/njcr.v12i1.}}</ref>. The reoptimized ''anti2'' structure had the total electronic energy of '''-234.61171Ha''' and the same symmetry C<sub>i</sub>. The comparison of two structures optimized using different levels of theory is summarized in '''Table 6'''. We can see that they have exactly the same dihedral angle of the central four C atoms at 180.00<sup>o</sup> and angles between three of them are very similar as 111.3<sup>o</sup> and 112.7<sup>o</sup> respectively. Thus the the geometry change using '''HF/3-21G''' or '''B3LYP/6-31G*''' is trivial.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Before re-optimization
! scope="col" | After re-optimization
|-
| <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2reoptimized</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2reoptimized.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| ''anti2 optimized at '''HF/3-21G''''' || ''anti2 optimized at '''B3LYP/6-31G*'''''
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 6.''' ''Two optimization of anti2 at '''HF/3-21G''' and '''B3LYP/6-31G*'''([[File:XJB_anti2.log]])''</div>

The frequency calculation was run as well and a list of energies is summarized in '''Table 7'''. A list of all the vibrational frequencies was also generated and no imaginary (negative) frequency was observed and these vibrations are visualized by simulating the infrared spectrum in '''Figure 2'''.

{| align="center"
|- align="center"
|
{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-234.46920
|-
| Sum of electronic and thermal Energies||-234.46186
|-
| Sum of electronic and thermal Enthalpie||-234.46091
|-
| Sum of electronic and thermal Free Energie||-234.50078
|}
|
[[Image:XJB_IR_anti2.jpg|300px]]
|- align="center"
| align="center" | '''Table 7.''' ''Frequency calculations of anti2 optimized at '''B3LYP/6-31G*'''
([[File:XJB_anti2_reoptimized.log]])''
| align="center" | '''Figure 2.''' ''The generated IR spectrum of optimized '''B3LYP/6-31G*''' structure''
|}

===Optimizing the "Chair" and "Boat" Transition Structures===

The study of the chair and boat transition states was carried out using three different approaches. The first one was called the Hessian which involved computing the force constant matrix; the second one was to freeze the reaction coordinate and then optimized the structure once the molecule was fully relaxed; the last one was to use QST2 method to specify the reactants and products then finding the transition structure.

<center>[[Image:XJB_2 ts.jpg|500px|center|thumb|'''Figure 3'''. ''The two transition states of Cope rearrangement'']]</center>

====The Chair Transition State====

In order to set up a transition state optimization, first an allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimized at the '''HF/3-21G''' level of theory. Then this half of the transition structure was duplicated with rotation to make the approximate resembling of the chair transition state with terminal ends of the fragments 2.2 Å apart. The chair transition structure optimization was then done by the first two different approaches.

<center>[[Image:XJB_guess ts.jpg|400px|center|thumb|'''Figure 4'''. ''The guessed chair transition structure of two allyl fragments'']]</center>

For the first optimization approach, the force constants were computed (the Hessian) and the chair transition state was optimized to a TS (Berny) at the '''HF/3-21G''' level of theory. The frequency calculation was carried out at the same time and gave an imaginary frequency at -818 cm<sup>-1</sup>. The electronic energy was calculated to be '''-231.61932Ha''' with C<sub>2h</sub> symmetry point group. The distance between the terminal carbons was 2.0203 Å. An animation of this transition state vibrations was generated to confirm the corresponding Cope rearrangement.

<center>[[File:XJB_chair ts animation.GIF|400px|center|thumb|'''Figure 5.''' ''Animation of the Cope rearrangement via the chair transition state'']]</center>

The chair transition state was then reoptimized using the '''B3LYP/6-31G*''' level of theory and carried out frequency calculations as well. The electronic energy calculated this time was '''-234.55698Ha''' with an imaginary frequency at -566 cm<sup>-1</sup>. We can see that with the geometries quite similar, the energies differ markedly calculated using two levels of theory. The summary and comparison of the calculations optimized at two levels of theory are summarized in '''Table 8'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46670||-234.41493
|-
| Sum of electronic and thermal Energies||-231.46134||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40807
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir1.jpg|200px]]||[[Image:XJB_chairir2.jpg|200px]]
|-
| ||[[File:XJB_chairts1.1.log]]||[[File:XJB_chairts1.2.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 8.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the first approach''</div>

The second approach is the frozen coordinate method using the Redundant Coord Editor to freeze the bond lengths of the terminal ends to 2.2 Å. First the calculation failed because of the recent update of GaussView. After editting the coordinate distance (B 5 1 2.2 F) and the frozen distance (B 5 1 F) the optimization went well this time. The structure was then optimized again using a normal guess Hessian modified including the two differentiating coordinates. This chair transition state was then reoptimized again using the '''B3LYP/6-31G*''' level of theory. The electronic energies were calculated to be the same as in the first approach and frequency calculations were summarized in '''Table 9'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46671||-234.41490
|-
| Sum of electronic and thermal Energies||-231.46135||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40806
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir3.jpg|200px]]||[[Image:XJB_chairir4.jpg|200px]]
|-
| ||[[File:XJB_chairts2.1.log]]||[[File:XJB_chairts2.2.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 9.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the second approach''</div>

As the same energy results given, the bond forming/breaking bond lengths in the chair transition structures in the above two different approaches were compared. The first approach gave the bond lengths of 2.0203 Å while the second approach gave 2.0207 Å. Considering the accuracy limit of bond length estimation in Gaussview, these two bond lengths can be seen as exactly the same. Two methods worked reasonably well because of the good assumption of transition structures. However for more complicated and larger molecule, the Hessian method may fail due to the difficulty to predict the transition geometry (different surface curvature). Thus better way to generate such transition structures is to freeze reaction coordinate.

{| align="center"
|- align="center"
|
[[Image:XJB_chairts bondlength1.jpg|350px|thumb|'''Figure 6.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by Hession'']]
|
[[Image:XJB_chairts bondlength2.jpg|350px|thumb|'''Figure 7.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by frozen coordinate'']]
|- align="center"
|}

====The Boat Transition State====

The boat transition structure was then optimized using the '''QST2''' method. The same numbering for the reactant and product molecules was important in order to specify for this reaction. The renumbering of the molecules is shown as in '''Figure 8''', but the following optimization failed and gave a twisted structure. The reactant and product geometries were then modified further by changing the central C-C-C-C dihedral angle to 0<sup>o</sup> and the inside two C-C-C (i.e. C2-C3-C4 and C3-C4-C5) angles to 100<sup>o</sup>. The job was run successfully and frequency calculations were done with the boat transition state generated. Only one imaginary frequency was detected at -840 cm<sup>-1</sup>. The electronic energy was '''-231.60280Ha''' with C<sub>2v</sub> symmetry. The distances between the terminal carbons of the allyl fragments were examined to be both 2.140 Å which were comparably longer than that of the chair transition structure.

{| align="center"
|- align="center"
|
[[Image:XJB_QST1.jpg|400px|thumb|'''Figure 8.''' ''Numbering of reactant and product'']]
|
[[Image:XJB_QST2.jpg|410px|thumb|'''Figure 9.''' ''Re-numbering of reactant and product'']]
|- align="center"
|}

<center>[[File:XJB_boat ts animation.GIF|400px|center|thumb|'''Figure 10.''' ''Animation of the Cope rearrangement via the boat transition state'']]</center>

The structure was then reoptimized at the higher '''B3LYP/6-31G*''' theory of level and the electronic energy was calculated to be '''-234.54309Ha''' with very similar geometry. The frequency calculation generated an imaginary frequency at -530 cm<sup>-1</sup>. The summary and comparison of two levels' calculations are carried out in the following table.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.45093||-234.40234
|-
| Sum of electronic and thermal Energies||-231.44530||-234.39601
|-
| Sum of electronic and thermal Enthalpie|| -231.44436||-234.39506
|-
| Sum of electronic and thermal Free Energie||-231.47977||-234.43175
|-
| Generated IR spectrum||[[Image:XJB_boatir1.jpg|200px]]||[[Image:XJB_boatir2.jpg|200px]]
|-
| ||[[File:XJB_boat ts1.log]]||[[File:XJB_boat ts2.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 10.''' ''Frequency calculations of the boat transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by QST2''</div>

====Intrinsic Reaction Coordinate====

Next a useful method called '''Intrinct Reaction Coordinate (IRC)''' was applied to the Hessian optimized chair transition structure to follow the mininmum energy path (MEP) from a transition structure down to its local minimum on a potential energy surface. IRC can be used to verify that the transition state we found actually connects the reactants and products, or to identify any reaction intermediates, or to generate an animation of the reaction<ref name="IRC">''The Journal of Chemical Physics.'', 2005, '''122''', pp 1-17. {{DOI|10.1063/1.1927521}}</ref>. The symmetrical reaction coordinate was computed only in the forward direction with force constant always calculated and the number of points along IRC 50. The last structure generated along IRC is given in '''Figure 11''' with the energy calculated as '''-231.69158Ha''' and the dihedral angle of the central four C atoms and the internal angle between three C atoms measured as ''D'' = -67.188<sup>o</sup> ''A'' = 111.78<sup>o</sup>. This energy does not match to any of the conformations discussed in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] thus a minimum geometry has not reached yet.

{| align="center"
|- align="center"
|
<jmol><jmolApplet><title>XJB_IRC.mol</title><color>white</color>
<size>200</size><script>zoom=5</script>
<uploadedFileContents>XJB_IRC.mol</uploadedFileContents>
</jmolApplet></jmol>
|
[[Image:XJB_irc.jpg|500px]]
|- align="center"
| align="center" | '''Figure 11.''' ''The energy minimum structure along IRC ([[File:XJB_IRC.log]])''
| align="center" | '''Figure 12.''' ''Total energy along IRC''
|}

Three improving options can be used to reach the minimum energy. The first one is to run a normal Hessian optimization at the last point on IRC; the second one involves using a larger number of points (in this case 100 was used); the last one is to redo the IRC computation specifying to compute force constant at every step. Three methods were all used and the results were summarized in '''Table 11'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Improved IRC method 1
! scope="col" | Improved IRC method 2
! scope="col" | Improved IRC method 3
|-
| [[Image:XJB_IRC1.jpg|300px|center|IRC1]]||[[Image:XJB_IRC2.jpg|300px|center|IRC3]] || [[Image:XJB_IRC3.jpg|300px|center|IRC3]]
|-
| ''D'' = -64.17<sup>o</sup> ''A'' = 112.04<sup>o</sup> || ''D'' = -66.83<sup>o</sup> ''A'' = 111.82<sup>o</sup> || ''D'' = -64.17<sup>o</sup> ''A'' = 112.04<sup>o</sup>
|-
| -231.69167Ha || -231.69150Ha || -231.69167Ha
|-
| [[File:XJB_IRC1.log]]||[[File:XJB_IRC2.log]]||[[File:XJB_IRC3.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 11.''' ''Comparison of three improvement methods of IRC computation upon the chair transition structure''</div>

The first method was very quick but maybe generated relatively low-accuracy result; the second approach involved the controlling of number of points which might result in ending up at the wrong direction and wrong structure; the third method was the most suitable option for this small system but it took longer time than the previous two. From '''Table 11''' it is concluded that method 1 and 3 gave the same lowest energy which was consistent to the energy of the ''gauche2'' conformation given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

====Activation Energies====

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | '''HF/3-21G''' at 0K
! scope="col" | '''HF/3-21G''' at 298.15K
! scope="col" | '''B3LYP/6-31G*''' at 0K
! scope="col" | '''B3LYP/6-31G*''' at 298.15K
! scope="col" | Expt. at 0K
|-
| '''ΔE (chair)'''||45.71||44.70||34.05||33.16||33.5 ± 0.5
|-
| '''ΔE (boat)'''||55.60||54.76||41.96||41.32||44.7 ± 0.5
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 12.''' ''Summary of activation energies for the Cope rearrangement via both transition states in kcal/mol''</div>

The activation energy calculation was done by calculating the difference in energies between the optimized reactant ''anti2'' and the optimized chair and boat transition states. It is observed that chair conformation gives generally lower activation energy than the boat structure thus more favoured transition state. The structure optimized at '''B3LYP/6-31G*''' level of theory at 0K has closer activation energy to the experimental result. Thus '''B3LYP/6-31G*''' gives a better estimation of this transition structure than '''HF/3-21G'''.

==The Diels Alder Cycloaddition==

In this exercise, AM1 semi-empirical method was used to compute the Diels Alder cycloaddition which is a concerted pericyclic reaction. The driving force of this reaction is the formation of a new σ bond between the π orbitals of the dieneophile and the diene which is stronger than the π bond. The reaction is not allowed without the HOMO-LUMO interation and a symmetry overlap.

===Cis Butadiene===

<center>[[Image:prototype Diels Alder.jpg|500px|center|thumb|'''Figure 13'''. ''The prototype reaction between butadiene and ethylene'']]</center>

To study a prototype Diels Alder cycloaddition reaction between butadiene and ethylene, first a cis butadiene molecule was built using Gaussview and optimized using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the butadiene were plotted. As seen the HOMO of cis-butadiene is anti-symmetric ('''a''') and the LUMO is symmetric ('''s''') with respect to plane.

{| align="center"
|- align="center"
|
[[Image:XJB_HOMO.jpg|280px|thumb|'''Figure 14.''' ''Anti-symmetric HOMO of cis-butadiene'']]
|
[[Image:XJB_LUMO.jpg|300px|thumb|'''Figure 15.''' ''Symmetric HOMO of cis-butadiene'']]
|- align="center"
|}

===The Transition State of prototype reaction between ethylene and butadiene===

The Hessian method was used to estimate transition state for the reaction between ethylene and butadiene because these two reactants are relatively small and simple molecules. The transition state was drawn by first building a structure as (a) in '''Figure 16''' then removing the -CH<sub>2</sub>-CH<sub>2</sub>- group to form the envelop structure as in (b). The '''clean''' option should not be used and the dashed bond length was guessed to be larger than 2.2 Å otherwise the geometry change would cause a failure in optimization.

<center>[[Image:XJB_dielsguessts.jpg|300px|center|thumb|'''Figure 16'''. ''The guessing transition state for the reaction between butadiene and ethylene'']]</center>

<center>[[File:XJB_butadiene and ethylene ts.GIF|400px|center|thumb|'''Figure 17.''' ''Animation of the butadiene and ethylene cycloaddition transition state'' [[File:XJB_prototype DA.log]]]]</center>

After the Hessian optimization, the transition structure has been successfully determined and an animation was generated in '''Figure 17'''. It has a C<sub>1</sub> symmetry point group and the bond lengths of two partly formed C-C bonds were measured to be 2.12Å. The typical sp<sup>3</sup> and sp<sup>2</sup> C-C bond lengths are 1.54 Å and 1.47 Å respectively<ref name="bondlength">''J. Phys. C: Solid State Phys.'', 1986, '''19''', pp 4613-4621.</ref>. The van der Waals radius of the C atom is 1.7 Å<ref name="bondlength2">''Z. Naturforsch.'', 2007, '''62b''', pp 235-243.</ref>. The measured bond distance of the transition structure is larger than typical bond lengths but still within the van der Waals radius, suggesting the bond forming interation between the carbon atoms.

From the animation ('''Figure 17''') the vibration directions of two molecules are opposite and they are approaching and leaving each other at the same time. This suggests the formation of the two bonds synchronous. The vibrational frequencies were also calculated and an infrared spectrum was simulated ('''Figure 20'''). An imaginary frequency of magnitude -955 cm<sup>-1</sup> was seen and the lowest positive frequency was observed at 147 cm<sup>-1</sup> which corresponded to the reactant vibration but not the bond formation.

{| align="center"
|
[[Image:XJB_HOMOts.jpg|300px|thumb|'''Figure 18'''. HOMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_LUMOts.jpg|300px|thumb|'''Figure 19'''. LUMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_IRts.jpg|300px|thumb|'''Figure 20'''. Generated IR of transition state of reaction between cis-butadiene and ethylene]]
|}

The HOMO and LUMO molecular orbitals of this transition structure were visualized in '''Figure 18''' and '''Figure 19'''. From these two figures it can be observed that the HOMO at the transition structure is anti-symmetric ('''a''') while the LUMO is symmetric ('''s'''). Thus this anti-symmetric transition state HOMO is formed by the HOMO of cis-butadiene and the LUMO of ethylene with the knowledge that they are both anti-symmetrical. This conclusion agrees with the previous talked conditions for allowed reaction.

===The cyclohexa-1,3-diene reaction with maleic anhydride===

<center>[[Image:XJB_mechanism3.jpg|500px|center|thumb|'''Figure 21'''. ''The endo and exo products of the Diels Alder reaction between cyclohexa-1,3-diene and maleic anhydride'']]</center>

The exo and endo transition states were drawn according to '''Figure 21''' with the new C-C bond formation line dashed and bond length estimation as 2.5 Å. Then the same Hessian method was applied to optimize the structures. The electronic energy of the endo structure was calculated to be '''-0.05150Ha''' and '''-0.05042Ha''' for the exo. This reaction is supposed to be kinetically controlled thus the lower energy endo transition state is more favoured.

The bond lengths of the partly formed C-C bond are 2.16 Å and 2.17 Å for endo and exo transition states respectively. Another measured C-C bond length is shown in '''Table 13''' as 2.89 Å for endo (C3-C8 and C2-C7) and 2.95 Å for endo (C7-C10 and C8-C9). The longer atom distance indicates the larger steric repulsion in the exo transition state because of the extra existence of hydrogen atoms on C7 and C8.

Woodward and Hoffmann also proposed a secondary orbital overlap statement as an explanation of the endo stereo-preference in Diels Alder reactions<ref name="soot">''J. Org. Chem.'', 1987, '''52(8)''', pp 1469–1474. {{DOI|10.1021/jo00384a016}}</ref>. This secondary orbital overlap does not involve bond changes but π orbital interactions as the highlighted carbon atoms of the endo transition state in '''Table 13'''. Such overlap stablises the structure and lowers the energy of the endo transition state while there is no such effect existed in the exo transition state.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | endo
! scope="col" | exo
|-
| '''Animation'''||[[File:XJB_endots.GIF|300px]]||[[File:XJB_exots.GIF|300px]]
|-
| '''Sum of electronic and zero-point Energies'''||0.133493||0.134880
|-
| '''Sum of electronic and thermal Energies'''||0.143683||0.144881
|-
| '''Sum of electronic and thermal Enthalpies'''||0.144627||0.145825
|-
| '''Sum of electronic and thermal Free Energies'''||0.097349||0.099117
|-
| '''Bond lengths'''||[[Image:XJB_endomeasure.jpg|300px]]||[[Image:XJB_exomeasure.jpg|300px]]
|-
| '''HOMO'''||[[Image:XJB_endoHOMO.jpg|300px]]||[[Image:XJB_exoHOMO.jpg|285px]]
|-
| '''Secondary orbital overlap effect'''||[[Image:XJB_endosoo.jpg|300px]]||[[Image:XJB_exosoo.jpg|300px]]
|-
| ||[[File:XJB_endo ts.log]]||[[File:XJB_exo ts.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 13.''' ''Summary of comparison of properties of endo and exo transition structures for reaction between cyclohexa-1,3-diene and maleic anhydride''</div>

===Further work===

The semiempirical/AM1 calculation is based on the "Neglect of Differential Diatomic Overlap (NDDO)" integral approximation, which neglects all two-electron integrals involving two-center charge distributions are neglected<ref>http://www.cup.uni-muenchen.de/ch/compchem/energy/semi1.html</ref>. The better optimization towards the endo and exo transition states should be done first by AM1 optimization followed by '''B3LYP/6-31G*''' reoptimization<ref>''J. Org. Chem.'', 2003, '''68 (19)''', pp 7158–7166. {{DOI|10.1021/jo0348827}}</ref>. The results are summarized in '''Table 14'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | endo
! scope="col" | exo
|-
| '''Structure'''||[[Image:XJB_endots2.jpg|300px]]||[[Image:XJB_exots2.jpg|300px]]
|-
| '''Electronic energy'''||-612.68340||-612.67931
|-
| ||[[File:endots reopt]]||[[File:exots reopt]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 14.''' ''Reoptimization using '''B3LYP/6-31G*''' of endo and exo transition structures for reaction between cyclohexa-1,3-diene and maleic anhydride''</div>

==Reference==

<references/>

Rep:Mod:XJB1130

2013-12-06T15:59:43Z

Jx1011: /* Further work */

==The Cope Rearrangement of 1,5-hexadiene==

The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement reaction ('''Figure 1'''). Its mechanism has been studied by a large number of experimental and computational researches, which is recently accepted that the reaction occurs via either a "chair" or a "boat" transition state structure in a concerted manner. The aim of this exercise is to locate the low-energy minimum and the transition structures on the potential energy surface by Gaussian calculation in order to study its mechanism. The reaction pathway and the activation energy can also be calculated by this calculation.

<center>[[Image:XJB_cope rearrangement.jpg|500px|center|thumb|'''Figure 1'''. ''The Cope rearrangement'']]</center>

===Optimizing the Reactants and Products===

Different conformations of 1,5-hexadiene can be optimized using '''Gaussview'''. The optimized structure of four conformers involved in this discussion and their corresponding energies optimized at '''HF/3-21G''' level of theory are concluded in the following table ('''Table 1''').

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | anti4
! scope="col" | anti2
! scope="col" | gauche
! scope="col" | gauche3
|-
| <jmol><jmolApplet><title>XJB_anti4.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti4.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche3.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| -231.69097Ha || -231.69254Ha || -231.68772Ha ||-231.69266Ha
|}

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 1.''' ''Four conformers of 1,5-hexadiene''</div>

This exercise first involved optimizing a molecule of 1,5-hexadiene with "anti" linkage for the cetral four C atoms at the '''HF/3-21G''' level of theory. The final optimization structure had the total electronic energy of '''-231.69097Ha''' with a '''C<sub>1</sub>''' symmetry. By comparing the energy and point group to that in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1], this structure is confirmed to be ''anti4''. The compositions of energies are also listed in '''Table 2''' as i) the sum of electronic and zero-point energies (potential energy at 0K including the zero-point vibrational energy), ii) the sum of electronic and thermal energies (energy including translational, rotational and vibrational at 298.15K and 1 atm), iii) the sum of electronic and thermal enthalpies (containning an additional correction for RT), and iv) the sum of electronic and thermal free energies (including the entropic contribution to the free energy). For what we are considering, the first two terms are the most useful for further comparisons.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Formula Description'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||E = E<sub>elec</sub> + ZPE||-231.53785
|-
| Sum of electronic and thermal Energies||E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>||-231.53096
|-
| Sum of electronic and thermal Enthalpie||H = E + RT||-231.53002
|-
| Sum of electronic and thermal Free Energie||G = H - TS||-231.56891
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 2.''' ''Frequency calculations of anti4 optimized at '''HF/3-21G''' ([[File:XJB_anti4.log]])''</div>

Next another 1,5-hexadiene molecule with "gauche" linkage for the central four atoms was drawn and optimized at the '''HF/3-21G''' level of theory as before. Energy for this conformation is expected to be higher than that for the ''anti4'' conformation due to the closer position of two alkene ends thus the steric interaction. The final structure had the electronic energy of '''-231.68772Ha''' which was indeed higher than that of the ''anti4'', indicating less favour towards this gauche structure. It had a '''C<sub>2</sub>''' symmetry and was confirmed to be the ''gauche'' conformation.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53439
|-
| Sum of electronic and thermal Energies||-231.52770
|-
| Sum of electronic and thermal Enthalpie||-231.52676
|-
| Sum of electronic and thermal Free Energie||-231.56483
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 3.''' ''Frequency calculations of gauche optimized at '''HF/3-21G''' ([[File:XJB_gauche.log]])''</div>

A lowest energy conformation ''gauche3'' of the reactant molecule was drawn based on and confirmed by [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. The optimization gave a final structure with an energy of '''-231.69266Ha''' with a symmetry of '''C<sub>1</sub>'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53949
|-
| Sum of electronic and thermal Energies||-231.53265
|-
| Sum of electronic and thermal Enthalpie||-231.5317
|-
| Sum of electronic and thermal Free Energie||-231.57064
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 4.''' ''Frequency calculations of gauche3 optimized at '''HF/3-21G''' ([[File:XJB_gauche3.log]])''</div>

The next step required us to draw the C<sub>i</sub> symmetry ''anti2'' conformation of 1,5-hexadiene and optimize it at the '''HF/3-21G''' level of theory. The optimized energy was calculated to be '''-231.69254Ha''' with point group of '''C<sub>i</sub>''', which was consistent with the one given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53954
|-
| Sum of electronic and thermal Energies||-231.53257
|-
| Sum of electronic and thermal Enthalpie||-231.53162
|-
| Sum of electronic and thermal Free Energie||-231.57092
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 5.''' ''Frequency calculations of anti2 optimized at '''HF/3-21G''' ([[File:XJB_anti2.log]])''</div>

The ''anti2'' structure was then reoptimized at '''B3LYP/6-31G*''' which was a improvement over the '''HF/3-21G''' basis set. It adds polarization to atoms and improves the modeling of core electrons thus producing more accurate description of orbitals<ref name="soo">''Nigerian Journal of Chemical Research'', 2007, '''12'''. {{DOI|10.4314/njcr.v12i1.}}</ref>. The reoptimized ''anti2'' structure had the total electronic energy of '''-234.61171Ha''' and the same symmetry C<sub>i</sub>. The comparison of two structures optimized using different levels of theory is summarized in '''Table 6'''. We can see that they have exactly the same dihedral angle of the central four C atoms at 180.00<sup>o</sup> and angles between three of them are very similar as 111.3<sup>o</sup> and 112.7<sup>o</sup> respectively. Thus the the geometry change using '''HF/3-21G''' or '''B3LYP/6-31G*''' is trivial.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Before re-optimization
! scope="col" | After re-optimization
|-
| <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2reoptimized</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2reoptimized.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| ''anti2 optimized at '''HF/3-21G''''' || ''anti2 optimized at '''B3LYP/6-31G*'''''
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 6.''' ''Two optimization of anti2 at '''HF/3-21G''' and '''B3LYP/6-31G*'''([[File:XJB_anti2.log]])''</div>

The frequency calculation was run as well and a list of energies is summarized in '''Table 7'''. A list of all the vibrational frequencies was also generated and no imaginary (negative) frequency was observed and these vibrations are visualized by simulating the infrared spectrum in '''Figure 2'''.

{| align="center"
|- align="center"
|
{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-234.46920
|-
| Sum of electronic and thermal Energies||-234.46186
|-
| Sum of electronic and thermal Enthalpie||-234.46091
|-
| Sum of electronic and thermal Free Energie||-234.50078
|}
|
[[Image:XJB_IR_anti2.jpg|300px]]
|- align="center"
| align="center" | '''Table 7.''' ''Frequency calculations of anti2 optimized at '''B3LYP/6-31G*'''
([[File:XJB_anti2_reoptimized.log]])''
| align="center" | '''Figure 2.''' ''The generated IR spectrum of optimized '''B3LYP/6-31G*''' structure''
|}

===Optimizing the "Chair" and "Boat" Transition Structures===

The study of the chair and boat transition states was carried out using three different approaches. The first one was called the Hessian which involved computing the force constant matrix; the second one was to freeze the reaction coordinate and then optimized the structure once the molecule was fully relaxed; the last one was to use QST2 method to specify the reactants and products then finding the transition structure.

<center>[[Image:XJB_2 ts.jpg|500px|center|thumb|'''Figure 3'''. ''The two transition states of Cope rearrangement'']]</center>

====The Chair Transition State====

In order to set up a transition state optimization, first an allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimized at the '''HF/3-21G''' level of theory. Then this half of the transition structure was duplicated with rotation to make the approximate resembling of the chair transition state with terminal ends of the fragments 2.2 Å apart. The chair transition structure optimization was then done by the first two different approaches.

<center>[[Image:XJB_guess ts.jpg|400px|center|thumb|'''Figure 4'''. ''The guessed chair transition structure of two allyl fragments'']]</center>

For the first optimization approach, the force constants were computed (the Hessian) and the chair transition state was optimized to a TS (Berny) at the '''HF/3-21G''' level of theory. The frequency calculation was carried out at the same time and gave an imaginary frequency at -818 cm<sup>-1</sup>. The electronic energy was calculated to be '''-231.61932Ha''' with C<sub>2h</sub> symmetry point group. The distance between the terminal carbons was 2.0203 Å. An animation of this transition state vibrations was generated to confirm the corresponding Cope rearrangement.

<center>[[File:XJB_chair ts animation.GIF|400px|center|thumb|'''Figure 5.''' ''Animation of the Cope rearrangement via the chair transition state'']]</center>

The chair transition state was then reoptimized using the '''B3LYP/6-31G*''' level of theory and carried out frequency calculations as well. The electronic energy calculated this time was '''-234.55698Ha''' with an imaginary frequency at -566 cm<sup>-1</sup>. We can see that with the geometries quite similar, the energies differ markedly calculated using two levels of theory. The summary and comparison of the calculations optimized at two levels of theory are summarized in '''Table 8'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46670||-234.41493
|-
| Sum of electronic and thermal Energies||-231.46134||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40807
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir1.jpg|200px]]||[[Image:XJB_chairir2.jpg|200px]]
|-
| ||[[File:XJB_chairts1.1.log]]||[[File:XJB_chairts1.2.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 8.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the first approach''</div>

The second approach is the frozen coordinate method using the Redundant Coord Editor to freeze the bond lengths of the terminal ends to 2.2 Å. First the calculation failed because of the recent update of GaussView. After editting the coordinate distance (B 5 1 2.2 F) and the frozen distance (B 5 1 F) the optimization went well this time. The structure was then optimized again using a normal guess Hessian modified including the two differentiating coordinates. This chair transition state was then reoptimized again using the '''B3LYP/6-31G*''' level of theory. The electronic energies were calculated to be the same as in the first approach and frequency calculations were summarized in '''Table 9'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46671||-234.41490
|-
| Sum of electronic and thermal Energies||-231.46135||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40806
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir3.jpg|200px]]||[[Image:XJB_chairir4.jpg|200px]]
|-
| ||[[File:XJB_chairts2.1.log]]||[[File:XJB_chairts2.2.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 9.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the second approach''</div>

As the same energy results given, the bond forming/breaking bond lengths in the chair transition structures in the above two different approaches were compared. The first approach gave the bond lengths of 2.0203 Å while the second approach gave 2.0207 Å. Considering the accuracy limit of bond length estimation in Gaussview, these two bond lengths can be seen as exactly the same. Two methods worked reasonably well because of the good assumption of transition structures. However for more complicated and larger molecule, the Hessian method may fail due to the difficulty to predict the transition geometry (different surface curvature). Thus better way to generate such transition structures is to freeze reaction coordinate.

{| align="center"
|- align="center"
|
[[Image:XJB_chairts bondlength1.jpg|350px|thumb|'''Figure 6.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by Hession'']]
|
[[Image:XJB_chairts bondlength2.jpg|350px|thumb|'''Figure 7.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by frozen coordinate'']]
|- align="center"
|}

====The Boat Transition State====

The boat transition structure was then optimized using the '''QST2''' method. The same numbering for the reactant and product molecules was important in order to specify for this reaction. The renumbering of the molecules is shown as in '''Figure 8''', but the following optimization failed and gave a twisted structure. The reactant and product geometries were then modified further by changing the central C-C-C-C dihedral angle to 0<sup>o</sup> and the inside two C-C-C (i.e. C2-C3-C4 and C3-C4-C5) angles to 100<sup>o</sup>. The job was run successfully and frequency calculations were done with the boat transition state generated. Only one imaginary frequency was detected at -840 cm<sup>-1</sup>. The electronic energy was '''-231.60280Ha''' with C<sub>2v</sub> symmetry. The distances between the terminal carbons of the allyl fragments were examined to be both 2.140 Å which were comparably longer than that of the chair transition structure.

{| align="center"
|- align="center"
|
[[Image:XJB_QST1.jpg|400px|thumb|'''Figure 8.''' ''Numbering of reactant and product'']]
|
[[Image:XJB_QST2.jpg|410px|thumb|'''Figure 9.''' ''Re-numbering of reactant and product'']]
|- align="center"
|}

<center>[[File:XJB_boat ts animation.GIF|400px|center|thumb|'''Figure 10.''' ''Animation of the Cope rearrangement via the boat transition state'']]</center>

The structure was then reoptimized at the higher '''B3LYP/6-31G*''' theory of level and the electronic energy was calculated to be '''-234.54309Ha''' with very similar geometry. The frequency calculation generated an imaginary frequency at -530 cm<sup>-1</sup>. The summary and comparison of two levels' calculations are carried out in the following table.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.45093||-234.40234
|-
| Sum of electronic and thermal Energies||-231.44530||-234.39601
|-
| Sum of electronic and thermal Enthalpie|| -231.44436||-234.39506
|-
| Sum of electronic and thermal Free Energie||-231.47977||-234.43175
|-
| Generated IR spectrum||[[Image:XJB_boatir1.jpg|200px]]||[[Image:XJB_boatir2.jpg|200px]]
|-
| ||[[File:XJB_boat ts1.log]]||[[File:XJB_boat ts2.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 10.''' ''Frequency calculations of the boat transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by QST2''</div>

====Intrinsic Reaction Coordinate====

Next a useful method called '''Intrinct Reaction Coordinate (IRC)''' was applied to the Hessian optimized chair transition structure to follow the mininmum energy path (MEP) from a transition structure down to its local minimum on a potential energy surface. IRC can be used to verify that the transition state we found actually connects the reactants and products, or to identify any reaction intermediates, or to generate an animation of the reaction<ref name="IRC">''The Journal of Chemical Physics.'', 2005, '''122''', pp 1-17. {{DOI|10.1063/1.1927521}}</ref>. The symmetrical reaction coordinate was computed only in the forward direction with force constant always calculated and the number of points along IRC 50. The last structure generated along IRC is given in '''Figure 11''' with the energy calculated as '''-231.69158Ha''' and the dihedral angle of the central four C atoms and the internal angle between three C atoms measured as ''D'' = -67.188<sup>o</sup> ''A'' = 111.78<sup>o</sup>. This energy does not match to any of the conformations discussed in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] thus a minimum geometry has not reached yet.

{| align="center"
|- align="center"
|
<jmol><jmolApplet><title>XJB_IRC.mol</title><color>white</color>
<size>200</size><script>zoom=5</script>
<uploadedFileContents>XJB_IRC.mol</uploadedFileContents>
</jmolApplet></jmol>
|
[[Image:XJB_irc.jpg|500px]]
|- align="center"
| align="center" | '''Figure 11.''' ''The energy minimum structure along IRC ([[File:XJB_IRC.log]])''
| align="center" | '''Figure 12.''' ''Total energy along IRC''
|}

Three improving options can be used to reach the minimum energy. The first one is to run a normal Hessian optimization at the last point on IRC; the second one involves using a larger number of points (in this case 100 was used); the last one is to redo the IRC computation specifying to compute force constant at every step. Three methods were all used and the results were summarized in '''Table 11'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Improved IRC method 1
! scope="col" | Improved IRC method 2
! scope="col" | Improved IRC method 3
|-
| [[Image:XJB_IRC1.jpg|300px|center|IRC1]]||[[Image:XJB_IRC2.jpg|300px|center|IRC3]] || [[Image:XJB_IRC3.jpg|300px|center|IRC3]]
|-
| ''D'' = -64.17<sup>o</sup> ''A'' = 112.04<sup>o</sup> || ''D'' = -66.83<sup>o</sup> ''A'' = 111.82<sup>o</sup> || ''D'' = -64.17<sup>o</sup> ''A'' = 112.04<sup>o</sup>
|-
| -231.69167Ha || -231.69150Ha || -231.69167Ha
|-
| [[File:XJB_IRC1.log]]||[[File:XJB_IRC2.log]]||[[File:XJB_IRC3.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 11.''' ''Comparison of three improvement methods of IRC computation upon the chair transition structure''</div>

The first method was very quick but maybe generated relatively low-accuracy result; the second approach involved the controlling of number of points which might result in ending up at the wrong direction and wrong structure; the third method was the most suitable option for this small system but it took longer time than the previous two. From '''Table 11''' it is concluded that method 1 and 3 gave the same lowest energy which was consistent to the energy of the ''gauche2'' conformation given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

====Activation Energies====

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | '''HF/3-21G''' at 0K
! scope="col" | '''HF/3-21G''' at 298.15K
! scope="col" | '''B3LYP/6-31G*''' at 0K
! scope="col" | '''B3LYP/6-31G*''' at 298.15K
! scope="col" | Expt. at 0K
|-
| '''ΔE (chair)'''||45.71||44.70||34.05||33.16||33.5 ± 0.5
|-
| '''ΔE (boat)'''||55.60||54.76||41.96||41.32||44.7 ± 0.5
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 12.''' ''Summary of activation energies for the Cope rearrangement via both transition states in kcal/mol''</div>

The activation energy calculation was done by calculating the difference in energies between the optimized reactant ''anti2'' and the optimized chair and boat transition states. It is observed that chair conformation gives generally lower activation energy than the boat structure thus more favoured transition state. The structure optimized at '''B3LYP/6-31G*''' level of theory at 0K has closer activation energy to the experimental result. Thus '''B3LYP/6-31G*''' gives a better estimation of this transition structure than '''HF/3-21G'''.

==The Diels Alder Cycloaddition==

In this exercise, AM1 semi-empirical method was used to compute the Diels Alder cycloaddition which is a concerted pericyclic reaction. The driving force of this reaction is the formation of a new σ bond between the π orbitals of the dieneophile and the diene which is stronger than the π bond. The reaction is not allowed without the HOMO-LUMO interation and a symmetry overlap.

===Cis Butadiene===

<center>[[Image:prototype Diels Alder.jpg|500px|center|thumb|'''Figure 13'''. ''The prototype reaction between butadiene and ethylene'']]</center>

To study a prototype Diels Alder cycloaddition reaction between butadiene and ethylene, first a cis butadiene molecule was built using Gaussview and optimized using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the butadiene were plotted. As seen the HOMO of cis-butadiene is anti-symmetric ('''a''') and the LUMO is symmetric ('''s''') with respect to plane.

{| align="center"
|- align="center"
|
[[Image:XJB_HOMO.jpg|280px|thumb|'''Figure 14.''' ''Anti-symmetric HOMO of cis-butadiene'']]
|
[[Image:XJB_LUMO.jpg|300px|thumb|'''Figure 15.''' ''Symmetric HOMO of cis-butadiene'']]
|- align="center"
|}

===The Transition State of prototype reaction between ethylene and butadiene===

The Hessian method was used to estimate transition state for the reaction between ethylene and butadiene because these two reactants are relatively small and simple molecules. The transition state was drawn by first building a structure as (a) in '''Figure 16''' then removing the -CH<sub>2</sub>-CH<sub>2</sub>- group to form the envelop structure as in (b). The '''clean''' option should not be used and the dashed bond length was guessed to be larger than 2.2 Å otherwise the geometry change would cause a failure in optimization.

<center>[[Image:XJB_dielsguessts.jpg|300px|center|thumb|'''Figure 16'''. ''The guessing transition state for the reaction between butadiene and ethylene'']]</center>

<center>[[File:XJB_butadiene and ethylene ts.GIF|400px|center|thumb|'''Figure 17.''' ''Animation of the butadiene and ethylene cycloaddition transition state'' [[File:XJB_prototype DA.log]]]]</center>

After the Hessian optimization, the transition structure has been successfully determined and an animation was generated in '''Figure 17'''. It has a C<sub>1</sub> symmetry point group and the bond lengths of two partly formed C-C bonds were measured to be 2.12Å. The typical sp<sup>3</sup> and sp<sup>2</sup> C-C bond lengths are 1.54 Å and 1.47 Å respectively<ref name="bondlength">''J. Phys. C: Solid State Phys.'', 1986, '''19''', pp 4613-4621.</ref>. The van der Waals radius of the C atom is 1.7 Å<ref name="bondlength2">''Z. Naturforsch.'', 2007, '''62b''', pp 235-243.</ref>. The measured bond distance of the transition structure is larger than typical bond lengths but still within the van der Waals radius, suggesting the bond forming interation between the carbon atoms.

From the animation ('''Figure 17''') the vibration directions of two molecules are opposite and they are approaching and leaving each other at the same time. This suggests the formation of the two bonds synchronous. The vibrational frequencies were also calculated and an infrared spectrum was simulated ('''Figure 20'''). An imaginary frequency of magnitude -955 cm<sup>-1</sup> was seen and the lowest positive frequency was observed at 147 cm<sup>-1</sup> which corresponded to the reactant vibration but not the bond formation.

{| align="center"
|
[[Image:XJB_HOMOts.jpg|300px|thumb|'''Figure 18'''. HOMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_LUMOts.jpg|300px|thumb|'''Figure 19'''. LUMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_IRts.jpg|300px|thumb|'''Figure 20'''. Generated IR of transition state of reaction between cis-butadiene and ethylene]]
|}

The HOMO and LUMO molecular orbitals of this transition structure were visualized in '''Figure 18''' and '''Figure 19'''. From these two figures it can be observed that the HOMO at the transition structure is anti-symmetric ('''a''') while the LUMO is symmetric ('''s'''). Thus this anti-symmetric transition state HOMO is formed by the HOMO of cis-butadiene and the LUMO of ethylene with the knowledge that they are both anti-symmetrical. This conclusion agrees with the previous talked conditions for allowed reaction.

===The cyclohexa-1,3-diene reaction with maleic anhydride===

<center>[[Image:XJB_mechanism3.jpg|500px|center|thumb|'''Figure 21'''. ''The endo and exo products of the Diels Alder reaction between cyclohexa-1,3-diene and maleic anhydride'']]</center>

The exo and endo transition states were drawn according to '''Figure 21''' with the new C-C bond formation line dashed and bond length estimation as 2.5 Å. Then the same Hessian method was applied to optimize the structures. The electronic energy of the endo structure was calculated to be '''-0.05150Ha''' and '''-0.05042Ha''' for the exo. This reaction is supposed to be kinetically controlled thus the lower energy endo transition state is more favoured.

The bond lengths of the partly formed C-C bond are 2.16 Å and 2.17 Å for endo and exo transition states respectively. Another measured C-C bond length is shown in '''Table 13''' as 2.89 Å for endo (C3-C8 and C2-C7) and 2.95 Å for endo (C7-C10 and C8-C9). The longer atom distance indicates the larger steric repulsion in the exo transition state because of the extra existence of hydrogen atoms on C7 and C8.

Woodward and Hoffmann also proposed a secondary orbital overlap statement as an explanation of the endo stereo-preference in Diels Alder reactions<ref name="soot">''J. Org. Chem.'', 1987, '''52(8)''', pp 1469–1474. {{DOI|10.1021/jo00384a016}}</ref>. This secondary orbital overlap does not involve bond changes but π orbital interactions as the highlighted carbon atoms of the endo transition state in '''Table 13'''. Such overlap stablises the structure and lowers the energy of the endo transition state while there is no such effect existed in the exo transition state.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | endo
! scope="col" | exo
|-
| '''Animation'''||[[File:XJB_endots.GIF|300px]]||[[File:XJB_exots.GIF|300px]]
|-
| '''Sum of electronic and zero-point Energies'''||0.133493||0.134880
|-
| '''Sum of electronic and thermal Energies'''||0.143683||0.144881
|-
| '''Sum of electronic and thermal Enthalpies'''||0.144627||0.145825
|-
| '''Sum of electronic and thermal Free Energies'''||0.097349||0.099117
|-
| '''Bond lengths'''||[[Image:XJB_endomeasure.jpg|300px]]||[[Image:XJB_exomeasure.jpg|300px]]
|-
| '''HOMO'''||[[Image:XJB_endoHOMO.jpg|300px]]||[[Image:XJB_exoHOMO.jpg|285px]]
|-
| '''Secondary orbital overlap effect'''||[[Image:XJB_endosoo.jpg|300px]]||[[Image:XJB_exosoo.jpg|300px]]
|-
| ||[[File:XJB_endo ts.log]]||[[File:XJB_exo ts.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 13.''' ''Summary of comparison of properties of endo and exo transition structures for reaction between cyclohexa-1,3-diene and maleic anhydride''</div>

===Further work===

The semiempirical/AM1 calculation is based on the "Neglect of Differential Diatomic Overlap (NDDO)" integral approximation, which neglects all two-electron integrals involving two-center charge distributions are neglected<ref>http://www.cup.uni-muenchen.de/ch/compchem/energy/semi1.html</ref>. The better optimization towards the endo and exo transition states should be done first by AM1 optimization followed by '''B3LYP/6-31G*''' reoptimization<ref>''J. Org. Chem.'', 2003, '''68 (19)''', pp 7158–7166. {{DOI|10.1021/jo0348827}}</ref>. The results are summarized in '''Table 14'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | endo
! scope="col" | exo
|-
| '''Structure'''||[[Image:XJB_endots2.jpg|300px]]||[[Image:XJB_exots2.jpg|300px]]
|-
| '''Electronic energy'''||-612.68340||-612.67931
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 14.''' ''Reoptimization using '''B3LYP/6-31G*''' of endo and exo transition structures for reaction between cyclohexa-1,3-diene and maleic anhydride''</div>

==Reference==

<references/>

File:XJB exo ts.log

2013-12-06T15:54:52Z

Jx1011:

File:XJB endo ts.log

2013-12-06T15:54:00Z

Jx1011:

Rep:Mod:XJB1130

2013-12-06T15:53:38Z

Jx1011: /* The cyclohexa-1,3-diene reaction with maleic anhydride */

==The Cope Rearrangement of 1,5-hexadiene==

The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement reaction ('''Figure 1'''). Its mechanism has been studied by a large number of experimental and computational researches, which is recently accepted that the reaction occurs via either a "chair" or a "boat" transition state structure in a concerted manner. The aim of this exercise is to locate the low-energy minimum and the transition structures on the potential energy surface by Gaussian calculation in order to study its mechanism. The reaction pathway and the activation energy can also be calculated by this calculation.

<center>[[Image:XJB_cope rearrangement.jpg|500px|center|thumb|'''Figure 1'''. ''The Cope rearrangement'']]</center>

===Optimizing the Reactants and Products===

Different conformations of 1,5-hexadiene can be optimized using '''Gaussview'''. The optimized structure of four conformers involved in this discussion and their corresponding energies optimized at '''HF/3-21G''' level of theory are concluded in the following table ('''Table 1''').

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | anti4
! scope="col" | anti2
! scope="col" | gauche
! scope="col" | gauche3
|-
| <jmol><jmolApplet><title>XJB_anti4.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti4.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche3.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| -231.69097Ha || -231.69254Ha || -231.68772Ha ||-231.69266Ha
|}

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 1.''' ''Four conformers of 1,5-hexadiene''</div>

This exercise first involved optimizing a molecule of 1,5-hexadiene with "anti" linkage for the cetral four C atoms at the '''HF/3-21G''' level of theory. The final optimization structure had the total electronic energy of '''-231.69097Ha''' with a '''C<sub>1</sub>''' symmetry. By comparing the energy and point group to that in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1], this structure is confirmed to be ''anti4''. The compositions of energies are also listed in '''Table 2''' as i) the sum of electronic and zero-point energies (potential energy at 0K including the zero-point vibrational energy), ii) the sum of electronic and thermal energies (energy including translational, rotational and vibrational at 298.15K and 1 atm), iii) the sum of electronic and thermal enthalpies (containning an additional correction for RT), and iv) the sum of electronic and thermal free energies (including the entropic contribution to the free energy). For what we are considering, the first two terms are the most useful for further comparisons.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Formula Description'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||E = E<sub>elec</sub> + ZPE||-231.53785
|-
| Sum of electronic and thermal Energies||E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>||-231.53096
|-
| Sum of electronic and thermal Enthalpie||H = E + RT||-231.53002
|-
| Sum of electronic and thermal Free Energie||G = H - TS||-231.56891
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 2.''' ''Frequency calculations of anti4 optimized at '''HF/3-21G''' ([[File:XJB_anti4.log]])''</div>

Next another 1,5-hexadiene molecule with "gauche" linkage for the central four atoms was drawn and optimized at the '''HF/3-21G''' level of theory as before. Energy for this conformation is expected to be higher than that for the ''anti4'' conformation due to the closer position of two alkene ends thus the steric interaction. The final structure had the electronic energy of '''-231.68772Ha''' which was indeed higher than that of the ''anti4'', indicating less favour towards this gauche structure. It had a '''C<sub>2</sub>''' symmetry and was confirmed to be the ''gauche'' conformation.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53439
|-
| Sum of electronic and thermal Energies||-231.52770
|-
| Sum of electronic and thermal Enthalpie||-231.52676
|-
| Sum of electronic and thermal Free Energie||-231.56483
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 3.''' ''Frequency calculations of gauche optimized at '''HF/3-21G''' ([[File:XJB_gauche.log]])''</div>

A lowest energy conformation ''gauche3'' of the reactant molecule was drawn based on and confirmed by [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. The optimization gave a final structure with an energy of '''-231.69266Ha''' with a symmetry of '''C<sub>1</sub>'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53949
|-
| Sum of electronic and thermal Energies||-231.53265
|-
| Sum of electronic and thermal Enthalpie||-231.5317
|-
| Sum of electronic and thermal Free Energie||-231.57064
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 4.''' ''Frequency calculations of gauche3 optimized at '''HF/3-21G''' ([[File:XJB_gauche3.log]])''</div>

The next step required us to draw the C<sub>i</sub> symmetry ''anti2'' conformation of 1,5-hexadiene and optimize it at the '''HF/3-21G''' level of theory. The optimized energy was calculated to be '''-231.69254Ha''' with point group of '''C<sub>i</sub>''', which was consistent with the one given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53954
|-
| Sum of electronic and thermal Energies||-231.53257
|-
| Sum of electronic and thermal Enthalpie||-231.53162
|-
| Sum of electronic and thermal Free Energie||-231.57092
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 5.''' ''Frequency calculations of anti2 optimized at '''HF/3-21G''' ([[File:XJB_anti2.log]])''</div>

The ''anti2'' structure was then reoptimized at '''B3LYP/6-31G*''' which was a improvement over the '''HF/3-21G''' basis set. It adds polarization to atoms and improves the modeling of core electrons thus producing more accurate description of orbitals<ref name="soo">''Nigerian Journal of Chemical Research'', 2007, '''12'''. {{DOI|10.4314/njcr.v12i1.}}</ref>. The reoptimized ''anti2'' structure had the total electronic energy of '''-234.61171Ha''' and the same symmetry C<sub>i</sub>. The comparison of two structures optimized using different levels of theory is summarized in '''Table 6'''. We can see that they have exactly the same dihedral angle of the central four C atoms at 180.00<sup>o</sup> and angles between three of them are very similar as 111.3<sup>o</sup> and 112.7<sup>o</sup> respectively. Thus the the geometry change using '''HF/3-21G''' or '''B3LYP/6-31G*''' is trivial.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Before re-optimization
! scope="col" | After re-optimization
|-
| <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2reoptimized</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2reoptimized.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| ''anti2 optimized at '''HF/3-21G''''' || ''anti2 optimized at '''B3LYP/6-31G*'''''
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 6.''' ''Two optimization of anti2 at '''HF/3-21G''' and '''B3LYP/6-31G*'''([[File:XJB_anti2.log]])''</div>

The frequency calculation was run as well and a list of energies is summarized in '''Table 7'''. A list of all the vibrational frequencies was also generated and no imaginary (negative) frequency was observed and these vibrations are visualized by simulating the infrared spectrum in '''Figure 2'''.

{| align="center"
|- align="center"
|
{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-234.46920
|-
| Sum of electronic and thermal Energies||-234.46186
|-
| Sum of electronic and thermal Enthalpie||-234.46091
|-
| Sum of electronic and thermal Free Energie||-234.50078
|}
|
[[Image:XJB_IR_anti2.jpg|300px]]
|- align="center"
| align="center" | '''Table 7.''' ''Frequency calculations of anti2 optimized at '''B3LYP/6-31G*'''
([[File:XJB_anti2_reoptimized.log]])''
| align="center" | '''Figure 2.''' ''The generated IR spectrum of optimized '''B3LYP/6-31G*''' structure''
|}

===Optimizing the "Chair" and "Boat" Transition Structures===

The study of the chair and boat transition states was carried out using three different approaches. The first one was called the Hessian which involved computing the force constant matrix; the second one was to freeze the reaction coordinate and then optimized the structure once the molecule was fully relaxed; the last one was to use QST2 method to specify the reactants and products then finding the transition structure.

<center>[[Image:XJB_2 ts.jpg|500px|center|thumb|'''Figure 3'''. ''The two transition states of Cope rearrangement'']]</center>

====The Chair Transition State====

In order to set up a transition state optimization, first an allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimized at the '''HF/3-21G''' level of theory. Then this half of the transition structure was duplicated with rotation to make the approximate resembling of the chair transition state with terminal ends of the fragments 2.2 Å apart. The chair transition structure optimization was then done by the first two different approaches.

<center>[[Image:XJB_guess ts.jpg|400px|center|thumb|'''Figure 4'''. ''The guessed chair transition structure of two allyl fragments'']]</center>

For the first optimization approach, the force constants were computed (the Hessian) and the chair transition state was optimized to a TS (Berny) at the '''HF/3-21G''' level of theory. The frequency calculation was carried out at the same time and gave an imaginary frequency at -818 cm<sup>-1</sup>. The electronic energy was calculated to be '''-231.61932Ha''' with C<sub>2h</sub> symmetry point group. The distance between the terminal carbons was 2.0203 Å. An animation of this transition state vibrations was generated to confirm the corresponding Cope rearrangement.

<center>[[File:XJB_chair ts animation.GIF|400px|center|thumb|'''Figure 5.''' ''Animation of the Cope rearrangement via the chair transition state'']]</center>

The chair transition state was then reoptimized using the '''B3LYP/6-31G*''' level of theory and carried out frequency calculations as well. The electronic energy calculated this time was '''-234.55698Ha''' with an imaginary frequency at -566 cm<sup>-1</sup>. We can see that with the geometries quite similar, the energies differ markedly calculated using two levels of theory. The summary and comparison of the calculations optimized at two levels of theory are summarized in '''Table 8'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46670||-234.41493
|-
| Sum of electronic and thermal Energies||-231.46134||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40807
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir1.jpg|200px]]||[[Image:XJB_chairir2.jpg|200px]]
|-
| ||[[File:XJB_chairts1.1.log]]||[[File:XJB_chairts1.2.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 8.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the first approach''</div>

The second approach is the frozen coordinate method using the Redundant Coord Editor to freeze the bond lengths of the terminal ends to 2.2 Å. First the calculation failed because of the recent update of GaussView. After editting the coordinate distance (B 5 1 2.2 F) and the frozen distance (B 5 1 F) the optimization went well this time. The structure was then optimized again using a normal guess Hessian modified including the two differentiating coordinates. This chair transition state was then reoptimized again using the '''B3LYP/6-31G*''' level of theory. The electronic energies were calculated to be the same as in the first approach and frequency calculations were summarized in '''Table 9'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46671||-234.41490
|-
| Sum of electronic and thermal Energies||-231.46135||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40806
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir3.jpg|200px]]||[[Image:XJB_chairir4.jpg|200px]]
|-
| ||[[File:XJB_chairts2.1.log]]||[[File:XJB_chairts2.2.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 9.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the second approach''</div>

As the same energy results given, the bond forming/breaking bond lengths in the chair transition structures in the above two different approaches were compared. The first approach gave the bond lengths of 2.0203 Å while the second approach gave 2.0207 Å. Considering the accuracy limit of bond length estimation in Gaussview, these two bond lengths can be seen as exactly the same. Two methods worked reasonably well because of the good assumption of transition structures. However for more complicated and larger molecule, the Hessian method may fail due to the difficulty to predict the transition geometry (different surface curvature). Thus better way to generate such transition structures is to freeze reaction coordinate.

{| align="center"
|- align="center"
|
[[Image:XJB_chairts bondlength1.jpg|350px|thumb|'''Figure 6.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by Hession'']]
|
[[Image:XJB_chairts bondlength2.jpg|350px|thumb|'''Figure 7.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by frozen coordinate'']]
|- align="center"
|}

====The Boat Transition State====

The boat transition structure was then optimized using the '''QST2''' method. The same numbering for the reactant and product molecules was important in order to specify for this reaction. The renumbering of the molecules is shown as in '''Figure 8''', but the following optimization failed and gave a twisted structure. The reactant and product geometries were then modified further by changing the central C-C-C-C dihedral angle to 0<sup>o</sup> and the inside two C-C-C (i.e. C2-C3-C4 and C3-C4-C5) angles to 100<sup>o</sup>. The job was run successfully and frequency calculations were done with the boat transition state generated. Only one imaginary frequency was detected at -840 cm<sup>-1</sup>. The electronic energy was '''-231.60280Ha''' with C<sub>2v</sub> symmetry. The distances between the terminal carbons of the allyl fragments were examined to be both 2.140 Å which were comparably longer than that of the chair transition structure.

{| align="center"
|- align="center"
|
[[Image:XJB_QST1.jpg|400px|thumb|'''Figure 8.''' ''Numbering of reactant and product'']]
|
[[Image:XJB_QST2.jpg|410px|thumb|'''Figure 9.''' ''Re-numbering of reactant and product'']]
|- align="center"
|}

<center>[[File:XJB_boat ts animation.GIF|400px|center|thumb|'''Figure 10.''' ''Animation of the Cope rearrangement via the boat transition state'']]</center>

The structure was then reoptimized at the higher '''B3LYP/6-31G*''' theory of level and the electronic energy was calculated to be '''-234.54309Ha''' with very similar geometry. The frequency calculation generated an imaginary frequency at -530 cm<sup>-1</sup>. The summary and comparison of two levels' calculations are carried out in the following table.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.45093||-234.40234
|-
| Sum of electronic and thermal Energies||-231.44530||-234.39601
|-
| Sum of electronic and thermal Enthalpie|| -231.44436||-234.39506
|-
| Sum of electronic and thermal Free Energie||-231.47977||-234.43175
|-
| Generated IR spectrum||[[Image:XJB_boatir1.jpg|200px]]||[[Image:XJB_boatir2.jpg|200px]]
|-
| ||[[File:XJB_boat ts1.log]]||[[File:XJB_boat ts2.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 10.''' ''Frequency calculations of the boat transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by QST2''</div>

====Intrinsic Reaction Coordinate====

Next a useful method called '''Intrinct Reaction Coordinate (IRC)''' was applied to the Hessian optimized chair transition structure to follow the mininmum energy path (MEP) from a transition structure down to its local minimum on a potential energy surface. IRC can be used to verify that the transition state we found actually connects the reactants and products, or to identify any reaction intermediates, or to generate an animation of the reaction<ref name="IRC">''The Journal of Chemical Physics.'', 2005, '''122''', pp 1-17. {{DOI|10.1063/1.1927521}}</ref>. The symmetrical reaction coordinate was computed only in the forward direction with force constant always calculated and the number of points along IRC 50. The last structure generated along IRC is given in '''Figure 11''' with the energy calculated as '''-231.69158Ha''' and the dihedral angle of the central four C atoms and the internal angle between three C atoms measured as ''D'' = -67.188<sup>o</sup> ''A'' = 111.78<sup>o</sup>. This energy does not match to any of the conformations discussed in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] thus a minimum geometry has not reached yet.

{| align="center"
|- align="center"
|
<jmol><jmolApplet><title>XJB_IRC.mol</title><color>white</color>
<size>200</size><script>zoom=5</script>
<uploadedFileContents>XJB_IRC.mol</uploadedFileContents>
</jmolApplet></jmol>
|
[[Image:XJB_irc.jpg|500px]]
|- align="center"
| align="center" | '''Figure 11.''' ''The energy minimum structure along IRC ([[File:XJB_IRC.log]])''
| align="center" | '''Figure 12.''' ''Total energy along IRC''
|}

Three improving options can be used to reach the minimum energy. The first one is to run a normal Hessian optimization at the last point on IRC; the second one involves using a larger number of points (in this case 100 was used); the last one is to redo the IRC computation specifying to compute force constant at every step. Three methods were all used and the results were summarized in '''Table 11'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Improved IRC method 1
! scope="col" | Improved IRC method 2
! scope="col" | Improved IRC method 3
|-
| [[Image:XJB_IRC1.jpg|300px|center|IRC1]]||[[Image:XJB_IRC2.jpg|300px|center|IRC3]] || [[Image:XJB_IRC3.jpg|300px|center|IRC3]]
|-
| ''D'' = -64.17<sup>o</sup> ''A'' = 112.04<sup>o</sup> || ''D'' = -66.83<sup>o</sup> ''A'' = 111.82<sup>o</sup> || ''D'' = -64.17<sup>o</sup> ''A'' = 112.04<sup>o</sup>
|-
| -231.69167Ha || -231.69150Ha || -231.69167Ha
|-
| [[File:XJB_IRC1.log]]||[[File:XJB_IRC2.log]]||[[File:XJB_IRC3.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 11.''' ''Comparison of three improvement methods of IRC computation upon the chair transition structure''</div>

The first method was very quick but maybe generated relatively low-accuracy result; the second approach involved the controlling of number of points which might result in ending up at the wrong direction and wrong structure; the third method was the most suitable option for this small system but it took longer time than the previous two. From '''Table 11''' it is concluded that method 1 and 3 gave the same lowest energy which was consistent to the energy of the ''gauche2'' conformation given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

====Activation Energies====

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | '''HF/3-21G''' at 0K
! scope="col" | '''HF/3-21G''' at 298.15K
! scope="col" | '''B3LYP/6-31G*''' at 0K
! scope="col" | '''B3LYP/6-31G*''' at 298.15K
! scope="col" | Expt. at 0K
|-
| '''ΔE (chair)'''||45.71||44.70||34.05||33.16||33.5 ± 0.5
|-
| '''ΔE (boat)'''||55.60||54.76||41.96||41.32||44.7 ± 0.5
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 12.''' ''Summary of activation energies for the Cope rearrangement via both transition states in kcal/mol''</div>

The activation energy calculation was done by calculating the difference in energies between the optimized reactant ''anti2'' and the optimized chair and boat transition states. It is observed that chair conformation gives generally lower activation energy than the boat structure thus more favoured transition state. The structure optimized at '''B3LYP/6-31G*''' level of theory at 0K has closer activation energy to the experimental result. Thus '''B3LYP/6-31G*''' gives a better estimation of this transition structure than '''HF/3-21G'''.

==The Diels Alder Cycloaddition==

In this exercise, AM1 semi-empirical method was used to compute the Diels Alder cycloaddition which is a concerted pericyclic reaction. The driving force of this reaction is the formation of a new σ bond between the π orbitals of the dieneophile and the diene which is stronger than the π bond. The reaction is not allowed without the HOMO-LUMO interation and a symmetry overlap.

===Cis Butadiene===

<center>[[Image:prototype Diels Alder.jpg|500px|center|thumb|'''Figure 13'''. ''The prototype reaction between butadiene and ethylene'']]</center>

To study a prototype Diels Alder cycloaddition reaction between butadiene and ethylene, first a cis butadiene molecule was built using Gaussview and optimized using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the butadiene were plotted. As seen the HOMO of cis-butadiene is anti-symmetric ('''a''') and the LUMO is symmetric ('''s''') with respect to plane.

{| align="center"
|- align="center"
|
[[Image:XJB_HOMO.jpg|280px|thumb|'''Figure 14.''' ''Anti-symmetric HOMO of cis-butadiene'']]
|
[[Image:XJB_LUMO.jpg|300px|thumb|'''Figure 15.''' ''Symmetric HOMO of cis-butadiene'']]
|- align="center"
|}

===The Transition State of prototype reaction between ethylene and butadiene===

The Hessian method was used to estimate transition state for the reaction between ethylene and butadiene because these two reactants are relatively small and simple molecules. The transition state was drawn by first building a structure as (a) in '''Figure 16''' then removing the -CH<sub>2</sub>-CH<sub>2</sub>- group to form the envelop structure as in (b). The '''clean''' option should not be used and the dashed bond length was guessed to be larger than 2.2 Å otherwise the geometry change would cause a failure in optimization.

<center>[[Image:XJB_dielsguessts.jpg|300px|center|thumb|'''Figure 16'''. ''The guessing transition state for the reaction between butadiene and ethylene'']]</center>

<center>[[File:XJB_butadiene and ethylene ts.GIF|400px|center|thumb|'''Figure 17.''' ''Animation of the butadiene and ethylene cycloaddition transition state'' [[File:XJB_prototype DA.log]]]]</center>

After the Hessian optimization, the transition structure has been successfully determined and an animation was generated in '''Figure 17'''. It has a C<sub>1</sub> symmetry point group and the bond lengths of two partly formed C-C bonds were measured to be 2.12Å. The typical sp<sup>3</sup> and sp<sup>2</sup> C-C bond lengths are 1.54 Å and 1.47 Å respectively<ref name="bondlength">''J. Phys. C: Solid State Phys.'', 1986, '''19''', pp 4613-4621.</ref>. The van der Waals radius of the C atom is 1.7 Å<ref name="bondlength2">''Z. Naturforsch.'', 2007, '''62b''', pp 235-243.</ref>. The measured bond distance of the transition structure is larger than typical bond lengths but still within the van der Waals radius, suggesting the bond forming interation between the carbon atoms.

From the animation ('''Figure 17''') the vibration directions of two molecules are opposite and they are approaching and leaving each other at the same time. This suggests the formation of the two bonds synchronous. The vibrational frequencies were also calculated and an infrared spectrum was simulated ('''Figure 20'''). An imaginary frequency of magnitude -955 cm<sup>-1</sup> was seen and the lowest positive frequency was observed at 147 cm<sup>-1</sup> which corresponded to the reactant vibration but not the bond formation.

{| align="center"
|
[[Image:XJB_HOMOts.jpg|300px|thumb|'''Figure 18'''. HOMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_LUMOts.jpg|300px|thumb|'''Figure 19'''. LUMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_IRts.jpg|300px|thumb|'''Figure 20'''. Generated IR of transition state of reaction between cis-butadiene and ethylene]]
|}

The HOMO and LUMO molecular orbitals of this transition structure were visualized in '''Figure 18''' and '''Figure 19'''. From these two figures it can be observed that the HOMO at the transition structure is anti-symmetric ('''a''') while the LUMO is symmetric ('''s'''). Thus this anti-symmetric transition state HOMO is formed by the HOMO of cis-butadiene and the LUMO of ethylene with the knowledge that they are both anti-symmetrical. This conclusion agrees with the previous talked conditions for allowed reaction.

===The cyclohexa-1,3-diene reaction with maleic anhydride===

<center>[[Image:XJB_mechanism3.jpg|500px|center|thumb|'''Figure 21'''. ''The endo and exo products of the Diels Alder reaction between cyclohexa-1,3-diene and maleic anhydride'']]</center>

The exo and endo transition states were drawn according to '''Figure 21''' with the new C-C bond formation line dashed and bond length estimation as 2.5 Å. Then the same Hessian method was applied to optimize the structures. The electronic energy of the endo structure was calculated to be '''-0.05150Ha''' and '''-0.05042Ha''' for the exo. This reaction is supposed to be kinetically controlled thus the lower energy endo transition state is more favoured.

The bond lengths of the partly formed C-C bond are 2.16 Å and 2.17 Å for endo and exo transition states respectively. Another measured C-C bond length is shown in '''Table 13''' as 2.89 Å for endo (C3-C8 and C2-C7) and 2.95 Å for endo (C7-C10 and C8-C9). The longer atom distance indicates the larger steric repulsion in the exo transition state because of the extra existence of hydrogen atoms on C7 and C8.

Woodward and Hoffmann also proposed a secondary orbital overlap statement as an explanation of the endo stereo-preference in Diels Alder reactions<ref name="soot">''J. Org. Chem.'', 1987, '''52(8)''', pp 1469–1474. {{DOI|10.1021/jo00384a016}}</ref>. This secondary orbital overlap does not involve bond changes but π orbital interactions as the highlighted carbon atoms of the endo transition state in '''Table 13'''. Such overlap stablises the structure and lowers the energy of the endo transition state while there is no such effect existed in the exo transition state.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | endo
! scope="col" | exo
|-
| '''Animation'''||[[File:XJB_endots.GIF|300px]]||[[File:XJB_exots.GIF|300px]]
|-
| '''Sum of electronic and zero-point Energies'''||0.133493||0.134880
|-
| '''Sum of electronic and thermal Energies'''||0.143683||0.144881
|-
| '''Sum of electronic and thermal Enthalpies'''||0.144627||0.145825
|-
| '''Sum of electronic and thermal Free Energies'''||0.097349||0.099117
|-
| '''Bond lengths'''||[[Image:XJB_endomeasure.jpg|300px]]||[[Image:XJB_exomeasure.jpg|300px]]
|-
| '''HOMO'''||[[Image:XJB_endoHOMO.jpg|300px]]||[[Image:XJB_exoHOMO.jpg|285px]]
|-
| '''Secondary orbital overlap effect'''||[[Image:XJB_endosoo.jpg|300px]]||[[Image:XJB_exosoo.jpg|300px]]
|-
| ||[[File:XJB_endo ts.log]]||[[File:XJB_exo ts.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 13.''' ''Summary of comparison of properties of endo and exo transition structures for reaction between cyclohexa-1,3-diene and maleic anhydride''</div>

===Further work===

The semiempirical/AM1 calculation is based on the "Neglect of Differential Diatomic Overlap (NDDO)" integral approximation, which neglects all two-electron integrals involving two-center charge distributions are neglected<ref>http://www.cup.uni-muenchen.de/ch/compchem/energy/semi1.html</ref>. The better optimization towards the endo and exo transition states should be done first by AM1 optimization followed by '''B3LYP/6-31G*''' reoptimization<ref>''J. Org. Chem.'', 2003, '''68 (19)''', pp 7158–7166. {{DOI|10.1021/jo0348827}}</ref>. The results are summarized in '''Table 14'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | endo
! scope="col" | exo
|-
| '''Structure'''||[[Image:XJB_endots2.jpg|300px]]||[[Image:XJB_exots2.jpg|300px]]
|-
| '''Electronic energy'''|| ||
|-
| '''Sum of electronic and zero-point Energies'''||0.133493||0.134880
|-
| '''Sum of electronic and thermal Energies'''||0.143683||0.144881
|-
| '''Sum of electronic and thermal Enthalpies'''||0.144627||0.145825
|-
| '''Sum of electronic and thermal Free Energies'''||0.097349||0.099117
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 14.''' ''Reoptimization using '''B3LYP/6-31G*''' of endo and exo transition structures for reaction between cyclohexa-1,3-diene and maleic anhydride''</div>

==Reference==

<references/>

Rep:Mod:XJB1130

2013-12-06T15:53:01Z

Jx1011: /* The cyclohexa-1,3-diene reaction with maleic anhydride */

==The Cope Rearrangement of 1,5-hexadiene==

The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement reaction ('''Figure 1'''). Its mechanism has been studied by a large number of experimental and computational researches, which is recently accepted that the reaction occurs via either a "chair" or a "boat" transition state structure in a concerted manner. The aim of this exercise is to locate the low-energy minimum and the transition structures on the potential energy surface by Gaussian calculation in order to study its mechanism. The reaction pathway and the activation energy can also be calculated by this calculation.

<center>[[Image:XJB_cope rearrangement.jpg|500px|center|thumb|'''Figure 1'''. ''The Cope rearrangement'']]</center>

===Optimizing the Reactants and Products===

Different conformations of 1,5-hexadiene can be optimized using '''Gaussview'''. The optimized structure of four conformers involved in this discussion and their corresponding energies optimized at '''HF/3-21G''' level of theory are concluded in the following table ('''Table 1''').

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | anti4
! scope="col" | anti2
! scope="col" | gauche
! scope="col" | gauche3
|-
| <jmol><jmolApplet><title>XJB_anti4.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti4.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche3.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| -231.69097Ha || -231.69254Ha || -231.68772Ha ||-231.69266Ha
|}

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 1.''' ''Four conformers of 1,5-hexadiene''</div>

This exercise first involved optimizing a molecule of 1,5-hexadiene with "anti" linkage for the cetral four C atoms at the '''HF/3-21G''' level of theory. The final optimization structure had the total electronic energy of '''-231.69097Ha''' with a '''C<sub>1</sub>''' symmetry. By comparing the energy and point group to that in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1], this structure is confirmed to be ''anti4''. The compositions of energies are also listed in '''Table 2''' as i) the sum of electronic and zero-point energies (potential energy at 0K including the zero-point vibrational energy), ii) the sum of electronic and thermal energies (energy including translational, rotational and vibrational at 298.15K and 1 atm), iii) the sum of electronic and thermal enthalpies (containning an additional correction for RT), and iv) the sum of electronic and thermal free energies (including the entropic contribution to the free energy). For what we are considering, the first two terms are the most useful for further comparisons.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Formula Description'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||E = E<sub>elec</sub> + ZPE||-231.53785
|-
| Sum of electronic and thermal Energies||E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>||-231.53096
|-
| Sum of electronic and thermal Enthalpie||H = E + RT||-231.53002
|-
| Sum of electronic and thermal Free Energie||G = H - TS||-231.56891
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 2.''' ''Frequency calculations of anti4 optimized at '''HF/3-21G''' ([[File:XJB_anti4.log]])''</div>

Next another 1,5-hexadiene molecule with "gauche" linkage for the central four atoms was drawn and optimized at the '''HF/3-21G''' level of theory as before. Energy for this conformation is expected to be higher than that for the ''anti4'' conformation due to the closer position of two alkene ends thus the steric interaction. The final structure had the electronic energy of '''-231.68772Ha''' which was indeed higher than that of the ''anti4'', indicating less favour towards this gauche structure. It had a '''C<sub>2</sub>''' symmetry and was confirmed to be the ''gauche'' conformation.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53439
|-
| Sum of electronic and thermal Energies||-231.52770
|-
| Sum of electronic and thermal Enthalpie||-231.52676
|-
| Sum of electronic and thermal Free Energie||-231.56483
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 3.''' ''Frequency calculations of gauche optimized at '''HF/3-21G''' ([[File:XJB_gauche.log]])''</div>

A lowest energy conformation ''gauche3'' of the reactant molecule was drawn based on and confirmed by [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. The optimization gave a final structure with an energy of '''-231.69266Ha''' with a symmetry of '''C<sub>1</sub>'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53949
|-
| Sum of electronic and thermal Energies||-231.53265
|-
| Sum of electronic and thermal Enthalpie||-231.5317
|-
| Sum of electronic and thermal Free Energie||-231.57064
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 4.''' ''Frequency calculations of gauche3 optimized at '''HF/3-21G''' ([[File:XJB_gauche3.log]])''</div>

The next step required us to draw the C<sub>i</sub> symmetry ''anti2'' conformation of 1,5-hexadiene and optimize it at the '''HF/3-21G''' level of theory. The optimized energy was calculated to be '''-231.69254Ha''' with point group of '''C<sub>i</sub>''', which was consistent with the one given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53954
|-
| Sum of electronic and thermal Energies||-231.53257
|-
| Sum of electronic and thermal Enthalpie||-231.53162
|-
| Sum of electronic and thermal Free Energie||-231.57092
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 5.''' ''Frequency calculations of anti2 optimized at '''HF/3-21G''' ([[File:XJB_anti2.log]])''</div>

The ''anti2'' structure was then reoptimized at '''B3LYP/6-31G*''' which was a improvement over the '''HF/3-21G''' basis set. It adds polarization to atoms and improves the modeling of core electrons thus producing more accurate description of orbitals<ref name="soo">''Nigerian Journal of Chemical Research'', 2007, '''12'''. {{DOI|10.4314/njcr.v12i1.}}</ref>. The reoptimized ''anti2'' structure had the total electronic energy of '''-234.61171Ha''' and the same symmetry C<sub>i</sub>. The comparison of two structures optimized using different levels of theory is summarized in '''Table 6'''. We can see that they have exactly the same dihedral angle of the central four C atoms at 180.00<sup>o</sup> and angles between three of them are very similar as 111.3<sup>o</sup> and 112.7<sup>o</sup> respectively. Thus the the geometry change using '''HF/3-21G''' or '''B3LYP/6-31G*''' is trivial.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Before re-optimization
! scope="col" | After re-optimization
|-
| <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2reoptimized</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2reoptimized.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| ''anti2 optimized at '''HF/3-21G''''' || ''anti2 optimized at '''B3LYP/6-31G*'''''
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 6.''' ''Two optimization of anti2 at '''HF/3-21G''' and '''B3LYP/6-31G*'''([[File:XJB_anti2.log]])''</div>

The frequency calculation was run as well and a list of energies is summarized in '''Table 7'''. A list of all the vibrational frequencies was also generated and no imaginary (negative) frequency was observed and these vibrations are visualized by simulating the infrared spectrum in '''Figure 2'''.

{| align="center"
|- align="center"
|
{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-234.46920
|-
| Sum of electronic and thermal Energies||-234.46186
|-
| Sum of electronic and thermal Enthalpie||-234.46091
|-
| Sum of electronic and thermal Free Energie||-234.50078
|}
|
[[Image:XJB_IR_anti2.jpg|300px]]
|- align="center"
| align="center" | '''Table 7.''' ''Frequency calculations of anti2 optimized at '''B3LYP/6-31G*'''
([[File:XJB_anti2_reoptimized.log]])''
| align="center" | '''Figure 2.''' ''The generated IR spectrum of optimized '''B3LYP/6-31G*''' structure''
|}

===Optimizing the "Chair" and "Boat" Transition Structures===

The study of the chair and boat transition states was carried out using three different approaches. The first one was called the Hessian which involved computing the force constant matrix; the second one was to freeze the reaction coordinate and then optimized the structure once the molecule was fully relaxed; the last one was to use QST2 method to specify the reactants and products then finding the transition structure.

<center>[[Image:XJB_2 ts.jpg|500px|center|thumb|'''Figure 3'''. ''The two transition states of Cope rearrangement'']]</center>

====The Chair Transition State====

In order to set up a transition state optimization, first an allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimized at the '''HF/3-21G''' level of theory. Then this half of the transition structure was duplicated with rotation to make the approximate resembling of the chair transition state with terminal ends of the fragments 2.2 Å apart. The chair transition structure optimization was then done by the first two different approaches.

<center>[[Image:XJB_guess ts.jpg|400px|center|thumb|'''Figure 4'''. ''The guessed chair transition structure of two allyl fragments'']]</center>

For the first optimization approach, the force constants were computed (the Hessian) and the chair transition state was optimized to a TS (Berny) at the '''HF/3-21G''' level of theory. The frequency calculation was carried out at the same time and gave an imaginary frequency at -818 cm<sup>-1</sup>. The electronic energy was calculated to be '''-231.61932Ha''' with C<sub>2h</sub> symmetry point group. The distance between the terminal carbons was 2.0203 Å. An animation of this transition state vibrations was generated to confirm the corresponding Cope rearrangement.

<center>[[File:XJB_chair ts animation.GIF|400px|center|thumb|'''Figure 5.''' ''Animation of the Cope rearrangement via the chair transition state'']]</center>

The chair transition state was then reoptimized using the '''B3LYP/6-31G*''' level of theory and carried out frequency calculations as well. The electronic energy calculated this time was '''-234.55698Ha''' with an imaginary frequency at -566 cm<sup>-1</sup>. We can see that with the geometries quite similar, the energies differ markedly calculated using two levels of theory. The summary and comparison of the calculations optimized at two levels of theory are summarized in '''Table 8'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46670||-234.41493
|-
| Sum of electronic and thermal Energies||-231.46134||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40807
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir1.jpg|200px]]||[[Image:XJB_chairir2.jpg|200px]]
|-
| ||[[File:XJB_chairts1.1.log]]||[[File:XJB_chairts1.2.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 8.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the first approach''</div>

The second approach is the frozen coordinate method using the Redundant Coord Editor to freeze the bond lengths of the terminal ends to 2.2 Å. First the calculation failed because of the recent update of GaussView. After editting the coordinate distance (B 5 1 2.2 F) and the frozen distance (B 5 1 F) the optimization went well this time. The structure was then optimized again using a normal guess Hessian modified including the two differentiating coordinates. This chair transition state was then reoptimized again using the '''B3LYP/6-31G*''' level of theory. The electronic energies were calculated to be the same as in the first approach and frequency calculations were summarized in '''Table 9'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46671||-234.41490
|-
| Sum of electronic and thermal Energies||-231.46135||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40806
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir3.jpg|200px]]||[[Image:XJB_chairir4.jpg|200px]]
|-
| ||[[File:XJB_chairts2.1.log]]||[[File:XJB_chairts2.2.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 9.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the second approach''</div>

As the same energy results given, the bond forming/breaking bond lengths in the chair transition structures in the above two different approaches were compared. The first approach gave the bond lengths of 2.0203 Å while the second approach gave 2.0207 Å. Considering the accuracy limit of bond length estimation in Gaussview, these two bond lengths can be seen as exactly the same. Two methods worked reasonably well because of the good assumption of transition structures. However for more complicated and larger molecule, the Hessian method may fail due to the difficulty to predict the transition geometry (different surface curvature). Thus better way to generate such transition structures is to freeze reaction coordinate.

{| align="center"
|- align="center"
|
[[Image:XJB_chairts bondlength1.jpg|350px|thumb|'''Figure 6.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by Hession'']]
|
[[Image:XJB_chairts bondlength2.jpg|350px|thumb|'''Figure 7.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by frozen coordinate'']]
|- align="center"
|}

====The Boat Transition State====

The boat transition structure was then optimized using the '''QST2''' method. The same numbering for the reactant and product molecules was important in order to specify for this reaction. The renumbering of the molecules is shown as in '''Figure 8''', but the following optimization failed and gave a twisted structure. The reactant and product geometries were then modified further by changing the central C-C-C-C dihedral angle to 0<sup>o</sup> and the inside two C-C-C (i.e. C2-C3-C4 and C3-C4-C5) angles to 100<sup>o</sup>. The job was run successfully and frequency calculations were done with the boat transition state generated. Only one imaginary frequency was detected at -840 cm<sup>-1</sup>. The electronic energy was '''-231.60280Ha''' with C<sub>2v</sub> symmetry. The distances between the terminal carbons of the allyl fragments were examined to be both 2.140 Å which were comparably longer than that of the chair transition structure.

{| align="center"
|- align="center"
|
[[Image:XJB_QST1.jpg|400px|thumb|'''Figure 8.''' ''Numbering of reactant and product'']]
|
[[Image:XJB_QST2.jpg|410px|thumb|'''Figure 9.''' ''Re-numbering of reactant and product'']]
|- align="center"
|}

<center>[[File:XJB_boat ts animation.GIF|400px|center|thumb|'''Figure 10.''' ''Animation of the Cope rearrangement via the boat transition state'']]</center>

The structure was then reoptimized at the higher '''B3LYP/6-31G*''' theory of level and the electronic energy was calculated to be '''-234.54309Ha''' with very similar geometry. The frequency calculation generated an imaginary frequency at -530 cm<sup>-1</sup>. The summary and comparison of two levels' calculations are carried out in the following table.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.45093||-234.40234
|-
| Sum of electronic and thermal Energies||-231.44530||-234.39601
|-
| Sum of electronic and thermal Enthalpie|| -231.44436||-234.39506
|-
| Sum of electronic and thermal Free Energie||-231.47977||-234.43175
|-
| Generated IR spectrum||[[Image:XJB_boatir1.jpg|200px]]||[[Image:XJB_boatir2.jpg|200px]]
|-
| ||[[File:XJB_boat ts1.log]]||[[File:XJB_boat ts2.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 10.''' ''Frequency calculations of the boat transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by QST2''</div>

====Intrinsic Reaction Coordinate====

Next a useful method called '''Intrinct Reaction Coordinate (IRC)''' was applied to the Hessian optimized chair transition structure to follow the mininmum energy path (MEP) from a transition structure down to its local minimum on a potential energy surface. IRC can be used to verify that the transition state we found actually connects the reactants and products, or to identify any reaction intermediates, or to generate an animation of the reaction<ref name="IRC">''The Journal of Chemical Physics.'', 2005, '''122''', pp 1-17. {{DOI|10.1063/1.1927521}}</ref>. The symmetrical reaction coordinate was computed only in the forward direction with force constant always calculated and the number of points along IRC 50. The last structure generated along IRC is given in '''Figure 11''' with the energy calculated as '''-231.69158Ha''' and the dihedral angle of the central four C atoms and the internal angle between three C atoms measured as ''D'' = -67.188<sup>o</sup> ''A'' = 111.78<sup>o</sup>. This energy does not match to any of the conformations discussed in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] thus a minimum geometry has not reached yet.

{| align="center"
|- align="center"
|
<jmol><jmolApplet><title>XJB_IRC.mol</title><color>white</color>
<size>200</size><script>zoom=5</script>
<uploadedFileContents>XJB_IRC.mol</uploadedFileContents>
</jmolApplet></jmol>
|
[[Image:XJB_irc.jpg|500px]]
|- align="center"
| align="center" | '''Figure 11.''' ''The energy minimum structure along IRC ([[File:XJB_IRC.log]])''
| align="center" | '''Figure 12.''' ''Total energy along IRC''
|}

Three improving options can be used to reach the minimum energy. The first one is to run a normal Hessian optimization at the last point on IRC; the second one involves using a larger number of points (in this case 100 was used); the last one is to redo the IRC computation specifying to compute force constant at every step. Three methods were all used and the results were summarized in '''Table 11'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Improved IRC method 1
! scope="col" | Improved IRC method 2
! scope="col" | Improved IRC method 3
|-
| [[Image:XJB_IRC1.jpg|300px|center|IRC1]]||[[Image:XJB_IRC2.jpg|300px|center|IRC3]] || [[Image:XJB_IRC3.jpg|300px|center|IRC3]]
|-
| ''D'' = -64.17<sup>o</sup> ''A'' = 112.04<sup>o</sup> || ''D'' = -66.83<sup>o</sup> ''A'' = 111.82<sup>o</sup> || ''D'' = -64.17<sup>o</sup> ''A'' = 112.04<sup>o</sup>
|-
| -231.69167Ha || -231.69150Ha || -231.69167Ha
|-
| [[File:XJB_IRC1.log]]||[[File:XJB_IRC2.log]]||[[File:XJB_IRC3.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 11.''' ''Comparison of three improvement methods of IRC computation upon the chair transition structure''</div>

The first method was very quick but maybe generated relatively low-accuracy result; the second approach involved the controlling of number of points which might result in ending up at the wrong direction and wrong structure; the third method was the most suitable option for this small system but it took longer time than the previous two. From '''Table 11''' it is concluded that method 1 and 3 gave the same lowest energy which was consistent to the energy of the ''gauche2'' conformation given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

====Activation Energies====

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | '''HF/3-21G''' at 0K
! scope="col" | '''HF/3-21G''' at 298.15K
! scope="col" | '''B3LYP/6-31G*''' at 0K
! scope="col" | '''B3LYP/6-31G*''' at 298.15K
! scope="col" | Expt. at 0K
|-
| '''ΔE (chair)'''||45.71||44.70||34.05||33.16||33.5 ± 0.5
|-
| '''ΔE (boat)'''||55.60||54.76||41.96||41.32||44.7 ± 0.5
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 12.''' ''Summary of activation energies for the Cope rearrangement via both transition states in kcal/mol''</div>

The activation energy calculation was done by calculating the difference in energies between the optimized reactant ''anti2'' and the optimized chair and boat transition states. It is observed that chair conformation gives generally lower activation energy than the boat structure thus more favoured transition state. The structure optimized at '''B3LYP/6-31G*''' level of theory at 0K has closer activation energy to the experimental result. Thus '''B3LYP/6-31G*''' gives a better estimation of this transition structure than '''HF/3-21G'''.

==The Diels Alder Cycloaddition==

In this exercise, AM1 semi-empirical method was used to compute the Diels Alder cycloaddition which is a concerted pericyclic reaction. The driving force of this reaction is the formation of a new σ bond between the π orbitals of the dieneophile and the diene which is stronger than the π bond. The reaction is not allowed without the HOMO-LUMO interation and a symmetry overlap.

===Cis Butadiene===

<center>[[Image:prototype Diels Alder.jpg|500px|center|thumb|'''Figure 13'''. ''The prototype reaction between butadiene and ethylene'']]</center>

To study a prototype Diels Alder cycloaddition reaction between butadiene and ethylene, first a cis butadiene molecule was built using Gaussview and optimized using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the butadiene were plotted. As seen the HOMO of cis-butadiene is anti-symmetric ('''a''') and the LUMO is symmetric ('''s''') with respect to plane.

{| align="center"
|- align="center"
|
[[Image:XJB_HOMO.jpg|280px|thumb|'''Figure 14.''' ''Anti-symmetric HOMO of cis-butadiene'']]
|
[[Image:XJB_LUMO.jpg|300px|thumb|'''Figure 15.''' ''Symmetric HOMO of cis-butadiene'']]
|- align="center"
|}

===The Transition State of prototype reaction between ethylene and butadiene===

The Hessian method was used to estimate transition state for the reaction between ethylene and butadiene because these two reactants are relatively small and simple molecules. The transition state was drawn by first building a structure as (a) in '''Figure 16''' then removing the -CH<sub>2</sub>-CH<sub>2</sub>- group to form the envelop structure as in (b). The '''clean''' option should not be used and the dashed bond length was guessed to be larger than 2.2 Å otherwise the geometry change would cause a failure in optimization.

<center>[[Image:XJB_dielsguessts.jpg|300px|center|thumb|'''Figure 16'''. ''The guessing transition state for the reaction between butadiene and ethylene'']]</center>

<center>[[File:XJB_butadiene and ethylene ts.GIF|400px|center|thumb|'''Figure 17.''' ''Animation of the butadiene and ethylene cycloaddition transition state'' [[File:XJB_prototype DA.log]]]]</center>

After the Hessian optimization, the transition structure has been successfully determined and an animation was generated in '''Figure 17'''. It has a C<sub>1</sub> symmetry point group and the bond lengths of two partly formed C-C bonds were measured to be 2.12Å. The typical sp<sup>3</sup> and sp<sup>2</sup> C-C bond lengths are 1.54 Å and 1.47 Å respectively<ref name="bondlength">''J. Phys. C: Solid State Phys.'', 1986, '''19''', pp 4613-4621.</ref>. The van der Waals radius of the C atom is 1.7 Å<ref name="bondlength2">''Z. Naturforsch.'', 2007, '''62b''', pp 235-243.</ref>. The measured bond distance of the transition structure is larger than typical bond lengths but still within the van der Waals radius, suggesting the bond forming interation between the carbon atoms.

From the animation ('''Figure 17''') the vibration directions of two molecules are opposite and they are approaching and leaving each other at the same time. This suggests the formation of the two bonds synchronous. The vibrational frequencies were also calculated and an infrared spectrum was simulated ('''Figure 20'''). An imaginary frequency of magnitude -955 cm<sup>-1</sup> was seen and the lowest positive frequency was observed at 147 cm<sup>-1</sup> which corresponded to the reactant vibration but not the bond formation.

{| align="center"
|
[[Image:XJB_HOMOts.jpg|300px|thumb|'''Figure 18'''. HOMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_LUMOts.jpg|300px|thumb|'''Figure 19'''. LUMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_IRts.jpg|300px|thumb|'''Figure 20'''. Generated IR of transition state of reaction between cis-butadiene and ethylene]]
|}

The HOMO and LUMO molecular orbitals of this transition structure were visualized in '''Figure 18''' and '''Figure 19'''. From these two figures it can be observed that the HOMO at the transition structure is anti-symmetric ('''a''') while the LUMO is symmetric ('''s'''). Thus this anti-symmetric transition state HOMO is formed by the HOMO of cis-butadiene and the LUMO of ethylene with the knowledge that they are both anti-symmetrical. This conclusion agrees with the previous talked conditions for allowed reaction.

===The cyclohexa-1,3-diene reaction with maleic anhydride===

<center>[[Image:XJB_mechanism3.jpg|500px|center|thumb|'''Figure 21'''. ''The endo and exo products of the Diels Alder reaction between cyclohexa-1,3-diene and maleic anhydride'']]</center>

The exo and endo transition states were drawn according to '''Figure 21''' with the new C-C bond formation line dashed and bond length estimation as 2.5 Å. Then the same Hessian method was applied to optimize the structures. The electronic energy of the endo structure was calculated to be '''-0.05150Ha''' and '''-0.05042Ha''' for the exo. This reaction is supposed to be kinetically controlled thus the lower energy endo transition state is more favoured.

The bond lengths of the partly formed C-C bond are 2.16 Å and 2.17 Å for endo and exo transition states respectively. Another measured C-C bond length is shown in '''Table 13''' as 2.89 Å for endo (C3-C8 and C2-C7) and 2.95 Å for endo (C7-C10 and C8-C9). The longer atom distance indicates the larger steric repulsion in the exo transition state because of the extra existence of hydrogen atoms on C7 and C8.

Woodward and Hoffmann also proposed a secondary orbital overlap statement as an explanation of the endo stereo-preference in Diels Alder reactions<ref name="soot">''J. Org. Chem.'', 1987, '''52(8)''', pp 1469–1474. {{DOI|10.1021/jo00384a016}}</ref>. This secondary orbital overlap does not involve bond changes but π orbital interactions as the highlighted carbon atoms of the endo transition state in '''Table 13'''. Such overlap stablises the structure and lowers the energy of the endo transition state while there is no such effect existed in the exo transition state.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | endo
! scope="col" | exo
|-
| '''Animation'''||[[File:XJB_endots.GIF|300px]]||[[File:XJB_exots.GIF|300px]]
|-
| '''Sum of electronic and zero-point Energies'''||0.133493||0.134880
|-
| '''Sum of electronic and thermal Energies'''||0.143683||0.144881
|-
| '''Sum of electronic and thermal Enthalpies'''||0.144627||0.145825
|-
| '''Sum of electronic and thermal Free Energies'''||0.097349||0.099117
|-
| '''Bond lengths'''||[[Image:XJB_endomeasure.jpg|300px]]||[[Image:XJB_exomeasure.jpg|300px]]
|-
| '''HOMO'''||[[Image:XJB_endoHOMO.jpg|300px]]||[[Image:XJB_exoHOMO.jpg|285px]]
|-
| '''Secondary orbital overlap effect'''||[[Image:XJB_endosoo.jpg|300px]]||[[Image:XJB_exosoo.jpg|300px]]
|-
| ||[[File:XJB_endo ts]]||[[File:XJB_exo ts]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 13.''' ''Summary of comparison of properties of endo and exo transition structures for reaction between cyclohexa-1,3-diene and maleic anhydride''</div>

===Further work===

The semiempirical/AM1 calculation is based on the "Neglect of Differential Diatomic Overlap (NDDO)" integral approximation, which neglects all two-electron integrals involving two-center charge distributions are neglected<ref>http://www.cup.uni-muenchen.de/ch/compchem/energy/semi1.html</ref>. The better optimization towards the endo and exo transition states should be done first by AM1 optimization followed by '''B3LYP/6-31G*''' reoptimization<ref>''J. Org. Chem.'', 2003, '''68 (19)''', pp 7158–7166. {{DOI|10.1021/jo0348827}}</ref>. The results are summarized in '''Table 14'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | endo
! scope="col" | exo
|-
| '''Structure'''||[[Image:XJB_endots2.jpg|300px]]||[[Image:XJB_exots2.jpg|300px]]
|-
| '''Electronic energy'''|| ||
|-
| '''Sum of electronic and zero-point Energies'''||0.133493||0.134880
|-
| '''Sum of electronic and thermal Energies'''||0.143683||0.144881
|-
| '''Sum of electronic and thermal Enthalpies'''||0.144627||0.145825
|-
| '''Sum of electronic and thermal Free Energies'''||0.097349||0.099117
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 14.''' ''Reoptimization using '''B3LYP/6-31G*''' of endo and exo transition structures for reaction between cyclohexa-1,3-diene and maleic anhydride''</div>

==Reference==

<references/>

Rep:Mod:XJB1130

2013-12-06T15:45:19Z

Jx1011: /* The Diels Alder Cycloaddition */

==The Cope Rearrangement of 1,5-hexadiene==

The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement reaction ('''Figure 1'''). Its mechanism has been studied by a large number of experimental and computational researches, which is recently accepted that the reaction occurs via either a "chair" or a "boat" transition state structure in a concerted manner. The aim of this exercise is to locate the low-energy minimum and the transition structures on the potential energy surface by Gaussian calculation in order to study its mechanism. The reaction pathway and the activation energy can also be calculated by this calculation.

<center>[[Image:XJB_cope rearrangement.jpg|500px|center|thumb|'''Figure 1'''. ''The Cope rearrangement'']]</center>

===Optimizing the Reactants and Products===

Different conformations of 1,5-hexadiene can be optimized using '''Gaussview'''. The optimized structure of four conformers involved in this discussion and their corresponding energies optimized at '''HF/3-21G''' level of theory are concluded in the following table ('''Table 1''').

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | anti4
! scope="col" | anti2
! scope="col" | gauche
! scope="col" | gauche3
|-
| <jmol><jmolApplet><title>XJB_anti4.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti4.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche3.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| -231.69097Ha || -231.69254Ha || -231.68772Ha ||-231.69266Ha
|}

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 1.''' ''Four conformers of 1,5-hexadiene''</div>

This exercise first involved optimizing a molecule of 1,5-hexadiene with "anti" linkage for the cetral four C atoms at the '''HF/3-21G''' level of theory. The final optimization structure had the total electronic energy of '''-231.69097Ha''' with a '''C<sub>1</sub>''' symmetry. By comparing the energy and point group to that in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1], this structure is confirmed to be ''anti4''. The compositions of energies are also listed in '''Table 2''' as i) the sum of electronic and zero-point energies (potential energy at 0K including the zero-point vibrational energy), ii) the sum of electronic and thermal energies (energy including translational, rotational and vibrational at 298.15K and 1 atm), iii) the sum of electronic and thermal enthalpies (containning an additional correction for RT), and iv) the sum of electronic and thermal free energies (including the entropic contribution to the free energy). For what we are considering, the first two terms are the most useful for further comparisons.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Formula Description'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||E = E<sub>elec</sub> + ZPE||-231.53785
|-
| Sum of electronic and thermal Energies||E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>||-231.53096
|-
| Sum of electronic and thermal Enthalpie||H = E + RT||-231.53002
|-
| Sum of electronic and thermal Free Energie||G = H - TS||-231.56891
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 2.''' ''Frequency calculations of anti4 optimized at '''HF/3-21G''' ([[File:XJB_anti4.log]])''</div>

Next another 1,5-hexadiene molecule with "gauche" linkage for the central four atoms was drawn and optimized at the '''HF/3-21G''' level of theory as before. Energy for this conformation is expected to be higher than that for the ''anti4'' conformation due to the closer position of two alkene ends thus the steric interaction. The final structure had the electronic energy of '''-231.68772Ha''' which was indeed higher than that of the ''anti4'', indicating less favour towards this gauche structure. It had a '''C<sub>2</sub>''' symmetry and was confirmed to be the ''gauche'' conformation.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53439
|-
| Sum of electronic and thermal Energies||-231.52770
|-
| Sum of electronic and thermal Enthalpie||-231.52676
|-
| Sum of electronic and thermal Free Energie||-231.56483
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 3.''' ''Frequency calculations of gauche optimized at '''HF/3-21G''' ([[File:XJB_gauche.log]])''</div>

A lowest energy conformation ''gauche3'' of the reactant molecule was drawn based on and confirmed by [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. The optimization gave a final structure with an energy of '''-231.69266Ha''' with a symmetry of '''C<sub>1</sub>'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53949
|-
| Sum of electronic and thermal Energies||-231.53265
|-
| Sum of electronic and thermal Enthalpie||-231.5317
|-
| Sum of electronic and thermal Free Energie||-231.57064
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 4.''' ''Frequency calculations of gauche3 optimized at '''HF/3-21G''' ([[File:XJB_gauche3.log]])''</div>

The next step required us to draw the C<sub>i</sub> symmetry ''anti2'' conformation of 1,5-hexadiene and optimize it at the '''HF/3-21G''' level of theory. The optimized energy was calculated to be '''-231.69254Ha''' with point group of '''C<sub>i</sub>''', which was consistent with the one given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53954
|-
| Sum of electronic and thermal Energies||-231.53257
|-
| Sum of electronic and thermal Enthalpie||-231.53162
|-
| Sum of electronic and thermal Free Energie||-231.57092
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 5.''' ''Frequency calculations of anti2 optimized at '''HF/3-21G''' ([[File:XJB_anti2.log]])''</div>

The ''anti2'' structure was then reoptimized at '''B3LYP/6-31G*''' which was a improvement over the '''HF/3-21G''' basis set. It adds polarization to atoms and improves the modeling of core electrons thus producing more accurate description of orbitals<ref name="soo">''Nigerian Journal of Chemical Research'', 2007, '''12'''. {{DOI|10.4314/njcr.v12i1.}}</ref>. The reoptimized ''anti2'' structure had the total electronic energy of '''-234.61171Ha''' and the same symmetry C<sub>i</sub>. The comparison of two structures optimized using different levels of theory is summarized in '''Table 6'''. We can see that they have exactly the same dihedral angle of the central four C atoms at 180.00<sup>o</sup> and angles between three of them are very similar as 111.3<sup>o</sup> and 112.7<sup>o</sup> respectively. Thus the the geometry change using '''HF/3-21G''' or '''B3LYP/6-31G*''' is trivial.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Before re-optimization
! scope="col" | After re-optimization
|-
| <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2reoptimized</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2reoptimized.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| ''anti2 optimized at '''HF/3-21G''''' || ''anti2 optimized at '''B3LYP/6-31G*'''''
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 6.''' ''Two optimization of anti2 at '''HF/3-21G''' and '''B3LYP/6-31G*'''([[File:XJB_anti2.log]])''</div>

The frequency calculation was run as well and a list of energies is summarized in '''Table 7'''. A list of all the vibrational frequencies was also generated and no imaginary (negative) frequency was observed and these vibrations are visualized by simulating the infrared spectrum in '''Figure 2'''.

{| align="center"
|- align="center"
|
{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-234.46920
|-
| Sum of electronic and thermal Energies||-234.46186
|-
| Sum of electronic and thermal Enthalpie||-234.46091
|-
| Sum of electronic and thermal Free Energie||-234.50078
|}
|
[[Image:XJB_IR_anti2.jpg|300px]]
|- align="center"
| align="center" | '''Table 7.''' ''Frequency calculations of anti2 optimized at '''B3LYP/6-31G*'''
([[File:XJB_anti2_reoptimized.log]])''
| align="center" | '''Figure 2.''' ''The generated IR spectrum of optimized '''B3LYP/6-31G*''' structure''
|}

===Optimizing the "Chair" and "Boat" Transition Structures===

The study of the chair and boat transition states was carried out using three different approaches. The first one was called the Hessian which involved computing the force constant matrix; the second one was to freeze the reaction coordinate and then optimized the structure once the molecule was fully relaxed; the last one was to use QST2 method to specify the reactants and products then finding the transition structure.

<center>[[Image:XJB_2 ts.jpg|500px|center|thumb|'''Figure 3'''. ''The two transition states of Cope rearrangement'']]</center>

====The Chair Transition State====

In order to set up a transition state optimization, first an allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimized at the '''HF/3-21G''' level of theory. Then this half of the transition structure was duplicated with rotation to make the approximate resembling of the chair transition state with terminal ends of the fragments 2.2 Å apart. The chair transition structure optimization was then done by the first two different approaches.

<center>[[Image:XJB_guess ts.jpg|400px|center|thumb|'''Figure 4'''. ''The guessed chair transition structure of two allyl fragments'']]</center>

For the first optimization approach, the force constants were computed (the Hessian) and the chair transition state was optimized to a TS (Berny) at the '''HF/3-21G''' level of theory. The frequency calculation was carried out at the same time and gave an imaginary frequency at -818 cm<sup>-1</sup>. The electronic energy was calculated to be '''-231.61932Ha''' with C<sub>2h</sub> symmetry point group. The distance between the terminal carbons was 2.0203 Å. An animation of this transition state vibrations was generated to confirm the corresponding Cope rearrangement.

<center>[[File:XJB_chair ts animation.GIF|400px|center|thumb|'''Figure 5.''' ''Animation of the Cope rearrangement via the chair transition state'']]</center>

The chair transition state was then reoptimized using the '''B3LYP/6-31G*''' level of theory and carried out frequency calculations as well. The electronic energy calculated this time was '''-234.55698Ha''' with an imaginary frequency at -566 cm<sup>-1</sup>. We can see that with the geometries quite similar, the energies differ markedly calculated using two levels of theory. The summary and comparison of the calculations optimized at two levels of theory are summarized in '''Table 8'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46670||-234.41493
|-
| Sum of electronic and thermal Energies||-231.46134||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40807
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir1.jpg|200px]]||[[Image:XJB_chairir2.jpg|200px]]
|-
| ||[[File:XJB_chairts1.1.log]]||[[File:XJB_chairts1.2.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 8.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the first approach''</div>

The second approach is the frozen coordinate method using the Redundant Coord Editor to freeze the bond lengths of the terminal ends to 2.2 Å. First the calculation failed because of the recent update of GaussView. After editting the coordinate distance (B 5 1 2.2 F) and the frozen distance (B 5 1 F) the optimization went well this time. The structure was then optimized again using a normal guess Hessian modified including the two differentiating coordinates. This chair transition state was then reoptimized again using the '''B3LYP/6-31G*''' level of theory. The electronic energies were calculated to be the same as in the first approach and frequency calculations were summarized in '''Table 9'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46671||-234.41490
|-
| Sum of electronic and thermal Energies||-231.46135||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40806
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir3.jpg|200px]]||[[Image:XJB_chairir4.jpg|200px]]
|-
| ||[[File:XJB_chairts2.1.log]]||[[File:XJB_chairts2.2.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 9.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the second approach''</div>

As the same energy results given, the bond forming/breaking bond lengths in the chair transition structures in the above two different approaches were compared. The first approach gave the bond lengths of 2.0203 Å while the second approach gave 2.0207 Å. Considering the accuracy limit of bond length estimation in Gaussview, these two bond lengths can be seen as exactly the same. Two methods worked reasonably well because of the good assumption of transition structures. However for more complicated and larger molecule, the Hessian method may fail due to the difficulty to predict the transition geometry (different surface curvature). Thus better way to generate such transition structures is to freeze reaction coordinate.

{| align="center"
|- align="center"
|
[[Image:XJB_chairts bondlength1.jpg|350px|thumb|'''Figure 6.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by Hession'']]
|
[[Image:XJB_chairts bondlength2.jpg|350px|thumb|'''Figure 7.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by frozen coordinate'']]
|- align="center"
|}

====The Boat Transition State====

The boat transition structure was then optimized using the '''QST2''' method. The same numbering for the reactant and product molecules was important in order to specify for this reaction. The renumbering of the molecules is shown as in '''Figure 8''', but the following optimization failed and gave a twisted structure. The reactant and product geometries were then modified further by changing the central C-C-C-C dihedral angle to 0<sup>o</sup> and the inside two C-C-C (i.e. C2-C3-C4 and C3-C4-C5) angles to 100<sup>o</sup>. The job was run successfully and frequency calculations were done with the boat transition state generated. Only one imaginary frequency was detected at -840 cm<sup>-1</sup>. The electronic energy was '''-231.60280Ha''' with C<sub>2v</sub> symmetry. The distances between the terminal carbons of the allyl fragments were examined to be both 2.140 Å which were comparably longer than that of the chair transition structure.

{| align="center"
|- align="center"
|
[[Image:XJB_QST1.jpg|400px|thumb|'''Figure 8.''' ''Numbering of reactant and product'']]
|
[[Image:XJB_QST2.jpg|410px|thumb|'''Figure 9.''' ''Re-numbering of reactant and product'']]
|- align="center"
|}

<center>[[File:XJB_boat ts animation.GIF|400px|center|thumb|'''Figure 10.''' ''Animation of the Cope rearrangement via the boat transition state'']]</center>

The structure was then reoptimized at the higher '''B3LYP/6-31G*''' theory of level and the electronic energy was calculated to be '''-234.54309Ha''' with very similar geometry. The frequency calculation generated an imaginary frequency at -530 cm<sup>-1</sup>. The summary and comparison of two levels' calculations are carried out in the following table.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.45093||-234.40234
|-
| Sum of electronic and thermal Energies||-231.44530||-234.39601
|-
| Sum of electronic and thermal Enthalpie|| -231.44436||-234.39506
|-
| Sum of electronic and thermal Free Energie||-231.47977||-234.43175
|-
| Generated IR spectrum||[[Image:XJB_boatir1.jpg|200px]]||[[Image:XJB_boatir2.jpg|200px]]
|-
| ||[[File:XJB_boat ts1.log]]||[[File:XJB_boat ts2.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 10.''' ''Frequency calculations of the boat transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by QST2''</div>

====Intrinsic Reaction Coordinate====

Next a useful method called '''Intrinct Reaction Coordinate (IRC)''' was applied to the Hessian optimized chair transition structure to follow the mininmum energy path (MEP) from a transition structure down to its local minimum on a potential energy surface. IRC can be used to verify that the transition state we found actually connects the reactants and products, or to identify any reaction intermediates, or to generate an animation of the reaction<ref name="IRC">''The Journal of Chemical Physics.'', 2005, '''122''', pp 1-17. {{DOI|10.1063/1.1927521}}</ref>. The symmetrical reaction coordinate was computed only in the forward direction with force constant always calculated and the number of points along IRC 50. The last structure generated along IRC is given in '''Figure 11''' with the energy calculated as '''-231.69158Ha''' and the dihedral angle of the central four C atoms and the internal angle between three C atoms measured as ''D'' = -67.188<sup>o</sup> ''A'' = 111.78<sup>o</sup>. This energy does not match to any of the conformations discussed in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] thus a minimum geometry has not reached yet.

{| align="center"
|- align="center"
|
<jmol><jmolApplet><title>XJB_IRC.mol</title><color>white</color>
<size>200</size><script>zoom=5</script>
<uploadedFileContents>XJB_IRC.mol</uploadedFileContents>
</jmolApplet></jmol>
|
[[Image:XJB_irc.jpg|500px]]
|- align="center"
| align="center" | '''Figure 11.''' ''The energy minimum structure along IRC ([[File:XJB_IRC.log]])''
| align="center" | '''Figure 12.''' ''Total energy along IRC''
|}

Three improving options can be used to reach the minimum energy. The first one is to run a normal Hessian optimization at the last point on IRC; the second one involves using a larger number of points (in this case 100 was used); the last one is to redo the IRC computation specifying to compute force constant at every step. Three methods were all used and the results were summarized in '''Table 11'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Improved IRC method 1
! scope="col" | Improved IRC method 2
! scope="col" | Improved IRC method 3
|-
| [[Image:XJB_IRC1.jpg|300px|center|IRC1]]||[[Image:XJB_IRC2.jpg|300px|center|IRC3]] || [[Image:XJB_IRC3.jpg|300px|center|IRC3]]
|-
| ''D'' = -64.17<sup>o</sup> ''A'' = 112.04<sup>o</sup> || ''D'' = -66.83<sup>o</sup> ''A'' = 111.82<sup>o</sup> || ''D'' = -64.17<sup>o</sup> ''A'' = 112.04<sup>o</sup>
|-
| -231.69167Ha || -231.69150Ha || -231.69167Ha
|-
| [[File:XJB_IRC1.log]]||[[File:XJB_IRC2.log]]||[[File:XJB_IRC3.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 11.''' ''Comparison of three improvement methods of IRC computation upon the chair transition structure''</div>

The first method was very quick but maybe generated relatively low-accuracy result; the second approach involved the controlling of number of points which might result in ending up at the wrong direction and wrong structure; the third method was the most suitable option for this small system but it took longer time than the previous two. From '''Table 11''' it is concluded that method 1 and 3 gave the same lowest energy which was consistent to the energy of the ''gauche2'' conformation given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

====Activation Energies====

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | '''HF/3-21G''' at 0K
! scope="col" | '''HF/3-21G''' at 298.15K
! scope="col" | '''B3LYP/6-31G*''' at 0K
! scope="col" | '''B3LYP/6-31G*''' at 298.15K
! scope="col" | Expt. at 0K
|-
| '''ΔE (chair)'''||45.71||44.70||34.05||33.16||33.5 ± 0.5
|-
| '''ΔE (boat)'''||55.60||54.76||41.96||41.32||44.7 ± 0.5
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 12.''' ''Summary of activation energies for the Cope rearrangement via both transition states in kcal/mol''</div>

The activation energy calculation was done by calculating the difference in energies between the optimized reactant ''anti2'' and the optimized chair and boat transition states. It is observed that chair conformation gives generally lower activation energy than the boat structure thus more favoured transition state. The structure optimized at '''B3LYP/6-31G*''' level of theory at 0K has closer activation energy to the experimental result. Thus '''B3LYP/6-31G*''' gives a better estimation of this transition structure than '''HF/3-21G'''.

==The Diels Alder Cycloaddition==

In this exercise, AM1 semi-empirical method was used to compute the Diels Alder cycloaddition which is a concerted pericyclic reaction. The driving force of this reaction is the formation of a new σ bond between the π orbitals of the dieneophile and the diene which is stronger than the π bond. The reaction is not allowed without the HOMO-LUMO interation and a symmetry overlap.

===Cis Butadiene===

<center>[[Image:prototype Diels Alder.jpg|500px|center|thumb|'''Figure 13'''. ''The prototype reaction between butadiene and ethylene'']]</center>

To study a prototype Diels Alder cycloaddition reaction between butadiene and ethylene, first a cis butadiene molecule was built using Gaussview and optimized using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the butadiene were plotted. As seen the HOMO of cis-butadiene is anti-symmetric ('''a''') and the LUMO is symmetric ('''s''') with respect to plane.

{| align="center"
|- align="center"
|
[[Image:XJB_HOMO.jpg|280px|thumb|'''Figure 14.''' ''Anti-symmetric HOMO of cis-butadiene'']]
|
[[Image:XJB_LUMO.jpg|300px|thumb|'''Figure 15.''' ''Symmetric HOMO of cis-butadiene'']]
|- align="center"
|}

===The Transition State of prototype reaction between ethylene and butadiene===

The Hessian method was used to estimate transition state for the reaction between ethylene and butadiene because these two reactants are relatively small and simple molecules. The transition state was drawn by first building a structure as (a) in '''Figure 16''' then removing the -CH<sub>2</sub>-CH<sub>2</sub>- group to form the envelop structure as in (b). The '''clean''' option should not be used and the dashed bond length was guessed to be larger than 2.2 Å otherwise the geometry change would cause a failure in optimization.

<center>[[Image:XJB_dielsguessts.jpg|300px|center|thumb|'''Figure 16'''. ''The guessing transition state for the reaction between butadiene and ethylene'']]</center>

<center>[[File:XJB_butadiene and ethylene ts.GIF|400px|center|thumb|'''Figure 17.''' ''Animation of the butadiene and ethylene cycloaddition transition state'' [[File:XJB_prototype DA.log]]]]</center>

After the Hessian optimization, the transition structure has been successfully determined and an animation was generated in '''Figure 17'''. It has a C<sub>1</sub> symmetry point group and the bond lengths of two partly formed C-C bonds were measured to be 2.12Å. The typical sp<sup>3</sup> and sp<sup>2</sup> C-C bond lengths are 1.54 Å and 1.47 Å respectively<ref name="bondlength">''J. Phys. C: Solid State Phys.'', 1986, '''19''', pp 4613-4621.</ref>. The van der Waals radius of the C atom is 1.7 Å<ref name="bondlength2">''Z. Naturforsch.'', 2007, '''62b''', pp 235-243.</ref>. The measured bond distance of the transition structure is larger than typical bond lengths but still within the van der Waals radius, suggesting the bond forming interation between the carbon atoms.

From the animation ('''Figure 17''') the vibration directions of two molecules are opposite and they are approaching and leaving each other at the same time. This suggests the formation of the two bonds synchronous. The vibrational frequencies were also calculated and an infrared spectrum was simulated ('''Figure 20'''). An imaginary frequency of magnitude -955 cm<sup>-1</sup> was seen and the lowest positive frequency was observed at 147 cm<sup>-1</sup> which corresponded to the reactant vibration but not the bond formation.

{| align="center"
|
[[Image:XJB_HOMOts.jpg|300px|thumb|'''Figure 18'''. HOMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_LUMOts.jpg|300px|thumb|'''Figure 19'''. LUMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_IRts.jpg|300px|thumb|'''Figure 20'''. Generated IR of transition state of reaction between cis-butadiene and ethylene]]
|}

The HOMO and LUMO molecular orbitals of this transition structure were visualized in '''Figure 18''' and '''Figure 19'''. From these two figures it can be observed that the HOMO at the transition structure is anti-symmetric ('''a''') while the LUMO is symmetric ('''s'''). Thus this anti-symmetric transition state HOMO is formed by the HOMO of cis-butadiene and the LUMO of ethylene with the knowledge that they are both anti-symmetrical. This conclusion agrees with the previous talked conditions for allowed reaction.

===The cyclohexa-1,3-diene reaction with maleic anhydride===

<center>[[Image:XJB_mechanism3.jpg|500px|center|thumb|'''Figure 21'''. ''The endo and exo products of the Diels Alder reaction between cyclohexa-1,3-diene and maleic anhydride'']]</center>

The exo and endo transition states were drawn according to '''Figure 21''' with the new C-C bond formation line dashed and bond length estimation as 2.5 Å. Then the same Hessian method was applied to optimize the structures. The electronic energy of the endo structure was calculated to be '''-0.05150Ha''' and '''-0.05042Ha''' for the exo. This reaction is supposed to be kinetically controlled thus the lower energy endo transition state is more favoured.

The bond lengths of the partly formed C-C bond are 2.16 Å and 2.17 Å for endo and exo transition states respectively. Another measured C-C bond length is shown in '''Table 13''' as 2.89 Å for endo (C3-C8 and C2-C7) and 2.95 Å for endo (C7-C10 and C8-C9). The longer atom distance indicates the larger steric repulsion in the exo transition state because of the extra existence of hydrogen atoms on C7 and C8.

Woodward and Hoffmann also proposed a secondary orbital overlap statement as an explanation of the endo stereo-preference in Diels Alder reactions<ref name="soot">''J. Org. Chem.'', 1987, '''52(8)''', pp 1469–1474. {{DOI|10.1021/jo00384a016}}</ref>. This secondary orbital overlap does not involve bond changes but π orbital interactions as the highlighted carbon atoms of the endo transition state in '''Table 13'''. Such overlap stablises the structure and lowers the energy of the endo transition state while there is no such effect existed in the exo transition state.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | endo
! scope="col" | exo
|-
| '''Animation'''||[[File:XJB_endots.GIF|300px]]||[[File:XJB_exots.GIF|300px]]
|-
| '''Sum of electronic and zero-point Energies'''||0.133493||0.134880
|-
| '''Sum of electronic and thermal Energies'''||0.143683||0.144881
|-
| '''Sum of electronic and thermal Enthalpies'''||0.144627||0.145825
|-
| '''Sum of electronic and thermal Free Energies'''||0.097349||0.099117
|-
| '''Bond lengths'''||[[Image:XJB_endomeasure.jpg|300px]]||[[Image:XJB_exomeasure.jpg|300px]]
|-
| '''HOMO'''||[[Image:XJB_endoHOMO.jpg|300px]]||[[Image:XJB_exoHOMO.jpg|285px]]
|-
| '''Secondary orbital overlap effect'''||[[Image:XJB_endosoo.jpg|300px]]||[[Image:XJB_exosoo.jpg|300px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 13.''' ''Summary of comparison of properties of endo and exo transition structures for reaction between cyclohexa-1,3-diene and maleic anhydride''</div>

===Further work===

The semiempirical/AM1 calculation is based on the "Neglect of Differential Diatomic Overlap (NDDO)" integral approximation, which neglects all two-electron integrals involving two-center charge distributions are neglected<ref>http://www.cup.uni-muenchen.de/ch/compchem/energy/semi1.html</ref>. The better optimization towards the endo and exo transition states should be done first by AM1 optimization followed by '''B3LYP/6-31G*''' reoptimization<ref>''J. Org. Chem.'', 2003, '''68 (19)''', pp 7158–7166. {{DOI|10.1021/jo0348827}}</ref>. The results are summarized in '''Table 14'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | endo
! scope="col" | exo
|-
| '''Structure'''||[[Image:XJB_endots2.jpg|300px]]||[[Image:XJB_exots2.jpg|300px]]
|-
| '''Electronic energy'''|| ||
|-
| '''Sum of electronic and zero-point Energies'''||0.133493||0.134880
|-
| '''Sum of electronic and thermal Energies'''||0.143683||0.144881
|-
| '''Sum of electronic and thermal Enthalpies'''||0.144627||0.145825
|-
| '''Sum of electronic and thermal Free Energies'''||0.097349||0.099117
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 14.''' ''Reoptimization using '''B3LYP/6-31G*''' of endo and exo transition structures for reaction between cyclohexa-1,3-diene and maleic anhydride''</div>

==Reference==

<references/>

Rep:Mod:XJB1130

2013-12-06T15:33:44Z

Jx1011: /* Further work */

==The Cope Rearrangement of 1,5-hexadiene==

The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement reaction ('''Figure 1'''). Its mechanism has been studied by a large number of experimental and computational researches, which is recently accepted that the reaction occurs via either a "chair" or a "boat" transition state structure in a concerted manner. The aim of this exercise is to locate the low-energy minimum and the transition structures on the potential energy surface by Gaussian calculation in order to study its mechanism. The reaction pathway and the activation energy can also be calculated by this calculation.

<center>[[Image:XJB_cope rearrangement.jpg|500px|center|thumb|'''Figure 1'''. ''The Cope rearrangement'']]</center>

===Optimizing the Reactants and Products===

Different conformations of 1,5-hexadiene can be optimized using '''Gaussview'''. The optimized structure of four conformers involved in this discussion and their corresponding energies optimized at '''HF/3-21G''' level of theory are concluded in the following table ('''Table 1''').

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | anti4
! scope="col" | anti2
! scope="col" | gauche
! scope="col" | gauche3
|-
| <jmol><jmolApplet><title>XJB_anti4.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti4.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche3.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| -231.69097Ha || -231.69254Ha || -231.68772Ha ||-231.69266Ha
|}

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 1.''' ''Four conformers of 1,5-hexadiene''</div>

This exercise first involved optimizing a molecule of 1,5-hexadiene with "anti" linkage for the cetral four C atoms at the '''HF/3-21G''' level of theory. The final optimization structure had the total electronic energy of '''-231.69097Ha''' with a '''C<sub>1</sub>''' symmetry. By comparing the energy and point group to that in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1], this structure is confirmed to be ''anti4''. The compositions of energies are also listed in '''Table 2''' as i) the sum of electronic and zero-point energies (potential energy at 0K including the zero-point vibrational energy), ii) the sum of electronic and thermal energies (energy including translational, rotational and vibrational at 298.15K and 1 atm), iii) the sum of electronic and thermal enthalpies (containning an additional correction for RT), and iv) the sum of electronic and thermal free energies (including the entropic contribution to the free energy). For what we are considering, the first two terms are the most useful for further comparisons.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Formula Description'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||E = E<sub>elec</sub> + ZPE||-231.53785
|-
| Sum of electronic and thermal Energies||E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>||-231.53096
|-
| Sum of electronic and thermal Enthalpie||H = E + RT||-231.53002
|-
| Sum of electronic and thermal Free Energie||G = H - TS||-231.56891
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 2.''' ''Frequency calculations of anti4 optimized at '''HF/3-21G''' ([[File:XJB_anti4.log]])''</div>

Next another 1,5-hexadiene molecule with "gauche" linkage for the central four atoms was drawn and optimized at the '''HF/3-21G''' level of theory as before. Energy for this conformation is expected to be higher than that for the ''anti4'' conformation due to the closer position of two alkene ends thus the steric interaction. The final structure had the electronic energy of '''-231.68772Ha''' which was indeed higher than that of the ''anti4'', indicating less favour towards this gauche structure. It had a '''C<sub>2</sub>''' symmetry and was confirmed to be the ''gauche'' conformation.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53439
|-
| Sum of electronic and thermal Energies||-231.52770
|-
| Sum of electronic and thermal Enthalpie||-231.52676
|-
| Sum of electronic and thermal Free Energie||-231.56483
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 3.''' ''Frequency calculations of gauche optimized at '''HF/3-21G''' ([[File:XJB_gauche.log]])''</div>

A lowest energy conformation ''gauche3'' of the reactant molecule was drawn based on and confirmed by [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. The optimization gave a final structure with an energy of '''-231.69266Ha''' with a symmetry of '''C<sub>1</sub>'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53949
|-
| Sum of electronic and thermal Energies||-231.53265
|-
| Sum of electronic and thermal Enthalpie||-231.5317
|-
| Sum of electronic and thermal Free Energie||-231.57064
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 4.''' ''Frequency calculations of gauche3 optimized at '''HF/3-21G''' ([[File:XJB_gauche3.log]])''</div>

The next step required us to draw the C<sub>i</sub> symmetry ''anti2'' conformation of 1,5-hexadiene and optimize it at the '''HF/3-21G''' level of theory. The optimized energy was calculated to be '''-231.69254Ha''' with point group of '''C<sub>i</sub>''', which was consistent with the one given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53954
|-
| Sum of electronic and thermal Energies||-231.53257
|-
| Sum of electronic and thermal Enthalpie||-231.53162
|-
| Sum of electronic and thermal Free Energie||-231.57092
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 5.''' ''Frequency calculations of anti2 optimized at '''HF/3-21G''' ([[File:XJB_anti2.log]])''</div>

The ''anti2'' structure was then reoptimized at '''B3LYP/6-31G*''' which was a improvement over the '''HF/3-21G''' basis set. It adds polarization to atoms and improves the modeling of core electrons thus producing more accurate description of orbitals<ref name="soo">''Nigerian Journal of Chemical Research'', 2007, '''12'''. {{DOI|10.4314/njcr.v12i1.}}</ref>. The reoptimized ''anti2'' structure had the total electronic energy of '''-234.61171Ha''' and the same symmetry C<sub>i</sub>. The comparison of two structures optimized using different levels of theory is summarized in '''Table 6'''. We can see that they have exactly the same dihedral angle of the central four C atoms at 180.00<sup>o</sup> and angles between three of them are very similar as 111.3<sup>o</sup> and 112.7<sup>o</sup> respectively. Thus the the geometry change using '''HF/3-21G''' or '''B3LYP/6-31G*''' is trivial.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Before re-optimization
! scope="col" | After re-optimization
|-
| <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2reoptimized</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2reoptimized.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| ''anti2 optimized at '''HF/3-21G''''' || ''anti2 optimized at '''B3LYP/6-31G*'''''
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 6.''' ''Two optimization of anti2 at '''HF/3-21G''' and '''B3LYP/6-31G*'''([[File:XJB_anti2.log]])''</div>

The frequency calculation was run as well and a list of energies is summarized in '''Table 7'''. A list of all the vibrational frequencies was also generated and no imaginary (negative) frequency was observed and these vibrations are visualized by simulating the infrared spectrum in '''Figure 2'''.

{| align="center"
|- align="center"
|
{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-234.46920
|-
| Sum of electronic and thermal Energies||-234.46186
|-
| Sum of electronic and thermal Enthalpie||-234.46091
|-
| Sum of electronic and thermal Free Energie||-234.50078
|}
|
[[Image:XJB_IR_anti2.jpg|300px]]
|- align="center"
| align="center" | '''Table 7.''' ''Frequency calculations of anti2 optimized at '''B3LYP/6-31G*'''
([[File:XJB_anti2_reoptimized.log]])''
| align="center" | '''Figure 2.''' ''The generated IR spectrum of optimized '''B3LYP/6-31G*''' structure''
|}

===Optimizing the "Chair" and "Boat" Transition Structures===

The study of the chair and boat transition states was carried out using three different approaches. The first one was called the Hessian which involved computing the force constant matrix; the second one was to freeze the reaction coordinate and then optimized the structure once the molecule was fully relaxed; the last one was to use QST2 method to specify the reactants and products then finding the transition structure.

<center>[[Image:XJB_2 ts.jpg|500px|center|thumb|'''Figure 3'''. ''The two transition states of Cope rearrangement'']]</center>

====The Chair Transition State====

In order to set up a transition state optimization, first an allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimized at the '''HF/3-21G''' level of theory. Then this half of the transition structure was duplicated with rotation to make the approximate resembling of the chair transition state with terminal ends of the fragments 2.2 Å apart. The chair transition structure optimization was then done by the first two different approaches.

<center>[[Image:XJB_guess ts.jpg|400px|center|thumb|'''Figure 4'''. ''The guessed chair transition structure of two allyl fragments'']]</center>

For the first optimization approach, the force constants were computed (the Hessian) and the chair transition state was optimized to a TS (Berny) at the '''HF/3-21G''' level of theory. The frequency calculation was carried out at the same time and gave an imaginary frequency at -818 cm<sup>-1</sup>. The electronic energy was calculated to be '''-231.61932Ha''' with C<sub>2h</sub> symmetry point group. The distance between the terminal carbons was 2.0203 Å. An animation of this transition state vibrations was generated to confirm the corresponding Cope rearrangement.

<center>[[File:XJB_chair ts animation.GIF|400px|center|thumb|'''Figure 5.''' ''Animation of the Cope rearrangement via the chair transition state'']]</center>

The chair transition state was then reoptimized using the '''B3LYP/6-31G*''' level of theory and carried out frequency calculations as well. The electronic energy calculated this time was '''-234.55698Ha''' with an imaginary frequency at -566 cm<sup>-1</sup>. We can see that with the geometries quite similar, the energies differ markedly calculated using two levels of theory. The summary and comparison of the calculations optimized at two levels of theory are summarized in '''Table 8'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46670||-234.41493
|-
| Sum of electronic and thermal Energies||-231.46134||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40807
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir1.jpg|200px]]||[[Image:XJB_chairir2.jpg|200px]]
|-
| ||[[File:XJB_chairts1.1.log]]||[[File:XJB_chairts1.2.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 8.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the first approach''</div>

The second approach is the frozen coordinate method using the Redundant Coord Editor to freeze the bond lengths of the terminal ends to 2.2 Å. First the calculation failed because of the recent update of GaussView. After editting the coordinate distance (B 5 1 2.2 F) and the frozen distance (B 5 1 F) the optimization went well this time. The structure was then optimized again using a normal guess Hessian modified including the two differentiating coordinates. This chair transition state was then reoptimized again using the '''B3LYP/6-31G*''' level of theory. The electronic energies were calculated to be the same as in the first approach and frequency calculations were summarized in '''Table 9'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46671||-234.41490
|-
| Sum of electronic and thermal Energies||-231.46135||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40806
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir3.jpg|200px]]||[[Image:XJB_chairir4.jpg|200px]]
|-
| ||[[File:XJB_chairts2.1.log]]||[[File:XJB_chairts2.2.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 9.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the second approach''</div>

As the same energy results given, the bond forming/breaking bond lengths in the chair transition structures in the above two different approaches were compared. The first approach gave the bond lengths of 2.0203 Å while the second approach gave 2.0207 Å. Considering the accuracy limit of bond length estimation in Gaussview, these two bond lengths can be seen as exactly the same. Two methods worked reasonably well because of the good assumption of transition structures. However for more complicated and larger molecule, the Hessian method may fail due to the difficulty to predict the transition geometry (different surface curvature). Thus better way to generate such transition structures is to freeze reaction coordinate.

{| align="center"
|- align="center"
|
[[Image:XJB_chairts bondlength1.jpg|350px|thumb|'''Figure 6.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by Hession'']]
|
[[Image:XJB_chairts bondlength2.jpg|350px|thumb|'''Figure 7.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by frozen coordinate'']]
|- align="center"
|}

====The Boat Transition State====

The boat transition structure was then optimized using the '''QST2''' method. The same numbering for the reactant and product molecules was important in order to specify for this reaction. The renumbering of the molecules is shown as in '''Figure 8''', but the following optimization failed and gave a twisted structure. The reactant and product geometries were then modified further by changing the central C-C-C-C dihedral angle to 0<sup>o</sup> and the inside two C-C-C (i.e. C2-C3-C4 and C3-C4-C5) angles to 100<sup>o</sup>. The job was run successfully and frequency calculations were done with the boat transition state generated. Only one imaginary frequency was detected at -840 cm<sup>-1</sup>. The electronic energy was '''-231.60280Ha''' with C<sub>2v</sub> symmetry. The distances between the terminal carbons of the allyl fragments were examined to be both 2.140 Å which were comparably longer than that of the chair transition structure.

{| align="center"
|- align="center"
|
[[Image:XJB_QST1.jpg|400px|thumb|'''Figure 8.''' ''Numbering of reactant and product'']]
|
[[Image:XJB_QST2.jpg|410px|thumb|'''Figure 9.''' ''Re-numbering of reactant and product'']]
|- align="center"
|}

<center>[[File:XJB_boat ts animation.GIF|400px|center|thumb|'''Figure 10.''' ''Animation of the Cope rearrangement via the boat transition state'']]</center>

The structure was then reoptimized at the higher '''B3LYP/6-31G*''' theory of level and the electronic energy was calculated to be '''-234.54309Ha''' with very similar geometry. The frequency calculation generated an imaginary frequency at -530 cm<sup>-1</sup>. The summary and comparison of two levels' calculations are carried out in the following table.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.45093||-234.40234
|-
| Sum of electronic and thermal Energies||-231.44530||-234.39601
|-
| Sum of electronic and thermal Enthalpie|| -231.44436||-234.39506
|-
| Sum of electronic and thermal Free Energie||-231.47977||-234.43175
|-
| Generated IR spectrum||[[Image:XJB_boatir1.jpg|200px]]||[[Image:XJB_boatir2.jpg|200px]]
|-
| ||[[File:XJB_boat ts1.log]]||[[File:XJB_boat ts2.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 10.''' ''Frequency calculations of the boat transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by QST2''</div>

====Intrinsic Reaction Coordinate====

Next a useful method called '''Intrinct Reaction Coordinate (IRC)''' was applied to the Hessian optimized chair transition structure to follow the mininmum energy path (MEP) from a transition structure down to its local minimum on a potential energy surface. IRC can be used to verify that the transition state we found actually connects the reactants and products, or to identify any reaction intermediates, or to generate an animation of the reaction<ref name="IRC">''The Journal of Chemical Physics.'', 2005, '''122''', pp 1-17. {{DOI|10.1063/1.1927521}}</ref>. The symmetrical reaction coordinate was computed only in the forward direction with force constant always calculated and the number of points along IRC 50. The last structure generated along IRC is given in '''Figure 11''' with the energy calculated as '''-231.69158Ha''' and the dihedral angle of the central four C atoms and the internal angle between three C atoms measured as ''D'' = -67.188<sup>o</sup> ''A'' = 111.78<sup>o</sup>. This energy does not match to any of the conformations discussed in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] thus a minimum geometry has not reached yet.

{| align="center"
|- align="center"
|
<jmol><jmolApplet><title>XJB_IRC.mol</title><color>white</color>
<size>200</size><script>zoom=5</script>
<uploadedFileContents>XJB_IRC.mol</uploadedFileContents>
</jmolApplet></jmol>
|
[[Image:XJB_irc.jpg|500px]]
|- align="center"
| align="center" | '''Figure 11.''' ''The energy minimum structure along IRC ([[File:XJB_IRC.log]])''
| align="center" | '''Figure 12.''' ''Total energy along IRC''
|}

Three improving options can be used to reach the minimum energy. The first one is to run a normal Hessian optimization at the last point on IRC; the second one involves using a larger number of points (in this case 100 was used); the last one is to redo the IRC computation specifying to compute force constant at every step. Three methods were all used and the results were summarized in '''Table 11'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Improved IRC method 1
! scope="col" | Improved IRC method 2
! scope="col" | Improved IRC method 3
|-
| [[Image:XJB_IRC1.jpg|300px|center|IRC1]]||[[Image:XJB_IRC2.jpg|300px|center|IRC3]] || [[Image:XJB_IRC3.jpg|300px|center|IRC3]]
|-
| ''D'' = -64.17<sup>o</sup> ''A'' = 112.04<sup>o</sup> || ''D'' = -66.83<sup>o</sup> ''A'' = 111.82<sup>o</sup> || ''D'' = -64.17<sup>o</sup> ''A'' = 112.04<sup>o</sup>
|-
| -231.69167Ha || -231.69150Ha || -231.69167Ha
|-
| [[File:XJB_IRC1.log]]||[[File:XJB_IRC2.log]]||[[File:XJB_IRC3.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 11.''' ''Comparison of three improvement methods of IRC computation upon the chair transition structure''</div>

The first method was very quick but maybe generated relatively low-accuracy result; the second approach involved the controlling of number of points which might result in ending up at the wrong direction and wrong structure; the third method was the most suitable option for this small system but it took longer time than the previous two. From '''Table 11''' it is concluded that method 1 and 3 gave the same lowest energy which was consistent to the energy of the ''gauche2'' conformation given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

====Activation Energies====

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | '''HF/3-21G''' at 0K
! scope="col" | '''HF/3-21G''' at 298.15K
! scope="col" | '''B3LYP/6-31G*''' at 0K
! scope="col" | '''B3LYP/6-31G*''' at 298.15K
! scope="col" | Expt. at 0K
|-
| '''ΔE (chair)'''||45.71||44.70||34.05||33.16||33.5 ± 0.5
|-
| '''ΔE (boat)'''||55.60||54.76||41.96||41.32||44.7 ± 0.5
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 12.''' ''Summary of activation energies for the Cope rearrangement via both transition states in kcal/mol''</div>

The activation energy calculation was done by calculating the difference in energies between the optimized reactant ''anti2'' and the optimized chair and boat transition states. It is observed that chair conformation gives generally lower activation energy than the boat structure thus more favoured transition state. The structure optimized at '''B3LYP/6-31G*''' level of theory at 0K has closer activation energy to the experimental result. Thus '''B3LYP/6-31G*''' gives a better estimation of this transition structure than '''HF/3-21G'''.

==The Diels Alder Cycloaddition==

In this exercise, AM1 semi-empirical method was used to compute the Diels Alder cycloaddition which is a concerted pericyclic reaction. The driving force of this reaction is the formation of a new σ bond between the π orbitals of the dieneophile and the diene which is stronger than the π bond. The reaction is not allowed without the HOMO-LUMO interation and a symmetry overlap.

===Cis Butadiene===

<center>[[Image:prototype Diels Alder.jpg|500px|center|thumb|'''Figure 13'''. ''The prototype reaction between butadiene and ethylene'']]</center>

To study a prototype Diels Alder cycloaddition reaction between butadiene and ethylene, first a cis butadiene molecule was built using Gaussview and optimized using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the butadiene were plotted. As seen the HOMO of cis-butadiene is anti-symmetric ('''a''') and the LUMO is symmetric ('''s''') with respect to plane.

{| align="center"
|- align="center"
|
[[Image:XJB_HOMO.jpg|280px|thumb|'''Figure 14.''' ''Anti-symmetric HOMO of cis-butadiene'']]
|
[[Image:XJB_LUMO.jpg|300px|thumb|'''Figure 15.''' ''Symmetric HOMO of cis-butadiene'']]
|- align="center"
|}

===The Transition State of prototype reaction between ethylene and butadiene===

The Hessian method was used to estimate transition state for the reaction between ethylene and butadiene because these two reactants are relatively small and simple molecules. The transition state was drawn by first building a structure as (a) in '''Figure 16''' then removing the -CH<sub>2</sub>-CH<sub>2</sub>- group to form the envelop structure as in (b). The '''clean''' option should not be used and the dashed bond length was guessed to be larger than 2.2 Å otherwise the geometry change would cause a failure in optimization.

<center>[[Image:XJB_dielsguessts.jpg|300px|center|thumb|'''Figure 16'''. ''The guessing transition state for the reaction between butadiene and ethylene'']]</center>

<center>[[File:XJB_butadiene and ethylene ts.GIF|400px|center|thumb|'''Figure 17.''' ''Animation of the butadiene and ethylene cycloaddition transition state'' [[File:XJB_prototype DA.log]]]]</center>

After the Hessian optimization, the transition structure has been successfully determined and an animation was generated in '''Figure 17'''. It has a C<sub>1</sub> symmetry point group and the bond lengths of two partly formed C-C bonds were measured to be 2.12Å. The typical sp<sup>3</sup> and sp<sup>2</sup> C-C bond lengths are 1.54 Å and 1.47 Å respectively<ref name="bondlength">''J. Phys. C: Solid State Phys.'', 1986, '''19''', pp 4613-4621.</ref>. The van der Waals radius of the C atom is 1.7 Å<ref name="bondlength2">''Z. Naturforsch.'', 2007, '''62b''', pp 235-243.</ref>. The measured bond distance of the transition structure is larger than typical bond lengths but still within the van der Waals radius, suggesting the bond forming interation between the carbon atoms.

From the animation ('''Figure 17''') the vibration directions of two molecules are opposite and they are approaching and leaving each other at the same time. This suggests the formation of the two bonds synchronous. The vibrational frequencies were also calculated and an infrared spectrum was simulated ('''Figure 20'''). An imaginary frequency of magnitude -955 cm<sup>-1</sup> was seen and the lowest positive frequency was observed at 147 cm<sup>-1</sup> which corresponded to the reactant vibration but not the bond formation.

{| align="center"
|
[[Image:XJB_HOMOts.jpg|300px|thumb|'''Figure 18'''. HOMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_LUMOts.jpg|300px|thumb|'''Figure 19'''. LUMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_IRts.jpg|300px|thumb|'''Figure 20'''. Generated IR of transition state of reaction between cis-butadiene and ethylene]]
|}

The HOMO and LUMO molecular orbitals of this transition structure were visualized in '''Figure 18''' and '''Figure 19'''. From these two figures it can be observed that the HOMO at the transition structure is anti-symmetric ('''a''') while the LUMO is symmetric ('''s'''). Thus this anti-symmetric transition state HOMO is formed by the HOMO of cis-butadiene and the LUMO of ethylene with the knowledge that they are both anti-symmetrical. This conclusion agrees with the previous talked conditions for allowed reaction.

===The cyclohexa-1,3-diene reaction with maleic anhydride===

<center>[[Image:XJB_mechanism3.jpg|500px|center|thumb|'''Figure 21'''. ''The endo and exo products of the Diels Alder reaction between cyclohexa-1,3-diene and maleic anhydride'']]</center>

The exo and endo transition states were drawn according to '''Figure 21''' with the new C-C bond formation line dashed and bond length estimation as 2.5 Å. Then the same Hessian method was applied to optimize the structures. The electronic energy of the endo structure was calculated to be '''-0.05150Ha''' and '''-0.05042Ha''' for the exo. This reaction is supposed to be kinetically controlled thus the lower energy endo transition state is more favoured.

The bond lengths of the partly formed C-C bond are 2.16 Å and 2.17 Å for endo and exo transition states respectively. Another measured C-C bond length is shown in '''Table 13''' as 2.89 Å for endo (C3-C8 and C2-C7) and 2.95 Å for endo (C7-C10 and C8-C9). The longer atom distance indicates the larger steric repulsion in the exo transition state because of the extra existence of hydrogen atoms on C7 and C8.

Woodward and Hoffmann also proposed a secondary orbital overlap statement as an explanation of the endo stereo-preference in Diels Alder reactions<ref name="soot">''J. Org. Chem.'', 1987, '''52(8)''', pp 1469–1474. {{DOI|10.1021/jo00384a016}}</ref>. This secondary orbital overlap does not involve bond changes but π orbital interactions as the highlighted carbon atoms of the endo transition state in '''Table 13'''. Such overlap stablises the structure and lowers the energy of the endo transition state while there is no such effect existed in the exo transition state.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | endo
! scope="col" | exo
|-
| '''Animation'''||[[File:XJB_endots.GIF|300px]]||[[File:XJB_exots.GIF|300px]]
|-
| '''Sum of electronic and zero-point Energies'''||0.133493||0.134880
|-
| '''Sum of electronic and thermal Energies'''||0.143683||0.144881
|-
| '''Sum of electronic and thermal Enthalpies'''||0.144627||0.145825
|-
| '''Sum of electronic and thermal Free Energies'''||0.097349||0.099117
|-
| '''Bond lengths'''||[[Image:XJB_endomeasure.jpg|300px]]||[[Image:XJB_exomeasure.jpg|300px]]
|-
| '''HOMO'''||[[Image:XJB_endoHOMO.jpg|300px]]||[[Image:XJB_exoHOMO.jpg|285px]]
|-
| '''Secondary orbital overlap effect'''||[[Image:XJB_endosoo.jpg|300px]]||[[Image:XJB_exosoo.jpg|300px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 13.''' ''Summary of comparison of properties of endo and exo transition structures for reaction between cyclohexa-1,3-diene and maleic anhydride''</div>

===Further work===

The semiempirical/AM1 calculation is based on the "Neglect of Differential Diatomic Overlap (NDDO)" integral approximation, which neglects all two-electron integrals involving two-center charge distributions are neglected<ref>http://www.cup.uni-muenchen.de/ch/compchem/energy/semi1.html</ref>. The better optimization towards the endo and exo transition states should be done first by AM1 optimization followed by B3LYP/6-31G* reoptimization<ref>''J. Org. Chem.'', 2003, '''68 (19)''', pp 7158–7166. {{DOI|10.1021/jo0348827}}</ref>.

==Reference==

<references/>

Rep:Mod:XJB1130

2013-12-06T14:10:25Z

Jx1011: /* The cyclohexa-1,3-diene reaction with maleic anhydride */

==The Cope Rearrangement of 1,5-hexadiene==

The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement reaction ('''Figure 1'''). Its mechanism has been studied by a large number of experimental and computational researches, which is recently accepted that the reaction occurs via either a "chair" or a "boat" transition state structure in a concerted manner. The aim of this exercise is to locate the low-energy minimum and the transition structures on the potential energy surface by Gaussian calculation in order to study its mechanism. The reaction pathway and the activation energy can also be calculated by this calculation.

<center>[[Image:XJB_cope rearrangement.jpg|500px|center|thumb|'''Figure 1'''. ''The Cope rearrangement'']]</center>

===Optimizing the Reactants and Products===

Different conformations of 1,5-hexadiene can be optimized using '''Gaussview'''. The optimized structure of four conformers involved in this discussion and their corresponding energies optimized at '''HF/3-21G''' level of theory are concluded in the following table ('''Table 1''').

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | anti4
! scope="col" | anti2
! scope="col" | gauche
! scope="col" | gauche3
|-
| <jmol><jmolApplet><title>XJB_anti4.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti4.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche3.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| -231.69097Ha || -231.69254Ha || -231.68772Ha ||-231.69266Ha
|}

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 1.''' ''Four conformers of 1,5-hexadiene''</div>

This exercise first involved optimizing a molecule of 1,5-hexadiene with "anti" linkage for the cetral four C atoms at the '''HF/3-21G''' level of theory. The final optimization structure had the total electronic energy of '''-231.69097Ha''' with a '''C<sub>1</sub>''' symmetry. By comparing the energy and point group to that in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1], this structure is confirmed to be ''anti4''. The compositions of energies are also listed in '''Table 2''' as i) the sum of electronic and zero-point energies (potential energy at 0K including the zero-point vibrational energy), ii) the sum of electronic and thermal energies (energy including translational, rotational and vibrational at 298.15K and 1 atm), iii) the sum of electronic and thermal enthalpies (containning an additional correction for RT), and iv) the sum of electronic and thermal free energies (including the entropic contribution to the free energy). For what we are considering, the first two terms are the most useful for further comparisons.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Formula Description'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||E = E<sub>elec</sub> + ZPE||-231.53785
|-
| Sum of electronic and thermal Energies||E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>||-231.53096
|-
| Sum of electronic and thermal Enthalpie||H = E + RT||-231.53002
|-
| Sum of electronic and thermal Free Energie||G = H - TS||-231.56891
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 2.''' ''Frequency calculations of anti4 optimized at '''HF/3-21G''' ([[File:XJB_anti4.log]])''</div>

Next another 1,5-hexadiene molecule with "gauche" linkage for the central four atoms was drawn and optimized at the '''HF/3-21G''' level of theory as before. Energy for this conformation is expected to be higher than that for the ''anti4'' conformation due to the closer position of two alkene ends thus the steric interaction. The final structure had the electronic energy of '''-231.68772Ha''' which was indeed higher than that of the ''anti4'', indicating less favour towards this gauche structure. It had a '''C<sub>2</sub>''' symmetry and was confirmed to be the ''gauche'' conformation.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53439
|-
| Sum of electronic and thermal Energies||-231.52770
|-
| Sum of electronic and thermal Enthalpie||-231.52676
|-
| Sum of electronic and thermal Free Energie||-231.56483
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 3.''' ''Frequency calculations of gauche optimized at '''HF/3-21G''' ([[File:XJB_gauche.log]])''</div>

A lowest energy conformation ''gauche3'' of the reactant molecule was drawn based on and confirmed by [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. The optimization gave a final structure with an energy of '''-231.69266Ha''' with a symmetry of '''C<sub>1</sub>'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53949
|-
| Sum of electronic and thermal Energies||-231.53265
|-
| Sum of electronic and thermal Enthalpie||-231.5317
|-
| Sum of electronic and thermal Free Energie||-231.57064
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 4.''' ''Frequency calculations of gauche3 optimized at '''HF/3-21G''' ([[File:XJB_gauche3.log]])''</div>

The next step required us to draw the C<sub>i</sub> symmetry ''anti2'' conformation of 1,5-hexadiene and optimize it at the '''HF/3-21G''' level of theory. The optimized energy was calculated to be '''-231.69254Ha''' with point group of '''C<sub>i</sub>''', which was consistent with the one given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53954
|-
| Sum of electronic and thermal Energies||-231.53257
|-
| Sum of electronic and thermal Enthalpie||-231.53162
|-
| Sum of electronic and thermal Free Energie||-231.57092
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 5.''' ''Frequency calculations of anti2 optimized at '''HF/3-21G''' ([[File:XJB_anti2.log]])''</div>

The ''anti2'' structure was then reoptimized at '''B3LYP/6-31G*''' which was a improvement over the '''HF/3-21G''' basis set. It adds polarization to atoms and improves the modeling of core electrons thus producing more accurate description of orbitals<ref name="soo">''Nigerian Journal of Chemical Research'', 2007, '''12'''. {{DOI|10.4314/njcr.v12i1.}}</ref>. The reoptimized ''anti2'' structure had the total electronic energy of '''-234.61171Ha''' and the same symmetry C<sub>i</sub>. The comparison of two structures optimized using different levels of theory is summarized in '''Table 6'''. We can see that they have exactly the same dihedral angle of the central four C atoms at 180.00<sup>o</sup> and angles between three of them are very similar as 111.3<sup>o</sup> and 112.7<sup>o</sup> respectively. Thus the the geometry change using '''HF/3-21G''' or '''B3LYP/6-31G*''' is trivial.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Before re-optimization
! scope="col" | After re-optimization
|-
| <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2reoptimized</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2reoptimized.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| ''anti2 optimized at '''HF/3-21G''''' || ''anti2 optimized at '''B3LYP/6-31G*'''''
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 6.''' ''Two optimization of anti2 at '''HF/3-21G''' and '''B3LYP/6-31G*'''([[File:XJB_anti2.log]])''</div>

The frequency calculation was run as well and a list of energies is summarized in '''Table 7'''. A list of all the vibrational frequencies was also generated and no imaginary (negative) frequency was observed and these vibrations are visualized by simulating the infrared spectrum in '''Figure 2'''.

{| align="center"
|- align="center"
|
{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-234.46920
|-
| Sum of electronic and thermal Energies||-234.46186
|-
| Sum of electronic and thermal Enthalpie||-234.46091
|-
| Sum of electronic and thermal Free Energie||-234.50078
|}
|
[[Image:XJB_IR_anti2.jpg|300px]]
|- align="center"
| align="center" | '''Table 7.''' ''Frequency calculations of anti2 optimized at '''B3LYP/6-31G*'''
([[File:XJB_anti2_reoptimized.log]])''
| align="center" | '''Figure 2.''' ''The generated IR spectrum of optimized '''B3LYP/6-31G*''' structure''
|}

===Optimizing the "Chair" and "Boat" Transition Structures===

The study of the chair and boat transition states was carried out using three different approaches. The first one was called the Hessian which involved computing the force constant matrix; the second one was to freeze the reaction coordinate and then optimized the structure once the molecule was fully relaxed; the last one was to use QST2 method to specify the reactants and products then finding the transition structure.

<center>[[Image:XJB_2 ts.jpg|500px|center|thumb|'''Figure 3'''. ''The two transition states of Cope rearrangement'']]</center>

====The Chair Transition State====

In order to set up a transition state optimization, first an allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimized at the '''HF/3-21G''' level of theory. Then this half of the transition structure was duplicated with rotation to make the approximate resembling of the chair transition state with terminal ends of the fragments 2.2 Å apart. The chair transition structure optimization was then done by the first two different approaches.

<center>[[Image:XJB_guess ts.jpg|400px|center|thumb|'''Figure 4'''. ''The guessed chair transition structure of two allyl fragments'']]</center>

For the first optimization approach, the force constants were computed (the Hessian) and the chair transition state was optimized to a TS (Berny) at the '''HF/3-21G''' level of theory. The frequency calculation was carried out at the same time and gave an imaginary frequency at -818 cm<sup>-1</sup>. The electronic energy was calculated to be '''-231.61932Ha''' with C<sub>2h</sub> symmetry point group. The distance between the terminal carbons was 2.0203 Å. An animation of this transition state vibrations was generated to confirm the corresponding Cope rearrangement.

<center>[[File:XJB_chair ts animation.GIF|400px|center|thumb|'''Figure 5.''' ''Animation of the Cope rearrangement via the chair transition state'']]</center>

The chair transition state was then reoptimized using the '''B3LYP/6-31G*''' level of theory and carried out frequency calculations as well. The electronic energy calculated this time was '''-234.55698Ha''' with an imaginary frequency at -566 cm<sup>-1</sup>. We can see that with the geometries quite similar, the energies differ markedly calculated using two levels of theory. The summary and comparison of the calculations optimized at two levels of theory are summarized in '''Table 8'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46670||-234.41493
|-
| Sum of electronic and thermal Energies||-231.46134||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40807
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir1.jpg|200px]]||[[Image:XJB_chairir2.jpg|200px]]
|-
| ||[[File:XJB_chairts1.1.log]]||[[File:XJB_chairts1.2.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 8.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the first approach''</div>

The second approach is the frozen coordinate method using the Redundant Coord Editor to freeze the bond lengths of the terminal ends to 2.2 Å. First the calculation failed because of the recent update of GaussView. After editting the coordinate distance (B 5 1 2.2 F) and the frozen distance (B 5 1 F) the optimization went well this time. The structure was then optimized again using a normal guess Hessian modified including the two differentiating coordinates. This chair transition state was then reoptimized again using the '''B3LYP/6-31G*''' level of theory. The electronic energies were calculated to be the same as in the first approach and frequency calculations were summarized in '''Table 9'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46671||-234.41490
|-
| Sum of electronic and thermal Energies||-231.46135||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40806
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir3.jpg|200px]]||[[Image:XJB_chairir4.jpg|200px]]
|-
| ||[[File:XJB_chairts2.1.log]]||[[File:XJB_chairts2.2.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 9.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the second approach''</div>

As the same energy results given, the bond forming/breaking bond lengths in the chair transition structures in the above two different approaches were compared. The first approach gave the bond lengths of 2.0203 Å while the second approach gave 2.0207 Å. Considering the accuracy limit of bond length estimation in Gaussview, these two bond lengths can be seen as exactly the same. Two methods worked reasonably well because of the good assumption of transition structures. However for more complicated and larger molecule, the Hessian method may fail due to the difficulty to predict the transition geometry (different surface curvature). Thus better way to generate such transition structures is to freeze reaction coordinate.

{| align="center"
|- align="center"
|
[[Image:XJB_chairts bondlength1.jpg|350px|thumb|'''Figure 6.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by Hession'']]
|
[[Image:XJB_chairts bondlength2.jpg|350px|thumb|'''Figure 7.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by frozen coordinate'']]
|- align="center"
|}

====The Boat Transition State====

The boat transition structure was then optimized using the '''QST2''' method. The same numbering for the reactant and product molecules was important in order to specify for this reaction. The renumbering of the molecules is shown as in '''Figure 8''', but the following optimization failed and gave a twisted structure. The reactant and product geometries were then modified further by changing the central C-C-C-C dihedral angle to 0<sup>o</sup> and the inside two C-C-C (i.e. C2-C3-C4 and C3-C4-C5) angles to 100<sup>o</sup>. The job was run successfully and frequency calculations were done with the boat transition state generated. Only one imaginary frequency was detected at -840 cm<sup>-1</sup>. The electronic energy was '''-231.60280Ha''' with C<sub>2v</sub> symmetry. The distances between the terminal carbons of the allyl fragments were examined to be both 2.140 Å which were comparably longer than that of the chair transition structure.

{| align="center"
|- align="center"
|
[[Image:XJB_QST1.jpg|400px|thumb|'''Figure 8.''' ''Numbering of reactant and product'']]
|
[[Image:XJB_QST2.jpg|410px|thumb|'''Figure 9.''' ''Re-numbering of reactant and product'']]
|- align="center"
|}

<center>[[File:XJB_boat ts animation.GIF|400px|center|thumb|'''Figure 10.''' ''Animation of the Cope rearrangement via the boat transition state'']]</center>

The structure was then reoptimized at the higher '''B3LYP/6-31G*''' theory of level and the electronic energy was calculated to be '''-234.54309Ha''' with very similar geometry. The frequency calculation generated an imaginary frequency at -530 cm<sup>-1</sup>. The summary and comparison of two levels' calculations are carried out in the following table.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.45093||-234.40234
|-
| Sum of electronic and thermal Energies||-231.44530||-234.39601
|-
| Sum of electronic and thermal Enthalpie|| -231.44436||-234.39506
|-
| Sum of electronic and thermal Free Energie||-231.47977||-234.43175
|-
| Generated IR spectrum||[[Image:XJB_boatir1.jpg|200px]]||[[Image:XJB_boatir2.jpg|200px]]
|-
| ||[[File:XJB_boat ts1.log]]||[[File:XJB_boat ts2.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 10.''' ''Frequency calculations of the boat transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by QST2''</div>

====Intrinsic Reaction Coordinate====

Next a useful method called '''Intrinct Reaction Coordinate (IRC)''' was applied to the Hessian optimized chair transition structure to follow the mininmum energy path (MEP) from a transition structure down to its local minimum on a potential energy surface. IRC can be used to verify that the transition state we found actually connects the reactants and products, or to identify any reaction intermediates, or to generate an animation of the reaction<ref name="IRC">''The Journal of Chemical Physics.'', 2005, '''122''', pp 1-17. {{DOI|10.1063/1.1927521}}</ref>. The symmetrical reaction coordinate was computed only in the forward direction with force constant always calculated and the number of points along IRC 50. The last structure generated along IRC is given in '''Figure 11''' with the energy calculated as '''-231.69158Ha''' and the dihedral angle of the central four C atoms and the internal angle between three C atoms measured as ''D'' = -67.188<sup>o</sup> ''A'' = 111.78<sup>o</sup>. This energy does not match to any of the conformations discussed in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] thus a minimum geometry has not reached yet.

{| align="center"
|- align="center"
|
<jmol><jmolApplet><title>XJB_IRC.mol</title><color>white</color>
<size>200</size><script>zoom=5</script>
<uploadedFileContents>XJB_IRC.mol</uploadedFileContents>
</jmolApplet></jmol>
|
[[Image:XJB_irc.jpg|500px]]
|- align="center"
| align="center" | '''Figure 11.''' ''The energy minimum structure along IRC ([[File:XJB_IRC.log]])''
| align="center" | '''Figure 12.''' ''Total energy along IRC''
|}

Three improving options can be used to reach the minimum energy. The first one is to run a normal Hessian optimization at the last point on IRC; the second one involves using a larger number of points (in this case 100 was used); the last one is to redo the IRC computation specifying to compute force constant at every step. Three methods were all used and the results were summarized in '''Table 11'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Improved IRC method 1
! scope="col" | Improved IRC method 2
! scope="col" | Improved IRC method 3
|-
| [[Image:XJB_IRC1.jpg|300px|center|IRC1]]||[[Image:XJB_IRC2.jpg|300px|center|IRC3]] || [[Image:XJB_IRC3.jpg|300px|center|IRC3]]
|-
| ''D'' = -64.17<sup>o</sup> ''A'' = 112.04<sup>o</sup> || ''D'' = -66.83<sup>o</sup> ''A'' = 111.82<sup>o</sup> || ''D'' = -64.17<sup>o</sup> ''A'' = 112.04<sup>o</sup>
|-
| -231.69167Ha || -231.69150Ha || -231.69167Ha
|-
| [[File:XJB_IRC1.log]]||[[File:XJB_IRC2.log]]||[[File:XJB_IRC3.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 11.''' ''Comparison of three improvement methods of IRC computation upon the chair transition structure''</div>

The first method was very quick but maybe generated relatively low-accuracy result; the second approach involved the controlling of number of points which might result in ending up at the wrong direction and wrong structure; the third method was the most suitable option for this small system but it took longer time than the previous two. From '''Table 11''' it is concluded that method 1 and 3 gave the same lowest energy which was consistent to the energy of the ''gauche2'' conformation given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

====Activation Energies====

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | '''HF/3-21G''' at 0K
! scope="col" | '''HF/3-21G''' at 298.15K
! scope="col" | '''B3LYP/6-31G*''' at 0K
! scope="col" | '''B3LYP/6-31G*''' at 298.15K
! scope="col" | Expt. at 0K
|-
| '''ΔE (chair)'''||45.71||44.70||34.05||33.16||33.5 ± 0.5
|-
| '''ΔE (boat)'''||55.60||54.76||41.96||41.32||44.7 ± 0.5
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 12.''' ''Summary of activation energies for the Cope rearrangement via both transition states in kcal/mol''</div>

The activation energy calculation was done by calculating the difference in energies between the optimized reactant ''anti2'' and the optimized chair and boat transition states. It is observed that chair conformation gives generally lower activation energy than the boat structure thus more favoured transition state. The structure optimized at '''B3LYP/6-31G*''' level of theory at 0K has closer activation energy to the experimental result. Thus '''B3LYP/6-31G*''' gives a better estimation of this transition structure than '''HF/3-21G'''.

==The Diels Alder Cycloaddition==

In this exercise, AM1 semi-empirical method was used to compute the Diels Alder cycloaddition which is a concerted pericyclic reaction. The driving force of this reaction is the formation of a new σ bond between the π orbitals of the dieneophile and the diene which is stronger than the π bond. The reaction is not allowed without the HOMO-LUMO interation and a symmetry overlap.

===Cis Butadiene===

<center>[[Image:prototype Diels Alder.jpg|500px|center|thumb|'''Figure 13'''. ''The prototype reaction between butadiene and ethylene'']]</center>

To study a prototype Diels Alder cycloaddition reaction between butadiene and ethylene, first a cis butadiene molecule was built using Gaussview and optimized using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the butadiene were plotted. As seen the HOMO of cis-butadiene is anti-symmetric ('''a''') and the LUMO is symmetric ('''s''') with respect to plane.

{| align="center"
|- align="center"
|
[[Image:XJB_HOMO.jpg|280px|thumb|'''Figure 14.''' ''Anti-symmetric HOMO of cis-butadiene'']]
|
[[Image:XJB_LUMO.jpg|300px|thumb|'''Figure 15.''' ''Symmetric HOMO of cis-butadiene'']]
|- align="center"
|}

===The Transition State of prototype reaction between ethylene and butadiene===

The Hessian method was used to estimate transition state for the reaction between ethylene and butadiene because these two reactants are relatively small and simple molecules. The transition state was drawn by first building a structure as (a) in '''Figure 16''' then removing the -CH<sub>2</sub>-CH<sub>2</sub>- group to form the envelop structure as in (b). The '''clean''' option should not be used and the dashed bond length was guessed to be larger than 2.2 Å otherwise the geometry change would cause a failure in optimization.

<center>[[Image:XJB_dielsguessts.jpg|300px|center|thumb|'''Figure 16'''. ''The guessing transition state for the reaction between butadiene and ethylene'']]</center>

<center>[[File:XJB_butadiene and ethylene ts.GIF|400px|center|thumb|'''Figure 17.''' ''Animation of the butadiene and ethylene cycloaddition transition state'' [[File:XJB_prototype DA.log]]]]</center>

After the Hessian optimization, the transition structure has been successfully determined and an animation was generated in '''Figure 17'''. It has a C<sub>1</sub> symmetry point group and the bond lengths of two partly formed C-C bonds were measured to be 2.12Å. The typical sp<sup>3</sup> and sp<sup>2</sup> C-C bond lengths are 1.54 Å and 1.47 Å respectively<ref name="bondlength">''J. Phys. C: Solid State Phys.'', 1986, '''19''', pp 4613-4621.</ref>. The van der Waals radius of the C atom is 1.7 Å<ref name="bondlength2">''Z. Naturforsch.'', 2007, '''62b''', pp 235-243.</ref>. The measured bond distance of the transition structure is larger than typical bond lengths but still within the van der Waals radius, suggesting the bond forming interation between the carbon atoms.

From the animation ('''Figure 17''') the vibration directions of two molecules are opposite and they are approaching and leaving each other at the same time. This suggests the formation of the two bonds synchronous. The vibrational frequencies were also calculated and an infrared spectrum was simulated ('''Figure 20'''). An imaginary frequency of magnitude -955 cm<sup>-1</sup> was seen and the lowest positive frequency was observed at 147 cm<sup>-1</sup> which corresponded to the reactant vibration but not the bond formation.

{| align="center"
|
[[Image:XJB_HOMOts.jpg|300px|thumb|'''Figure 18'''. HOMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_LUMOts.jpg|300px|thumb|'''Figure 19'''. LUMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_IRts.jpg|300px|thumb|'''Figure 20'''. Generated IR of transition state of reaction between cis-butadiene and ethylene]]
|}

The HOMO and LUMO molecular orbitals of this transition structure were visualized in '''Figure 18''' and '''Figure 19'''. From these two figures it can be observed that the HOMO at the transition structure is anti-symmetric ('''a''') while the LUMO is symmetric ('''s'''). Thus this anti-symmetric transition state HOMO is formed by the HOMO of cis-butadiene and the LUMO of ethylene with the knowledge that they are both anti-symmetrical. This conclusion agrees with the previous talked conditions for allowed reaction.

===The cyclohexa-1,3-diene reaction with maleic anhydride===

<center>[[Image:XJB_mechanism3.jpg|500px|center|thumb|'''Figure 21'''. ''The endo and exo products of the Diels Alder reaction between cyclohexa-1,3-diene and maleic anhydride'']]</center>

The exo and endo transition states were drawn according to '''Figure 21''' with the new C-C bond formation line dashed and bond length estimation as 2.5 Å. Then the same Hessian method was applied to optimize the structures. The electronic energy of the endo structure was calculated to be '''-0.05150Ha''' and '''-0.05042Ha''' for the exo. This reaction is supposed to be kinetically controlled thus the lower energy endo transition state is more favoured.

The bond lengths of the partly formed C-C bond are 2.16 Å and 2.17 Å for endo and exo transition states respectively. Another measured C-C bond length is shown in '''Table 13''' as 2.89 Å for endo (C3-C8 and C2-C7) and 2.95 Å for endo (C7-C10 and C8-C9). The longer atom distance indicates the larger steric repulsion in the exo transition state because of the extra existence of hydrogen atoms on C7 and C8.

Woodward and Hoffmann also proposed a secondary orbital overlap statement as an explanation of the endo stereo-preference in Diels Alder reactions<ref name="soot">''J. Org. Chem.'', 1987, '''52(8)''', pp 1469–1474. {{DOI|10.1021/jo00384a016}}</ref>. This secondary orbital overlap does not involve bond changes but π orbital interactions as the highlighted carbon atoms of the endo transition state in '''Table 13'''. Such overlap stablises the structure and lowers the energy of the endo transition state while there is no such effect existed in the exo transition state.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | endo
! scope="col" | exo
|-
| '''Animation'''||[[File:XJB_endots.GIF|300px]]||[[File:XJB_exots.GIF|300px]]
|-
| '''Sum of electronic and zero-point Energies'''||0.133493||0.134880
|-
| '''Sum of electronic and thermal Energies'''||0.143683||0.144881
|-
| '''Sum of electronic and thermal Enthalpies'''||0.144627||0.145825
|-
| '''Sum of electronic and thermal Free Energies'''||0.097349||0.099117
|-
| '''Bond lengths'''||[[Image:XJB_endomeasure.jpg|300px]]||[[Image:XJB_exomeasure.jpg|300px]]
|-
| '''HOMO'''||[[Image:XJB_endoHOMO.jpg|300px]]||[[Image:XJB_exoHOMO.jpg|285px]]
|-
| '''Secondary orbital overlap effect'''||[[Image:XJB_endosoo.jpg|300px]]||[[Image:XJB_exosoo.jpg|300px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 13.''' ''Summary of comparison of properties of endo and exo transition structures for reaction between cyclohexa-1,3-diene and maleic anhydride''</div>

===Further work===

==Reference==

<references/>

File:XJB prototype DA.log

2013-12-06T14:06:09Z

Jx1011:

Rep:Mod:XJB1130

2013-12-06T14:01:27Z

Jx1011: /* The Diels Alder Cycloaddition */

==The Cope Rearrangement of 1,5-hexadiene==

The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement reaction ('''Figure 1'''). Its mechanism has been studied by a large number of experimental and computational researches, which is recently accepted that the reaction occurs via either a "chair" or a "boat" transition state structure in a concerted manner. The aim of this exercise is to locate the low-energy minimum and the transition structures on the potential energy surface by Gaussian calculation in order to study its mechanism. The reaction pathway and the activation energy can also be calculated by this calculation.

<center>[[Image:XJB_cope rearrangement.jpg|500px|center|thumb|'''Figure 1'''. ''The Cope rearrangement'']]</center>

===Optimizing the Reactants and Products===

Different conformations of 1,5-hexadiene can be optimized using '''Gaussview'''. The optimized structure of four conformers involved in this discussion and their corresponding energies optimized at '''HF/3-21G''' level of theory are concluded in the following table ('''Table 1''').

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | anti4
! scope="col" | anti2
! scope="col" | gauche
! scope="col" | gauche3
|-
| <jmol><jmolApplet><title>XJB_anti4.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti4.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche3.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| -231.69097Ha || -231.69254Ha || -231.68772Ha ||-231.69266Ha
|}

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 1.''' ''Four conformers of 1,5-hexadiene''</div>

This exercise first involved optimizing a molecule of 1,5-hexadiene with "anti" linkage for the cetral four C atoms at the '''HF/3-21G''' level of theory. The final optimization structure had the total electronic energy of '''-231.69097Ha''' with a '''C<sub>1</sub>''' symmetry. By comparing the energy and point group to that in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1], this structure is confirmed to be ''anti4''. The compositions of energies are also listed in '''Table 2''' as i) the sum of electronic and zero-point energies (potential energy at 0K including the zero-point vibrational energy), ii) the sum of electronic and thermal energies (energy including translational, rotational and vibrational at 298.15K and 1 atm), iii) the sum of electronic and thermal enthalpies (containning an additional correction for RT), and iv) the sum of electronic and thermal free energies (including the entropic contribution to the free energy). For what we are considering, the first two terms are the most useful for further comparisons.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Formula Description'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||E = E<sub>elec</sub> + ZPE||-231.53785
|-
| Sum of electronic and thermal Energies||E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>||-231.53096
|-
| Sum of electronic and thermal Enthalpie||H = E + RT||-231.53002
|-
| Sum of electronic and thermal Free Energie||G = H - TS||-231.56891
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 2.''' ''Frequency calculations of anti4 optimized at '''HF/3-21G''' ([[File:XJB_anti4.log]])''</div>

Next another 1,5-hexadiene molecule with "gauche" linkage for the central four atoms was drawn and optimized at the '''HF/3-21G''' level of theory as before. Energy for this conformation is expected to be higher than that for the ''anti4'' conformation due to the closer position of two alkene ends thus the steric interaction. The final structure had the electronic energy of '''-231.68772Ha''' which was indeed higher than that of the ''anti4'', indicating less favour towards this gauche structure. It had a '''C<sub>2</sub>''' symmetry and was confirmed to be the ''gauche'' conformation.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53439
|-
| Sum of electronic and thermal Energies||-231.52770
|-
| Sum of electronic and thermal Enthalpie||-231.52676
|-
| Sum of electronic and thermal Free Energie||-231.56483
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 3.''' ''Frequency calculations of gauche optimized at '''HF/3-21G''' ([[File:XJB_gauche.log]])''</div>

A lowest energy conformation ''gauche3'' of the reactant molecule was drawn based on and confirmed by [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. The optimization gave a final structure with an energy of '''-231.69266Ha''' with a symmetry of '''C<sub>1</sub>'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53949
|-
| Sum of electronic and thermal Energies||-231.53265
|-
| Sum of electronic and thermal Enthalpie||-231.5317
|-
| Sum of electronic and thermal Free Energie||-231.57064
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 4.''' ''Frequency calculations of gauche3 optimized at '''HF/3-21G''' ([[File:XJB_gauche3.log]])''</div>

The next step required us to draw the C<sub>i</sub> symmetry ''anti2'' conformation of 1,5-hexadiene and optimize it at the '''HF/3-21G''' level of theory. The optimized energy was calculated to be '''-231.69254Ha''' with point group of '''C<sub>i</sub>''', which was consistent with the one given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53954
|-
| Sum of electronic and thermal Energies||-231.53257
|-
| Sum of electronic and thermal Enthalpie||-231.53162
|-
| Sum of electronic and thermal Free Energie||-231.57092
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 5.''' ''Frequency calculations of anti2 optimized at '''HF/3-21G''' ([[File:XJB_anti2.log]])''</div>

The ''anti2'' structure was then reoptimized at '''B3LYP/6-31G*''' which was a improvement over the '''HF/3-21G''' basis set. It adds polarization to atoms and improves the modeling of core electrons thus producing more accurate description of orbitals<ref name="soo">''Nigerian Journal of Chemical Research'', 2007, '''12'''. {{DOI|10.4314/njcr.v12i1.}}</ref>. The reoptimized ''anti2'' structure had the total electronic energy of '''-234.61171Ha''' and the same symmetry C<sub>i</sub>. The comparison of two structures optimized using different levels of theory is summarized in '''Table 6'''. We can see that they have exactly the same dihedral angle of the central four C atoms at 180.00<sup>o</sup> and angles between three of them are very similar as 111.3<sup>o</sup> and 112.7<sup>o</sup> respectively. Thus the the geometry change using '''HF/3-21G''' or '''B3LYP/6-31G*''' is trivial.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Before re-optimization
! scope="col" | After re-optimization
|-
| <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2reoptimized</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2reoptimized.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| ''anti2 optimized at '''HF/3-21G''''' || ''anti2 optimized at '''B3LYP/6-31G*'''''
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 6.''' ''Two optimization of anti2 at '''HF/3-21G''' and '''B3LYP/6-31G*'''([[File:XJB_anti2.log]])''</div>

The frequency calculation was run as well and a list of energies is summarized in '''Table 7'''. A list of all the vibrational frequencies was also generated and no imaginary (negative) frequency was observed and these vibrations are visualized by simulating the infrared spectrum in '''Figure 2'''.

{| align="center"
|- align="center"
|
{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-234.46920
|-
| Sum of electronic and thermal Energies||-234.46186
|-
| Sum of electronic and thermal Enthalpie||-234.46091
|-
| Sum of electronic and thermal Free Energie||-234.50078
|}
|
[[Image:XJB_IR_anti2.jpg|300px]]
|- align="center"
| align="center" | '''Table 7.''' ''Frequency calculations of anti2 optimized at '''B3LYP/6-31G*'''
([[File:XJB_anti2_reoptimized.log]])''
| align="center" | '''Figure 2.''' ''The generated IR spectrum of optimized '''B3LYP/6-31G*''' structure''
|}

===Optimizing the "Chair" and "Boat" Transition Structures===

The study of the chair and boat transition states was carried out using three different approaches. The first one was called the Hessian which involved computing the force constant matrix; the second one was to freeze the reaction coordinate and then optimized the structure once the molecule was fully relaxed; the last one was to use QST2 method to specify the reactants and products then finding the transition structure.

<center>[[Image:XJB_2 ts.jpg|500px|center|thumb|'''Figure 3'''. ''The two transition states of Cope rearrangement'']]</center>

====The Chair Transition State====

In order to set up a transition state optimization, first an allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimized at the '''HF/3-21G''' level of theory. Then this half of the transition structure was duplicated with rotation to make the approximate resembling of the chair transition state with terminal ends of the fragments 2.2 Å apart. The chair transition structure optimization was then done by the first two different approaches.

<center>[[Image:XJB_guess ts.jpg|400px|center|thumb|'''Figure 4'''. ''The guessed chair transition structure of two allyl fragments'']]</center>

For the first optimization approach, the force constants were computed (the Hessian) and the chair transition state was optimized to a TS (Berny) at the '''HF/3-21G''' level of theory. The frequency calculation was carried out at the same time and gave an imaginary frequency at -818 cm<sup>-1</sup>. The electronic energy was calculated to be '''-231.61932Ha''' with C<sub>2h</sub> symmetry point group. The distance between the terminal carbons was 2.0203 Å. An animation of this transition state vibrations was generated to confirm the corresponding Cope rearrangement.

<center>[[File:XJB_chair ts animation.GIF|400px|center|thumb|'''Figure 5.''' ''Animation of the Cope rearrangement via the chair transition state'']]</center>

The chair transition state was then reoptimized using the '''B3LYP/6-31G*''' level of theory and carried out frequency calculations as well. The electronic energy calculated this time was '''-234.55698Ha''' with an imaginary frequency at -566 cm<sup>-1</sup>. We can see that with the geometries quite similar, the energies differ markedly calculated using two levels of theory. The summary and comparison of the calculations optimized at two levels of theory are summarized in '''Table 8'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46670||-234.41493
|-
| Sum of electronic and thermal Energies||-231.46134||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40807
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir1.jpg|200px]]||[[Image:XJB_chairir2.jpg|200px]]
|-
| ||[[File:XJB_chairts1.1.log]]||[[File:XJB_chairts1.2.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 8.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the first approach''</div>

The second approach is the frozen coordinate method using the Redundant Coord Editor to freeze the bond lengths of the terminal ends to 2.2 Å. First the calculation failed because of the recent update of GaussView. After editting the coordinate distance (B 5 1 2.2 F) and the frozen distance (B 5 1 F) the optimization went well this time. The structure was then optimized again using a normal guess Hessian modified including the two differentiating coordinates. This chair transition state was then reoptimized again using the '''B3LYP/6-31G*''' level of theory. The electronic energies were calculated to be the same as in the first approach and frequency calculations were summarized in '''Table 9'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46671||-234.41490
|-
| Sum of electronic and thermal Energies||-231.46135||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40806
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir3.jpg|200px]]||[[Image:XJB_chairir4.jpg|200px]]
|-
| ||[[File:XJB_chairts2.1.log]]||[[File:XJB_chairts2.2.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 9.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the second approach''</div>

As the same energy results given, the bond forming/breaking bond lengths in the chair transition structures in the above two different approaches were compared. The first approach gave the bond lengths of 2.0203 Å while the second approach gave 2.0207 Å. Considering the accuracy limit of bond length estimation in Gaussview, these two bond lengths can be seen as exactly the same. Two methods worked reasonably well because of the good assumption of transition structures. However for more complicated and larger molecule, the Hessian method may fail due to the difficulty to predict the transition geometry (different surface curvature). Thus better way to generate such transition structures is to freeze reaction coordinate.

{| align="center"
|- align="center"
|
[[Image:XJB_chairts bondlength1.jpg|350px|thumb|'''Figure 6.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by Hession'']]
|
[[Image:XJB_chairts bondlength2.jpg|350px|thumb|'''Figure 7.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by frozen coordinate'']]
|- align="center"
|}

====The Boat Transition State====

The boat transition structure was then optimized using the '''QST2''' method. The same numbering for the reactant and product molecules was important in order to specify for this reaction. The renumbering of the molecules is shown as in '''Figure 8''', but the following optimization failed and gave a twisted structure. The reactant and product geometries were then modified further by changing the central C-C-C-C dihedral angle to 0<sup>o</sup> and the inside two C-C-C (i.e. C2-C3-C4 and C3-C4-C5) angles to 100<sup>o</sup>. The job was run successfully and frequency calculations were done with the boat transition state generated. Only one imaginary frequency was detected at -840 cm<sup>-1</sup>. The electronic energy was '''-231.60280Ha''' with C<sub>2v</sub> symmetry. The distances between the terminal carbons of the allyl fragments were examined to be both 2.140 Å which were comparably longer than that of the chair transition structure.

{| align="center"
|- align="center"
|
[[Image:XJB_QST1.jpg|400px|thumb|'''Figure 8.''' ''Numbering of reactant and product'']]
|
[[Image:XJB_QST2.jpg|410px|thumb|'''Figure 9.''' ''Re-numbering of reactant and product'']]
|- align="center"
|}

<center>[[File:XJB_boat ts animation.GIF|400px|center|thumb|'''Figure 10.''' ''Animation of the Cope rearrangement via the boat transition state'']]</center>

The structure was then reoptimized at the higher '''B3LYP/6-31G*''' theory of level and the electronic energy was calculated to be '''-234.54309Ha''' with very similar geometry. The frequency calculation generated an imaginary frequency at -530 cm<sup>-1</sup>. The summary and comparison of two levels' calculations are carried out in the following table.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.45093||-234.40234
|-
| Sum of electronic and thermal Energies||-231.44530||-234.39601
|-
| Sum of electronic and thermal Enthalpie|| -231.44436||-234.39506
|-
| Sum of electronic and thermal Free Energie||-231.47977||-234.43175
|-
| Generated IR spectrum||[[Image:XJB_boatir1.jpg|200px]]||[[Image:XJB_boatir2.jpg|200px]]
|-
| ||[[File:XJB_boat ts1.log]]||[[File:XJB_boat ts2.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 10.''' ''Frequency calculations of the boat transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by QST2''</div>

====Intrinsic Reaction Coordinate====

Next a useful method called '''Intrinct Reaction Coordinate (IRC)''' was applied to the Hessian optimized chair transition structure to follow the mininmum energy path (MEP) from a transition structure down to its local minimum on a potential energy surface. IRC can be used to verify that the transition state we found actually connects the reactants and products, or to identify any reaction intermediates, or to generate an animation of the reaction<ref name="IRC">''The Journal of Chemical Physics.'', 2005, '''122''', pp 1-17. {{DOI|10.1063/1.1927521}}</ref>. The symmetrical reaction coordinate was computed only in the forward direction with force constant always calculated and the number of points along IRC 50. The last structure generated along IRC is given in '''Figure 11''' with the energy calculated as '''-231.69158Ha''' and the dihedral angle of the central four C atoms and the internal angle between three C atoms measured as ''D'' = -67.188<sup>o</sup> ''A'' = 111.78<sup>o</sup>. This energy does not match to any of the conformations discussed in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] thus a minimum geometry has not reached yet.

{| align="center"
|- align="center"
|
<jmol><jmolApplet><title>XJB_IRC.mol</title><color>white</color>
<size>200</size><script>zoom=5</script>
<uploadedFileContents>XJB_IRC.mol</uploadedFileContents>
</jmolApplet></jmol>
|
[[Image:XJB_irc.jpg|500px]]
|- align="center"
| align="center" | '''Figure 11.''' ''The energy minimum structure along IRC ([[File:XJB_IRC.log]])''
| align="center" | '''Figure 12.''' ''Total energy along IRC''
|}

Three improving options can be used to reach the minimum energy. The first one is to run a normal Hessian optimization at the last point on IRC; the second one involves using a larger number of points (in this case 100 was used); the last one is to redo the IRC computation specifying to compute force constant at every step. Three methods were all used and the results were summarized in '''Table 11'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Improved IRC method 1
! scope="col" | Improved IRC method 2
! scope="col" | Improved IRC method 3
|-
| [[Image:XJB_IRC1.jpg|300px|center|IRC1]]||[[Image:XJB_IRC2.jpg|300px|center|IRC3]] || [[Image:XJB_IRC3.jpg|300px|center|IRC3]]
|-
| ''D'' = -64.17<sup>o</sup> ''A'' = 112.04<sup>o</sup> || ''D'' = -66.83<sup>o</sup> ''A'' = 111.82<sup>o</sup> || ''D'' = -64.17<sup>o</sup> ''A'' = 112.04<sup>o</sup>
|-
| -231.69167Ha || -231.69150Ha || -231.69167Ha
|-
| [[File:XJB_IRC1.log]]||[[File:XJB_IRC2.log]]||[[File:XJB_IRC3.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 11.''' ''Comparison of three improvement methods of IRC computation upon the chair transition structure''</div>

The first method was very quick but maybe generated relatively low-accuracy result; the second approach involved the controlling of number of points which might result in ending up at the wrong direction and wrong structure; the third method was the most suitable option for this small system but it took longer time than the previous two. From '''Table 11''' it is concluded that method 1 and 3 gave the same lowest energy which was consistent to the energy of the ''gauche2'' conformation given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

====Activation Energies====

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | '''HF/3-21G''' at 0K
! scope="col" | '''HF/3-21G''' at 298.15K
! scope="col" | '''B3LYP/6-31G*''' at 0K
! scope="col" | '''B3LYP/6-31G*''' at 298.15K
! scope="col" | Expt. at 0K
|-
| '''ΔE (chair)'''||45.71||44.70||34.05||33.16||33.5 ± 0.5
|-
| '''ΔE (boat)'''||55.60||54.76||41.96||41.32||44.7 ± 0.5
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 12.''' ''Summary of activation energies for the Cope rearrangement via both transition states in kcal/mol''</div>

The activation energy calculation was done by calculating the difference in energies between the optimized reactant ''anti2'' and the optimized chair and boat transition states. It is observed that chair conformation gives generally lower activation energy than the boat structure thus more favoured transition state. The structure optimized at '''B3LYP/6-31G*''' level of theory at 0K has closer activation energy to the experimental result. Thus '''B3LYP/6-31G*''' gives a better estimation of this transition structure than '''HF/3-21G'''.

==The Diels Alder Cycloaddition==

In this exercise, AM1 semi-empirical method was used to compute the Diels Alder cycloaddition which is a concerted pericyclic reaction. The driving force of this reaction is the formation of a new σ bond between the π orbitals of the dieneophile and the diene which is stronger than the π bond. The reaction is not allowed without the HOMO-LUMO interation and a symmetry overlap.

===Cis Butadiene===

<center>[[Image:prototype Diels Alder.jpg|500px|center|thumb|'''Figure 13'''. ''The prototype reaction between butadiene and ethylene'']]</center>

To study a prototype Diels Alder cycloaddition reaction between butadiene and ethylene, first a cis butadiene molecule was built using Gaussview and optimized using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the butadiene were plotted. As seen the HOMO of cis-butadiene is anti-symmetric ('''a''') and the LUMO is symmetric ('''s''') with respect to plane.

{| align="center"
|- align="center"
|
[[Image:XJB_HOMO.jpg|280px|thumb|'''Figure 14.''' ''Anti-symmetric HOMO of cis-butadiene'']]
|
[[Image:XJB_LUMO.jpg|300px|thumb|'''Figure 15.''' ''Symmetric HOMO of cis-butadiene'']]
|- align="center"
|}

===The Transition State of prototype reaction between ethylene and butadiene===

The Hessian method was used to estimate transition state for the reaction between ethylene and butadiene because these two reactants are relatively small and simple molecules. The transition state was drawn by first building a structure as (a) in '''Figure 16''' then removing the -CH<sub>2</sub>-CH<sub>2</sub>- group to form the envelop structure as in (b). The '''clean''' option should not be used and the dashed bond length was guessed to be larger than 2.2 Å otherwise the geometry change would cause a failure in optimization.

<center>[[Image:XJB_dielsguessts.jpg|300px|center|thumb|'''Figure 16'''. ''The guessing transition state for the reaction between butadiene and ethylene'']]</center>

<center>[[File:XJB_butadiene and ethylene ts.GIF|400px|center|thumb|'''Figure 17.''' ''Animation of the butadiene and ethylene cycloaddition transition state'' [[File:XJB_prototype DA.log]]]]</center>

After the Hessian optimization, the transition structure has been successfully determined and an animation was generated in '''Figure 17'''. It has a C<sub>1</sub> symmetry point group and the bond lengths of two partly formed C-C bonds were measured to be 2.12Å. The typical sp<sup>3</sup> and sp<sup>2</sup> C-C bond lengths are 1.54 Å and 1.47 Å respectively<ref name="bondlength">''J. Phys. C: Solid State Phys.'', 1986, '''19''', pp 4613-4621.</ref>. The van der Waals radius of the C atom is 1.7 Å<ref name="bondlength2">''Z. Naturforsch.'', 2007, '''62b''', pp 235-243.</ref>. The measured bond distance of the transition structure is larger than typical bond lengths but still within the van der Waals radius, suggesting the bond forming interation between the carbon atoms.

From the animation ('''Figure 17''') the vibration directions of two molecules are opposite and they are approaching and leaving each other at the same time. This suggests the formation of the two bonds synchronous. The vibrational frequencies were also calculated and an infrared spectrum was simulated ('''Figure 20'''). An imaginary frequency of magnitude -955 cm<sup>-1</sup> was seen and the lowest positive frequency was observed at 147 cm<sup>-1</sup> which corresponded to the reactant vibration but not the bond formation.

{| align="center"
|
[[Image:XJB_HOMOts.jpg|300px|thumb|'''Figure 18'''. HOMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_LUMOts.jpg|300px|thumb|'''Figure 19'''. LUMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_IRts.jpg|300px|thumb|'''Figure 20'''. Generated IR of transition state of reaction between cis-butadiene and ethylene]]
|}

The HOMO and LUMO molecular orbitals of this transition structure were visualized in '''Figure 18''' and '''Figure 19'''. From these two figures it can be observed that the HOMO at the transition structure is anti-symmetric ('''a''') while the LUMO is symmetric ('''s'''). Thus this anti-symmetric transition state HOMO is formed by the HOMO of cis-butadiene and the LUMO of ethylene with the knowledge that they are both anti-symmetrical. This conclusion agrees with the previous talked conditions for allowed reaction.

===The cyclohexa-1,3-diene reaction with maleic anhydride===

<center>[[Image:XJB_mechanism3.jpg|500px|center|thumb|'''Figure 21'''. ''The endo and exo products of the Diels Alder reaction between cyclohexa-1,3-diene and maleic anhydride'']]</center>

The exo and endo transition states were drawn according to '''Figure 21''' with the new C-C bond formation line dashed and bond length estimation as 2.5 Å. Then the same Hessian method was applied to optimize the structures. The electronic energy of the endo structure was calculated to be '''-0.05150Ha''' and '''-0.05042Ha''' for the exo. This reaction is supposed to be kinetically controlled thus the lower energy endo transition state is more favoured.

The bond lengths of the partly formed C-C bond are 2.16 Å and 2.17 Å for endo and exo transition states respectively. Another measured C-C bond length is shown in '''Table 13''' as 2.89 Å for endo (C3-C8 and C2-C7) and 2.95 Å for endo (C7-C10 and C8-C9). The longer atom distance indicates the larger steric repulsion in the exo transition state because of the extra existence of hydrogen atoms on C7 and C8.

Woodward and Hoffmann also proposed a secondary orbital overlap statement as an explanation of the endo stereo-preference in Diels Alder reactions<ref name="soot">''J. Org. Chem.'', 1987, '''52(8)''', pp 1469–1474. {{DOI|10.1021/jo00384a016}}</ref>. This secondary orbital overlap does not involve bond changes but π orbital interactions as the highlighted carbon atoms of the endo transition state in '''Table 13'''. Such overlap stablises the structure and lowers the energy of the endo transition state while there is no such effect existed in the exo transition state.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | endo
! scope="col" | exo
|-
| '''Animation'''||[[File:XJB_endots.GIF|300px]]||[[File:XJB_exots.GIF|300px]]
|-
| '''Sum of electronic and zero-point Energies'''||0.133493||0.134880
|-
| '''Sum of electronic and thermal Energies'''||0.143683||0.144881
|-
| '''Sum of electronic and thermal Enthalpies'''||0.144627||0.145825
|-
| '''Sum of electronic and thermal Free Energies'''||0.097349||0.099117
|-
| '''Bond lengths'''||[[Image:XJB_endomeasure.jpg|300px]]||[[Image:XJB_exomeasure.jpg|300px]]
|-
| '''HOMO'''||[[Image:XJB_endoHOMO.jpg|300px]]||[[Image:XJB_exoHOMO.jpg|285px]]
|-
| '''Secondary orbital overlap effect'''||[[Image:XJB_endosoo.jpg|300px]]||[[Image:XJB_exosoo.jpg|300px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 13.''' ''Summary of comparison of properties of endo and exo transition structures for reaction between cyclohexa-1,3-diene and maleic anhydride''</div>

==Reference==

<references/>

Rep:Mod:XJB1130

2013-12-06T13:18:35Z

Jx1011:

==The Cope Rearrangement of 1,5-hexadiene==

The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement reaction ('''Figure 1'''). Its mechanism has been studied by a large number of experimental and computational researches, which is recently accepted that the reaction occurs via either a "chair" or a "boat" transition state structure in a concerted manner. The aim of this exercise is to locate the low-energy minimum and the transition structures on the potential energy surface by Gaussian calculation in order to study its mechanism. The reaction pathway and the activation energy can also be calculated by this calculation.

<center>[[Image:XJB_cope rearrangement.jpg|500px|center|thumb|'''Figure 1'''. ''The Cope rearrangement'']]</center>

===Optimizing the Reactants and Products===

Different conformations of 1,5-hexadiene can be optimized using '''Gaussview'''. The optimized structure of four conformers involved in this discussion and their corresponding energies optimized at '''HF/3-21G''' level of theory are concluded in the following table ('''Table 1''').

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | anti4
! scope="col" | anti2
! scope="col" | gauche
! scope="col" | gauche3
|-
| <jmol><jmolApplet><title>XJB_anti4.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti4.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche3.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| -231.69097Ha || -231.69254Ha || -231.68772Ha ||-231.69266Ha
|}

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 1.''' ''Four conformers of 1,5-hexadiene''</div>

This exercise first involved optimizing a molecule of 1,5-hexadiene with "anti" linkage for the cetral four C atoms at the '''HF/3-21G''' level of theory. The final optimization structure had the total electronic energy of '''-231.69097Ha''' with a '''C<sub>1</sub>''' symmetry. By comparing the energy and point group to that in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1], this structure is confirmed to be ''anti4''. The compositions of energies are also listed in '''Table 2''' as i) the sum of electronic and zero-point energies (potential energy at 0K including the zero-point vibrational energy), ii) the sum of electronic and thermal energies (energy including translational, rotational and vibrational at 298.15K and 1 atm), iii) the sum of electronic and thermal enthalpies (containning an additional correction for RT), and iv) the sum of electronic and thermal free energies (including the entropic contribution to the free energy). For what we are considering, the first two terms are the most useful for further comparisons.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Formula Description'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||E = E<sub>elec</sub> + ZPE||-231.53785
|-
| Sum of electronic and thermal Energies||E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>||-231.53096
|-
| Sum of electronic and thermal Enthalpie||H = E + RT||-231.53002
|-
| Sum of electronic and thermal Free Energie||G = H - TS||-231.56891
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 2.''' ''Frequency calculations of anti4 optimized at '''HF/3-21G''' ([[File:XJB_anti4.log]])''</div>

Next another 1,5-hexadiene molecule with "gauche" linkage for the central four atoms was drawn and optimized at the '''HF/3-21G''' level of theory as before. Energy for this conformation is expected to be higher than that for the ''anti4'' conformation due to the closer position of two alkene ends thus the steric interaction. The final structure had the electronic energy of '''-231.68772Ha''' which was indeed higher than that of the ''anti4'', indicating less favour towards this gauche structure. It had a '''C<sub>2</sub>''' symmetry and was confirmed to be the ''gauche'' conformation.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53439
|-
| Sum of electronic and thermal Energies||-231.52770
|-
| Sum of electronic and thermal Enthalpie||-231.52676
|-
| Sum of electronic and thermal Free Energie||-231.56483
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 3.''' ''Frequency calculations of gauche optimized at '''HF/3-21G''' ([[File:XJB_gauche.log]])''</div>

A lowest energy conformation ''gauche3'' of the reactant molecule was drawn based on and confirmed by [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. The optimization gave a final structure with an energy of '''-231.69266Ha''' with a symmetry of '''C<sub>1</sub>'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53949
|-
| Sum of electronic and thermal Energies||-231.53265
|-
| Sum of electronic and thermal Enthalpie||-231.5317
|-
| Sum of electronic and thermal Free Energie||-231.57064
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 4.''' ''Frequency calculations of gauche3 optimized at '''HF/3-21G''' ([[File:XJB_gauche3.log]])''</div>

The next step required us to draw the C<sub>i</sub> symmetry ''anti2'' conformation of 1,5-hexadiene and optimize it at the '''HF/3-21G''' level of theory. The optimized energy was calculated to be '''-231.69254Ha''' with point group of '''C<sub>i</sub>''', which was consistent with the one given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53954
|-
| Sum of electronic and thermal Energies||-231.53257
|-
| Sum of electronic and thermal Enthalpie||-231.53162
|-
| Sum of electronic and thermal Free Energie||-231.57092
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 5.''' ''Frequency calculations of anti2 optimized at '''HF/3-21G''' ([[File:XJB_anti2.log]])''</div>

The ''anti2'' structure was then reoptimized at '''B3LYP/6-31G*''' which was a improvement over the '''HF/3-21G''' basis set. It adds polarization to atoms and improves the modeling of core electrons thus producing more accurate description of orbitals<ref name="soo">''Nigerian Journal of Chemical Research'', 2007, '''12'''. {{DOI|10.4314/njcr.v12i1.}}</ref>. The reoptimized ''anti2'' structure had the total electronic energy of '''-234.61171Ha''' and the same symmetry C<sub>i</sub>. The comparison of two structures optimized using different levels of theory is summarized in '''Table 6'''. We can see that they have exactly the same dihedral angle of the central four C atoms at 180.00<sup>o</sup> and angles between three of them are very similar as 111.3<sup>o</sup> and 112.7<sup>o</sup> respectively. Thus the the geometry change using '''HF/3-21G''' or '''B3LYP/6-31G*''' is trivial.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Before re-optimization
! scope="col" | After re-optimization
|-
| <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2reoptimized</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2reoptimized.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| ''anti2 optimized at '''HF/3-21G''''' || ''anti2 optimized at '''B3LYP/6-31G*'''''
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 6.''' ''Two optimization of anti2 at '''HF/3-21G''' and '''B3LYP/6-31G*'''([[File:XJB_anti2.log]])''</div>

The frequency calculation was run as well and a list of energies is summarized in '''Table 7'''. A list of all the vibrational frequencies was also generated and no imaginary (negative) frequency was observed and these vibrations are visualized by simulating the infrared spectrum in '''Figure 2'''.

{| align="center"
|- align="center"
|
{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-234.46920
|-
| Sum of electronic and thermal Energies||-234.46186
|-
| Sum of electronic and thermal Enthalpie||-234.46091
|-
| Sum of electronic and thermal Free Energie||-234.50078
|}
|
[[Image:XJB_IR_anti2.jpg|300px]]
|- align="center"
| align="center" | '''Table 7.''' ''Frequency calculations of anti2 optimized at '''B3LYP/6-31G*'''
([[File:XJB_anti2_reoptimized.log]])''
| align="center" | '''Figure 2.''' ''The generated IR spectrum of optimized '''B3LYP/6-31G*''' structure''
|}

===Optimizing the "Chair" and "Boat" Transition Structures===

The study of the chair and boat transition states was carried out using three different approaches. The first one was called the Hessian which involved computing the force constant matrix; the second one was to freeze the reaction coordinate and then optimized the structure once the molecule was fully relaxed; the last one was to use QST2 method to specify the reactants and products then finding the transition structure.

<center>[[Image:XJB_2 ts.jpg|500px|center|thumb|'''Figure 3'''. ''The two transition states of Cope rearrangement'']]</center>

====The Chair Transition State====

In order to set up a transition state optimization, first an allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimized at the '''HF/3-21G''' level of theory. Then this half of the transition structure was duplicated with rotation to make the approximate resembling of the chair transition state with terminal ends of the fragments 2.2 Å apart. The chair transition structure optimization was then done by the first two different approaches.

<center>[[Image:XJB_guess ts.jpg|400px|center|thumb|'''Figure 4'''. ''The guessed chair transition structure of two allyl fragments'']]</center>

For the first optimization approach, the force constants were computed (the Hessian) and the chair transition state was optimized to a TS (Berny) at the '''HF/3-21G''' level of theory. The frequency calculation was carried out at the same time and gave an imaginary frequency at -818 cm<sup>-1</sup>. The electronic energy was calculated to be '''-231.61932Ha''' with C<sub>2h</sub> symmetry point group. The distance between the terminal carbons was 2.0203 Å. An animation of this transition state vibrations was generated to confirm the corresponding Cope rearrangement.

<center>[[File:XJB_chair ts animation.GIF|400px|center|thumb|'''Figure 5.''' ''Animation of the Cope rearrangement via the chair transition state'']]</center>

The chair transition state was then reoptimized using the '''B3LYP/6-31G*''' level of theory and carried out frequency calculations as well. The electronic energy calculated this time was '''-234.55698Ha''' with an imaginary frequency at -566 cm<sup>-1</sup>. We can see that with the geometries quite similar, the energies differ markedly calculated using two levels of theory. The summary and comparison of the calculations optimized at two levels of theory are summarized in '''Table 8'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46670||-234.41493
|-
| Sum of electronic and thermal Energies||-231.46134||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40807
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir1.jpg|200px]]||[[Image:XJB_chairir2.jpg|200px]]
|-
| ||[[File:XJB_chairts1.1.log]]||[[File:XJB_chairts1.2.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 8.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the first approach''</div>

The second approach is the frozen coordinate method using the Redundant Coord Editor to freeze the bond lengths of the terminal ends to 2.2 Å. First the calculation failed because of the recent update of GaussView. After editting the coordinate distance (B 5 1 2.2 F) and the frozen distance (B 5 1 F) the optimization went well this time. The structure was then optimized again using a normal guess Hessian modified including the two differentiating coordinates. This chair transition state was then reoptimized again using the '''B3LYP/6-31G*''' level of theory. The electronic energies were calculated to be the same as in the first approach and frequency calculations were summarized in '''Table 9'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46671||-234.41490
|-
| Sum of electronic and thermal Energies||-231.46135||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40806
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir3.jpg|200px]]||[[Image:XJB_chairir4.jpg|200px]]
|-
| ||[[File:XJB_chairts2.1.log]]||[[File:XJB_chairts2.2.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 9.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the second approach''</div>

As the same energy results given, the bond forming/breaking bond lengths in the chair transition structures in the above two different approaches were compared. The first approach gave the bond lengths of 2.0203 Å while the second approach gave 2.0207 Å. Considering the accuracy limit of bond length estimation in Gaussview, these two bond lengths can be seen as exactly the same. Two methods worked reasonably well because of the good assumption of transition structures. However for more complicated and larger molecule, the Hessian method may fail due to the difficulty to predict the transition geometry (different surface curvature). Thus better way to generate such transition structures is to freeze reaction coordinate.

{| align="center"
|- align="center"
|
[[Image:XJB_chairts bondlength1.jpg|350px|thumb|'''Figure 6.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by Hession'']]
|
[[Image:XJB_chairts bondlength2.jpg|350px|thumb|'''Figure 7.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by frozen coordinate'']]
|- align="center"
|}

====The Boat Transition State====

The boat transition structure was then optimized using the '''QST2''' method. The same numbering for the reactant and product molecules was important in order to specify for this reaction. The renumbering of the molecules is shown as in '''Figure 8''', but the following optimization failed and gave a twisted structure. The reactant and product geometries were then modified further by changing the central C-C-C-C dihedral angle to 0<sup>o</sup> and the inside two C-C-C (i.e. C2-C3-C4 and C3-C4-C5) angles to 100<sup>o</sup>. The job was run successfully and frequency calculations were done with the boat transition state generated. Only one imaginary frequency was detected at -840 cm<sup>-1</sup>. The electronic energy was '''-231.60280Ha''' with C<sub>2v</sub> symmetry. The distances between the terminal carbons of the allyl fragments were examined to be both 2.140 Å which were comparably longer than that of the chair transition structure.

{| align="center"
|- align="center"
|
[[Image:XJB_QST1.jpg|400px|thumb|'''Figure 8.''' ''Numbering of reactant and product'']]
|
[[Image:XJB_QST2.jpg|410px|thumb|'''Figure 9.''' ''Re-numbering of reactant and product'']]
|- align="center"
|}

<center>[[File:XJB_boat ts animation.GIF|400px|center|thumb|'''Figure 10.''' ''Animation of the Cope rearrangement via the boat transition state'']]</center>

The structure was then reoptimized at the higher '''B3LYP/6-31G*''' theory of level and the electronic energy was calculated to be '''-234.54309Ha''' with very similar geometry. The frequency calculation generated an imaginary frequency at -530 cm<sup>-1</sup>. The summary and comparison of two levels' calculations are carried out in the following table.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.45093||-234.40234
|-
| Sum of electronic and thermal Energies||-231.44530||-234.39601
|-
| Sum of electronic and thermal Enthalpie|| -231.44436||-234.39506
|-
| Sum of electronic and thermal Free Energie||-231.47977||-234.43175
|-
| Generated IR spectrum||[[Image:XJB_boatir1.jpg|200px]]||[[Image:XJB_boatir2.jpg|200px]]
|-
| ||[[File:XJB_boat ts1.log]]||[[File:XJB_boat ts2.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 10.''' ''Frequency calculations of the boat transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by QST2''</div>

====Intrinsic Reaction Coordinate====

Next a useful method called '''Intrinct Reaction Coordinate (IRC)''' was applied to the Hessian optimized chair transition structure to follow the mininmum energy path (MEP) from a transition structure down to its local minimum on a potential energy surface. IRC can be used to verify that the transition state we found actually connects the reactants and products, or to identify any reaction intermediates, or to generate an animation of the reaction<ref name="IRC">''The Journal of Chemical Physics.'', 2005, '''122''', pp 1-17. {{DOI|10.1063/1.1927521}}</ref>. The symmetrical reaction coordinate was computed only in the forward direction with force constant always calculated and the number of points along IRC 50. The last structure generated along IRC is given in '''Figure 11''' with the energy calculated as '''-231.69158Ha''' and the dihedral angle of the central four C atoms and the internal angle between three C atoms measured as ''D'' = -67.188<sup>o</sup> ''A'' = 111.78<sup>o</sup>. This energy does not match to any of the conformations discussed in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1] thus a minimum geometry has not reached yet.

{| align="center"
|- align="center"
|
<jmol><jmolApplet><title>XJB_IRC.mol</title><color>white</color>
<size>200</size><script>zoom=5</script>
<uploadedFileContents>XJB_IRC.mol</uploadedFileContents>
</jmolApplet></jmol>
|
[[Image:XJB_irc.jpg|500px]]
|- align="center"
| align="center" | '''Figure 11.''' ''The energy minimum structure along IRC ([[File:XJB_IRC.log]])''
| align="center" | '''Figure 12.''' ''Total energy along IRC''
|}

Three improving options can be used to reach the minimum energy. The first one is to run a normal Hessian optimization at the last point on IRC; the second one involves using a larger number of points (in this case 100 was used); the last one is to redo the IRC computation specifying to compute force constant at every step. Three methods were all used and the results were summarized in '''Table 11'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Improved IRC method 1
! scope="col" | Improved IRC method 2
! scope="col" | Improved IRC method 3
|-
| [[Image:XJB_IRC1.jpg|300px|center|IRC1]]||[[Image:XJB_IRC2.jpg|300px|center|IRC3]] || [[Image:XJB_IRC3.jpg|300px|center|IRC3]]
|-
| ''D'' = -64.17<sup>o</sup> ''A'' = 112.04<sup>o</sup> || ''D'' = -66.83<sup>o</sup> ''A'' = 111.82<sup>o</sup> || ''D'' = -64.17<sup>o</sup> ''A'' = 112.04<sup>o</sup>
|-
| -231.69167Ha || -231.69150Ha || -231.69167Ha
|-
| [[File:XJB_IRC1.log]]||[[File:XJB_IRC2.log]]||[[File:XJB_IRC3.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 11.''' ''Comparison of three improvement methods of IRC computation upon the chair transition structure''</div>

The first method was very quick but maybe generated relatively low-accuracy result; the second approach involved the controlling of number of points which might result in ending up at the wrong direction and wrong structure; the third method was the most suitable option for this small system but it took longer time than the previous two. From '''Table 11''' it is concluded that method 1 and 3 gave the same lowest energy which was consistent to the energy of the ''gauche2'' conformation given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

====Activation Energies====

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | '''HF/3-21G''' at 0K
! scope="col" | '''HF/3-21G''' at 298.15K
! scope="col" | '''B3LYP/6-31G*''' at 0K
! scope="col" | '''B3LYP/6-31G*''' at 298.15K
! scope="col" | Expt. at 0K
|-
| '''ΔE (chair)'''||45.71||44.70||34.05||33.16||33.5 ± 0.5
|-
| '''ΔE (boat)'''||55.60||54.76||41.96||41.32||44.7 ± 0.5
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 12.''' ''Summary of activation energies for the Cope rearrangement via both transition states in kcal/mol''</div>

The activation energy calculation was done by calculating the difference in energies between the optimized reactant ''anti2'' and the optimized chair and boat transition states. It is observed that chair conformation gives generally lower activation energy than the boat structure thus more favoured transition state. The structure optimized at '''B3LYP/6-31G*''' level of theory at 0K has closer activation energy to the experimental result. Thus '''B3LYP/6-31G*''' gives a better estimation of this transition structure than '''HF/3-21G'''.

==The Diels Alder Cycloaddition==

In this exercise, AM1 semi-empirical method is used to compute the Diels Alder cycloaddition which is a concerted pericyclic reaction. The driving force of this reaction is the formation of a new σ bond between the π orbitals of the dieneophile and the diene. The reaction is allowed with HOMO-LUMO interation and a symmetry overlap.

===Cis Butadiene===

<center>[[Image:prototype Diels Alder.jpg|500px|center|thumb|'''Figure 13'''. ''The prototype reaction between butadiene and ethylene'']]</center>

To study a prototype Diels Alder cycloaddition reaction between butadiene and ethylene, first a cis butadiene molecule was built using Gaussview and optimized using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the butadiene were plotted. As seen the HOMO of cis-butadiene is anti-symmetric ('''a''') and the LUMO is symmetric ('''s''') with respect to plane.

{| align="center"
|- align="center"
|
[[Image:XJB_HOMO.jpg|280px|thumb|'''Figure 14.''' ''Anti-symmetric HOMO of cis-butadiene'']]
|
[[Image:XJB_LUMO.jpg|300px|thumb|'''Figure 15.''' ''Symmetric HOMO of cis-butadiene'']]
|- align="center"
|}

===The Transition State of prototype reaction between ethylene and butadiene===

The Hessian method was used to estimate transition state for the reaction between ethylene and butadiene because these two reactants are relatively small and simple molecules. The transition state was drawn by first building a structure as (a) in '''Figure 16''' then removing the -CH<sub>2</sub>-CH<sub>2</sub>- group to form the envelop structure as in (b). The '''clean''' option should not be used and the dashed bond length was guessed to be larger than 2.2 Å otherwise the geometry change would cause a failure in optimization.

<center>[[Image:XJB_dielsguessts.jpg|300px|center|thumb|'''Figure 16'''. ''The guessing transition state for the reaction between butadiene and ethylene'']]</center>

<center>[[File:XJB_butadiene and ethylene ts.GIF|400px|center|thumb|'''Figure 17.''' ''Animation of the butadiene and ethylene cycloaddition transition state'']]</center>

After the Hessian optimization, the transition structure has been successfully determined and an animation was generated in '''Figure 17'''. It has a C<sub>1</sub> symmetry point group and the bond lengths of two partly formed C-C bonds were measured to be 2.12Å. The typical sp<sup>3</sup> and sp<sup>2</sup> C-C bond lengths are 1.54Å and 1.47Å respectively[ref]. The van der Waals radius of the C atom is 1.7 Å[ref]. The measured bond distance of the transition structure is larger than typical bond lengths but still within the van der Waals radius, suggesting the bond forming interation between the carbon atoms.

From the animation ('''Figure 17''') the vibration directions of two molecules are opposite and they are approaching and leaving each other at the same time. This suggests the formation of the two bonds synchronous. The vibrational frequencies were also calculated and an infrared spectrum was simulated ('''Figure 20'''). An imaginary frequency of magnitude -955 cm<sup>-1</sup> was seen and the lowest positive frequency was observed at 147 cm<sup>-1</sup> which corresponded to the reactant vibration but not the bond formation.

{| align="center"
|
[[Image:XJB_HOMOts.jpg|300px|thumb|'''Figure 18'''. HOMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_LUMOts.jpg|300px|thumb|'''Figure 19'''. LUMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_IRts.jpg|300px|thumb|'''Figure 20'''. Generated IR of transition state of reaction between cis-butadiene and ethylene]]
|}

The HOMO and LUMO molecular orbitals of this transition structure were visualized in '''Figure 18''' and '''Figure 19'''. From these two figures it can be observed that the HOMO at the transition structure is anti-symmetric ('''a''') while the LUMO is symmetric ('''s'''). Thus this anti-symmetric transition state HOMO is formed by the HOMO of cis-butadiene and the LUMO of ethylene with the knowledge that they are both anti-symmetrical. This conclusion agrees with the previous talked conditions for allowed reaction.

===The cyclohexa-1,3-diene reaction with maleic anhydride===

<center>[[Image:XJB_mechanism3.jpg|500px|center|thumb|'''Figure 21'''. ''The endo and exo products of the Diels Alder reaction between cyclohexa-1,3-diene and maleic anhydride'']]</center>

The exo and endo transition states were drawn according to '''Figure 21''' with the new C-C bond formation line dashed and bond length estimation as 2.5. Then the same Hessian method was applied to optimize the structures. The electronic energy of the endo structure was calculated to be '''-0.05150Ha''' and '''-0.05042Ha''' for the exo. This reaction is supposed to be kinetically controlled thus the lower energy endo transition state is more favoured.

The bond lengths of the partly formed C-C bond are 2.16 Å and 2.17 Å for endo and exo transition states respectively. Another measured C-C bond length is shown in '''Table 13''' as 2.89 Å for endo (C3-C8 and C2-C7) and 2.95 Å for endo (C7-C10 and C8-C9). The longer atom distance indicates the larger steric repulsion in the exo transition state because of the extra existence of hydrogen atoms on C7 and C8.

Woodward and Hoffmann also proposed a secondary orbital overlap statement as an explanation of the endo stereo-preference in Diels Alder reactions<ref name="soot">''J. Org. Chem.'', 1987, '''52(8)''', pp 1469–1474. {{DOI|10.1021/jo00384a016}}</ref>. This secondary orbital overlap does not involve bond changes but π orbital interactions as the highlighted carbon atoms of the endo transition state in '''Table 13'''. Such overlap stablises the structure and lowers the energy of the endo transition state while there is no such effect existed in the exo transition state.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | endo
! scope="col" | exo
|-
| '''Animation'''||[[File:XJB_endots.GIF|300px]]||[[File:XJB_exots.GIF|300px]]
|-
| '''Sum of electronic and zero-point Energies'''||0.133493||0.134880
|-
| '''Sum of electronic and thermal Energies'''||0.143683||0.144881
|-
| '''Sum of electronic and thermal Enthalpies'''||0.144627||0.145825
|-
| '''Sum of electronic and thermal Free Energies'''||0.097349||0.099117
|-
| '''Bond lengths'''||[[Image:XJB_endomeasure.jpg|300px]]||[[Image:XJB_exomeasure.jpg|300px]]
|-
| '''HOMO'''||[[Image:XJB_endoHOMO.jpg|300px]]||[[Image:XJB_exoHOMO.jpg|285px]]
|-
| '''Secondary orbital overlap effect'''||[[Image:XJB_endosoo.jpg|300px]]||[[Image:XJB_exosoo.jpg|300px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 13.''' ''Summary of comparison of properties of endo and exo transition structures for reaction between cyclohexa-1,3-diene and maleic anhydride''</div>

==Reference==

<references/>

Rep:Mod:XJB1130

2013-12-06T13:06:49Z

Jx1011: /* The Chair Transition State */

==The Cope Rearrangement of 1,5-hexadiene==

The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement reaction ('''Figure 1'''). Its mechanism has been studied by a large number of experimental and computational researches, which is recently accepted that the reaction occurs via either a "chair" or a "boat" transition state structure in a concerted manner. The aim of this exercise is to locate the low-energy minimum and the transition structures on the potential energy surface by Gaussian calculation in order to study its mechanism. The reaction pathway and the activation energy can also be calculated by this calculation.

<center>[[Image:XJB_cope rearrangement.jpg|500px|center|thumb|'''Figure 1'''. ''The Cope rearrangement'']]</center>

===Optimizing the Reactants and Products===

Different conformations of 1,5-hexadiene can be optimized using '''Gaussview'''. The optimized structure of four conformers involved in this discussion and their corresponding energies optimized at '''HF/3-21G''' level of theory are concluded in the following table ('''Table 1''').

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | anti4
! scope="col" | anti2
! scope="col" | gauche
! scope="col" | gauche3
|-
| <jmol><jmolApplet><title>XJB_anti4.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti4.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche3.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| -231.69097Ha || -231.69254Ha || -231.68772Ha ||-231.69266Ha
|}

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 1.''' ''Four conformers of 1,5-hexadiene''</div>

This exercise first involved optimizing a molecule of 1,5-hexadiene with "anti" linkage for the cetral four C atoms at the '''HF/3-21G''' level of theory. The final optimization structure had the total electronic energy of '''-231.69097Ha''' with a '''C<sub>1</sub>''' symmetry. By comparing the energy and point group to that in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1], this structure is confirmed to be ''anti4''. The compositions of energies are also listed in '''Table 2''' as i) the sum of electronic and zero-point energies (potential energy at 0K including the zero-point vibrational energy), ii) the sum of electronic and thermal energies (energy including translational, rotational and vibrational at 298.15K and 1 atm), iii) the sum of electronic and thermal enthalpies (containning an additional correction for RT), and iv) the sum of electronic and thermal free energies (including the entropic contribution to the free energy). For what we are considering, the first two terms are the most useful for further comparisons.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Formula Description'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||E = E<sub>elec</sub> + ZPE||-231.53785
|-
| Sum of electronic and thermal Energies||E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>||-231.53096
|-
| Sum of electronic and thermal Enthalpie||H = E + RT||-231.53002
|-
| Sum of electronic and thermal Free Energie||G = H - TS||-231.56891
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 2.''' ''Frequency calculations of anti4 optimized at '''HF/3-21G''' ([[File:XJB_anti4.log]])''</div>

Next another 1,5-hexadiene molecule with "gauche" linkage for the central four atoms was drawn and optimized at the '''HF/3-21G''' level of theory as before. Energy for this conformation is expected to be higher than that for the ''anti4'' conformation due to the closer position of two alkene ends thus the steric interaction. The final structure had the electronic energy of '''-231.68772Ha''' which was indeed higher than that of the ''anti4'', indicating less favour towards this gauche structure. It had a '''C<sub>2</sub>''' symmetry and was confirmed to be the ''gauche'' conformation.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53439
|-
| Sum of electronic and thermal Energies||-231.52770
|-
| Sum of electronic and thermal Enthalpie||-231.52676
|-
| Sum of electronic and thermal Free Energie||-231.56483
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 3.''' ''Frequency calculations of gauche optimized at '''HF/3-21G''' ([[File:XJB_gauche.log]])''</div>

A lowest energy conformation ''gauche3'' of the reactant molecule was drawn based on and confirmed by [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. The optimization gave a final structure with an energy of '''-231.69266Ha''' with a symmetry of '''C<sub>1</sub>'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53949
|-
| Sum of electronic and thermal Energies||-231.53265
|-
| Sum of electronic and thermal Enthalpie||-231.5317
|-
| Sum of electronic and thermal Free Energie||-231.57064
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 4.''' ''Frequency calculations of gauche3 optimized at '''HF/3-21G''' ([[File:XJB_gauche3.log]])''</div>

The next step required us to draw the C<sub>i</sub> symmetry ''anti2'' conformation of 1,5-hexadiene and optimize it at the '''HF/3-21G''' level of theory. The optimized energy was calculated to be '''-231.69254Ha''' with point group of '''C<sub>i</sub>''', which was consistent with the one given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53954
|-
| Sum of electronic and thermal Energies||-231.53257
|-
| Sum of electronic and thermal Enthalpie||-231.53162
|-
| Sum of electronic and thermal Free Energie||-231.57092
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 5.''' ''Frequency calculations of anti2 optimized at '''HF/3-21G''' ([[File:XJB_anti2.log]])''</div>

The ''anti2'' structure was then reoptimized at '''B3LYP/6-31G*''' which was a improvement over the '''HF/3-21G''' basis set. It adds polarization to atoms and improves the modeling of core electrons thus producing more accurate description of orbitals<ref name="soo">''Nigerian Journal of Chemical Research'', 2007, '''12'''. {{DOI|10.4314/njcr.v12i1.}}</ref>. The reoptimized ''anti2'' structure had the total electronic energy of '''-234.61171Ha''' and the same symmetry C<sub>i</sub>. The comparison of two structures optimized using different levels of theory is summarized in '''Table 6'''. We can see that they have exactly the same dihedral angle of the central four C atoms at 180.00<sup>o</sup> and angles between three of them are very similar as 111.3<sup>o</sup> and 112.7<sup>o</sup> respectively. Thus the the geometry change using '''HF/3-21G''' or '''B3LYP/6-31G*''' is trivial.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Before re-optimization
! scope="col" | After re-optimization
|-
| <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2reoptimized</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2reoptimized.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| ''anti2 optimized at '''HF/3-21G''''' || ''anti2 optimized at '''B3LYP/6-31G*'''''
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 6.''' ''Two optimization of anti2 at '''HF/3-21G''' and '''B3LYP/6-31G*'''([[File:XJB_anti2.log]])''</div>

The frequency calculation was run as well and a list of energies is summarized in '''Table 7'''. A list of all the vibrational frequencies was also generated and no imaginary (negative) frequency was observed and these vibrations are visualized by simulating the infrared spectrum in '''Figure 2'''.

{| align="center"
|- align="center"
|
{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-234.46920
|-
| Sum of electronic and thermal Energies||-234.46186
|-
| Sum of electronic and thermal Enthalpie||-234.46091
|-
| Sum of electronic and thermal Free Energie||-234.50078
|}
|
[[Image:XJB_IR_anti2.jpg|300px]]
|- align="center"
| align="center" | '''Table 7.''' ''Frequency calculations of anti2 optimized at '''B3LYP/6-31G*'''
([[File:XJB_anti2_reoptimized.log]])''
| align="center" | '''Figure 2.''' ''The generated IR spectrum of optimized '''B3LYP/6-31G*''' structure''
|}

===Optimizing the "Chair" and "Boat" Transition Structures===

The study of the chair and boat transition states was carried out using three different approaches. The first one was called the Hessian which involved computing the force constant matrix; the second one was to freeze the reaction coordinate and then optimized the structure once the molecule was fully relaxed; the last one was to use QST2 method to specify the reactants and products then finding the transition structure.

<center>[[Image:XJB_2 ts.jpg|500px|center|thumb|'''Figure 3'''. ''The two transition states of Cope rearrangement'']]</center>

====The Chair Transition State====

In order to set up a transition state optimization, first an allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimized at the '''HF/3-21G''' level of theory. Then this half of the transition structure was duplicated with rotation to make the approximate resembling of the chair transition state with terminal ends of the fragments 2.2 Å apart. The chair transition structure optimization was then done by the first two different approaches.

<center>[[Image:XJB_guess ts.jpg|400px|center|thumb|'''Figure 4'''. ''The guessed chair transition structure of two allyl fragments'']]</center>

For the first optimization approach, the force constants were computed (the Hessian) and the chair transition state was optimized to a TS (Berny) at the '''HF/3-21G''' level of theory. The frequency calculation was carried out at the same time and gave an imaginary frequency at -818 cm<sup>-1</sup>. The electronic energy was calculated to be '''-231.61932Ha''' with C<sub>2h</sub> symmetry point group. The distance between the terminal carbons was 2.0203 Å. An animation of this transition state vibrations was generated to confirm the corresponding Cope rearrangement.

<center>[[File:XJB_chair ts animation.GIF|400px|center|thumb|'''Figure 5.''' ''Animation of the Cope rearrangement via the chair transition state'']]</center>

The chair transition state was then reoptimized using the '''B3LYP/6-31G*''' level of theory and carried out frequency calculations as well. The electronic energy calculated this time was '''-234.55698Ha''' with an imaginary frequency at -566 cm<sup>-1</sup>. We can see that with the geometries quite similar, the energies differ markedly calculated using two levels of theory. The summary and comparison of the calculations optimized at two levels of theory are summarized in '''Table 8'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46670||-234.41493
|-
| Sum of electronic and thermal Energies||-231.46134||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40807
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir1.jpg|200px]]||[[Image:XJB_chairir2.jpg|200px]]
|-
| ||[[File:XJB_chairts1.1.log]]||[[File:XJB_chairts1.2.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 8.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the first approach''</div>

The second approach is the frozen coordinate method using the Redundant Coord Editor to freeze the bond lengths of the terminal ends to 2.2 Å. First the calculation failed because of the recent update of GaussView. After editting the coordinate distance (B 5 1 2.2 F) and the frozen distance (B 5 1 F) the optimization went well this time. The structure was then optimized again using a normal guess Hessian modified including the two differentiating coordinates. This chair transition state was then reoptimized again using the '''B3LYP/6-31G*''' level of theory. The electronic energies were calculated to be the same as in the first approach and frequency calculations were summarized in '''Table 9'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46671||-234.41490
|-
| Sum of electronic and thermal Energies||-231.46135||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40806
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir3.jpg|200px]]||[[Image:XJB_chairir4.jpg|200px]]
|-
| ||[[File:XJB_chairts2.1.log]]||[[File:XJB_chairts2.2.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 9.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the second approach''</div>

As the same energy results given, the bond forming/breaking bond lengths in the chair transition structures in the above two different approaches were compared. The first approach gave the bond lengths of 2.0203 Å while the second approach gave 2.0207 Å. Considering the accuracy limit of bond length estimation in Gaussview, these two bond lengths can be seen as exactly the same. Two methods worked reasonably well because of the good assumption of transition structures. However for more complicated and larger molecule, the Hessian method may fail due to the difficulty to predict the transition geometry (different surface curvature). Thus better way to generate such transition structures is to freeze reaction coordinate.

{| align="center"
|- align="center"
|
[[Image:XJB_chairts bondlength1.jpg|350px|thumb|'''Figure 6.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by Hession'']]
|
[[Image:XJB_chairts bondlength2.jpg|350px|thumb|'''Figure 7.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by frozen coordinate'']]
|- align="center"
|}

====The Boat Transition State====

The boat transition structure was then optimized using the '''QST2''' method. The same numbering for the reactant and product molecules was important in order to specify for this reaction. The renumbering of the molecules is shown as in '''Figure 8''', but the following optimization failed and gave a twisted structure. The reactant and product geometries were then modified further by changing the central C-C-C-C dihedral angle to 0<sup>o</sup> and the inside two C-C-C (i.e. C2-C3-C4 and C3-C4-C5) angles to 100<sup>o</sup>. The job was run successfully and frequency calculations were done with the boat transition state generated. Only one imaginary frequency was detected at -840 cm<sup>-1</sup>. The electronic energy was '''-231.60280Ha''' with C<sub>2v</sub> symmetry. The distances between the terminal carbons of the allyl fragments were examined to be both 2.140 Å which were comparably longer than that of the chair transition structure.

{| align="center"
|- align="center"
|
[[Image:XJB_QST1.jpg|400px|thumb|'''Figure 8.''' ''Numbering of reactant and product'']]
|
[[Image:XJB_QST2.jpg|410px|thumb|'''Figure 9.''' ''Re-numbering of reactant and product'']]
|- align="center"
|}

<center>[[File:XJB_boat ts animation.GIF|400px|center|thumb|'''Figure 10.''' ''Animation of the Cope rearrangement via the boat transition state'']]</center>

The structure was then reoptimized at the higher '''B3LYP/6-31G*''' theory of level and the electronic energy was calculated to be '''-234.54309Ha''' with very similar geometry. The frequency calculation generated an imaginary frequency at -530 cm<sup>-1</sup>. The summary and comparison of two levels' calculations are carried out in the following table.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.45093||-234.40234
|-
| Sum of electronic and thermal Energies||-231.44530||-234.39601
|-
| Sum of electronic and thermal Enthalpie|| -231.44436||-234.39506
|-
| Sum of electronic and thermal Free Energie||-231.47977||-234.43175
|-
| Generated IR spectrum||[[Image:XJB_boatir1.jpg|200px]]||[[Image:XJB_boatir2.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 10.''' ''Frequency calculations of the boat transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by QST2
([[File:XJB_boat ts1.log]], [[File:XJB_boat ts2.log]])''</div>

====Intrinsic Reaction Coordinate====

Next a useful method called '''Intrinct Reaction Coordinate (IRC)''' is applied to the Hessian optimized chair transition structure to follow the mininmum energy path (MEP) from a transition structure down to its local minimum on a potential energy surface. IRC can be used to verify that the transition state you found actually connects the reactants and products you think it does, or to identify any reaction intermediates you haven't thought about, or to generate an animation of the reaction [ref]. The symmetrical reaction coordinate was computed only in the forward direction with force constant always calculated and the number of points along IRC 50. The last structure generated along IRC is given in '''Figure 11''' with the energy calculated as '''-231.69158Ha''' and the dihedral angle and internal angle measured as D = -67.188<sup>o</sup> A = 111.78<sup>o</sup>. This energy does not match to any of the conformations discussed in Appendix 1 thus a minimum geometry has not reached yet.

{| align="center"
|- align="center"
|
<jmol><jmolApplet><title>XJB_IRC.mol</title><color>white</color>
<size>200</size><script>zoom=5</script>
<uploadedFileContents>XJB_IRC.mol</uploadedFileContents>
</jmolApplet></jmol>
|
[[Image:XJB_irc.jpg|500px]]
|- align="center"
| align="center" | '''Figure 11.''' ''The energy minimum structure along IRC''
| align="center" | '''Figure 12.''' ''Total energy along IRC''
|}

Three improving options can be used to reach the minimum energy. The first one is to run a normal Hessian optimization of the last point on IRC; the second one involves using a larger number of points (in this case 100 was used); the last one is to redo the IRC computation specifying to compute force constant at every step. Three methods were all used and the results were summarized in '''Table 11'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Improved IRC method 1
! scope="col" | Improved IRC method 2
! scope="col" | Improved IRC method 3
|-
| [[Image:XJB_IRC1.jpg|300px|center|IRC1]]||[[Image:XJB_IRC2.jpg|300px|center|IRC3]] || [[Image:XJB_IRC3.jpg|300px|center|IRC3]]
|-
| D = -64.17<sup>o</sup> A = 112.04<sup>o</sup> || D = -66.83<sup>o</sup> A = 111.82<sup>o</sup> || D = -64.17<sup>o</sup> A = 112.04<sup>o</sup>
|-
| -231.69167Ha || -231.69150Ha || -231.69167Ha
|-
| [[File:XJB_IRC1.log]]||[[File:XJB_IRC2.log]]||[[File:XJB_IRC3.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 11.''' ''Comparison of three improvement methods of IRC computation upon the chair transition structure''</div>

The first method was very quick but maybe generated relatively low-accuracy result; the second approach involved the controlling of number of points which might result in ending up at the wrong direction and wrong structure; the third method was the most suitable option for this small system but it took longer time than the previous two. From '''Table 11''' it is concluded that method 1 and 3 gave the same lowest energy which was consistent to the energy of the ''gauche2'' conformation given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

====Activation Energies====

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | '''HF/3-21G''' at 0K
! scope="col" | '''HF/3-21G''' at 298.15K
! scope="col" | '''B3LYP/6-31G*''' at 0K
! scope="col" | '''B3LYP/6-31G*''' at 298.15K
! scope="col" | Expt. at 0K
|-
| '''ΔE (chair)'''||45.71||44.70||34.05||33.16||33.5 ± 0.5
|-
| '''ΔE (boat)'''||55.60||54.76||41.96||41.32||44.7 ± 0.5
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 12.''' ''Summary of activation energies for the Cope rearrangement via both transition states in kcal/mol''</div>

The activation energy calculation was done by calculating the difference in energies between the optimized reactant ''anti2'' and the optimized chair and boat transition states. It is observed that chair conformation gives generally lower activation energy thus the more favoured transition state. The structure optimized at '''B3LYP/6-31G*''' level of theory at 0K has closer activation energy to the experimental result. Thus '''B3LYP/6-31G*''' gives a better estimation of transition state than '''HF/3-21G'''.

==The Diels Alder Cycloaddition==

In this exercise, AM1 semi-empirical method is used to compute the Diels Alder cycloaddition which is a concerted pericyclic reaction. The driving force of this reaction is the formation of a new σ bond between the π orbitals of the dieneophile and the diene. The reaction is allowed with HOMO-LUMO interation and a symmetry overlap.

===Cis Butadiene===

<center>[[Image:prototype Diels Alder.jpg|500px|center|thumb|'''Figure 13'''. ''The prototype reaction between butadiene and ethylene'']]</center>

To study a prototype Diels Alder cycloaddition reaction between butadiene and ethylene, first a cis butadiene molecule was built using Gaussview and optimized using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the butadiene were plotted. As seen the HOMO of cis-butadiene is anti-symmetric ('''a''') and the LUMO is symmetric ('''s''') with respect to plane.

{| align="center"
|- align="center"
|
[[Image:XJB_HOMO.jpg|280px|thumb|'''Figure 14.''' ''Anti-symmetric HOMO of cis-butadiene'']]
|
[[Image:XJB_LUMO.jpg|300px|thumb|'''Figure 15.''' ''Symmetric HOMO of cis-butadiene'']]
|- align="center"
|}

===The Transition State of prototype reaction between ethylene and butadiene===

The Hessian method was used to estimate transition state for the reaction between ethylene and butadiene because these two reactants are relatively small and simple molecules. The transition state was drawn by first building a structure as (a) in '''Figure 16''' then removing the -CH<sub>2</sub>-CH<sub>2</sub>- group to form the envelop structure as in (b). The '''clean''' option should not be used and the dashed bond length was guessed to be larger than 2.2 Å otherwise the geometry change would cause a failure in optimization.

<center>[[Image:XJB_dielsguessts.jpg|300px|center|thumb|'''Figure 16'''. ''The guessing transition state for the reaction between butadiene and ethylene'']]</center>

<center>[[File:XJB_butadiene and ethylene ts.GIF|400px|center|thumb|'''Figure 17.''' ''Animation of the butadiene and ethylene cycloaddition transition state'']]</center>

After the Hessian optimization, the transition structure has been successfully determined and an animation was generated in '''Figure 17'''. It has a C<sub>1</sub> symmetry point group and the bond lengths of two partly formed C-C bonds were measured to be 2.12Å. The typical sp<sup>3</sup> and sp<sup>2</sup> C-C bond lengths are 1.54Å and 1.47Å respectively[ref]. The van der Waals radius of the C atom is 1.7 Å[ref]. The measured bond distance of the transition structure is larger than typical bond lengths but still within the van der Waals radius, suggesting the bond forming interation between the carbon atoms.

From the animation ('''Figure 17''') the vibration directions of two molecules are opposite and they are approaching and leaving each other at the same time. This suggests the formation of the two bonds synchronous. The vibrational frequencies were also calculated and an infrared spectrum was simulated ('''Figure 20'''). An imaginary frequency of magnitude -955 cm<sup>-1</sup> was seen and the lowest positive frequency was observed at 147 cm<sup>-1</sup> which corresponded to the reactant vibration but not the bond formation.

{| align="center"
|
[[Image:XJB_HOMOts.jpg|300px|thumb|'''Figure 18'''. HOMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_LUMOts.jpg|300px|thumb|'''Figure 19'''. LUMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_IRts.jpg|300px|thumb|'''Figure 20'''. Generated IR of transition state of reaction between cis-butadiene and ethylene]]
|}

The HOMO and LUMO molecular orbitals of this transition structure were visualized in '''Figure 18''' and '''Figure 19'''. From these two figures it can be observed that the HOMO at the transition structure is anti-symmetric ('''a''') while the LUMO is symmetric ('''s'''). Thus this anti-symmetric transition state HOMO is formed by the HOMO of cis-butadiene and the LUMO of ethylene with the knowledge that they are both anti-symmetrical. This conclusion agrees with the previous talked conditions for allowed reaction.

===The cyclohexa-1,3-diene reaction with maleic anhydride===

<center>[[Image:XJB_mechanism3.jpg|500px|center|thumb|'''Figure 21'''. ''The endo and exo products of the Diels Alder reaction between cyclohexa-1,3-diene and maleic anhydride'']]</center>

The exo and endo transition states were drawn according to '''Figure 21''' with the new C-C bond formation line dashed and bond length estimation as 2.5. Then the same Hessian method was applied to optimize the structures. The electronic energy of the endo structure was calculated to be '''-0.05150Ha''' and '''-0.05042Ha''' for the exo. This reaction is supposed to be kinetically controlled thus the lower energy endo transition state is more favoured.

The bond lengths of the partly formed C-C bond are 2.16 Å and 2.17 Å for endo and exo transition states respectively. Another measured C-C bond length is shown in '''Table 13''' as 2.89 Å for endo (C3-C8 and C2-C7) and 2.95 Å for endo (C7-C10 and C8-C9). The longer atom distance indicates the larger steric repulsion in the exo transition state because of the extra existence of hydrogen atoms on C7 and C8.

Woodward and Hoffmann also proposed a secondary orbital overlap statement as an explanation of the endo stereo-preference in Diels Alder reactions<ref name="soot">''J. Org. Chem.'', 1987, '''52(8)''', pp 1469–1474. {{DOI|10.1021/jo00384a016}}</ref>. This secondary orbital overlap does not involve bond changes but π orbital interactions as the highlighted carbon atoms of the endo transition state in '''Table 13'''. Such overlap stablises the structure and lowers the energy of the endo transition state while there is no such effect existed in the exo transition state.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | endo
! scope="col" | exo
|-
| '''Animation'''||[[File:XJB_endots.GIF|300px]]||[[File:XJB_exots.GIF|300px]]
|-
| '''Sum of electronic and zero-point Energies'''||0.133493||0.134880
|-
| '''Sum of electronic and thermal Energies'''||0.143683||0.144881
|-
| '''Sum of electronic and thermal Enthalpies'''||0.144627||0.145825
|-
| '''Sum of electronic and thermal Free Energies'''||0.097349||0.099117
|-
| '''Bond lengths'''||[[Image:XJB_endomeasure.jpg|300px]]||[[Image:XJB_exomeasure.jpg|300px]]
|-
| '''HOMO'''||[[Image:XJB_endoHOMO.jpg|300px]]||[[Image:XJB_exoHOMO.jpg|285px]]
|-
| '''Secondary orbital overlap effect'''||[[Image:XJB_endosoo.jpg|300px]]||[[Image:XJB_exosoo.jpg|300px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 13.''' ''Summary of comparison of properties of endo and exo transition structures for reaction between cyclohexa-1,3-diene and maleic anhydride''</div>

==Reference==

<references/>

Rep:Mod:XJB1130

2013-12-06T13:05:05Z

Jx1011:

==The Cope Rearrangement of 1,5-hexadiene==

The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement reaction ('''Figure 1'''). Its mechanism has been studied by a large number of experimental and computational researches, which is recently accepted that the reaction occurs via either a "chair" or a "boat" transition state structure in a concerted manner. The aim of this exercise is to locate the low-energy minimum and the transition structures on the potential energy surface by Gaussian calculation in order to study its mechanism. The reaction pathway and the activation energy can also be calculated by this calculation.

<center>[[Image:XJB_cope rearrangement.jpg|500px|center|thumb|'''Figure 1'''. ''The Cope rearrangement'']]</center>

===Optimizing the Reactants and Products===

Different conformations of 1,5-hexadiene can be optimized using '''Gaussview'''. The optimized structure of four conformers involved in this discussion and their corresponding energies optimized at '''HF/3-21G''' level of theory are concluded in the following table ('''Table 1''').

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | anti4
! scope="col" | anti2
! scope="col" | gauche
! scope="col" | gauche3
|-
| <jmol><jmolApplet><title>XJB_anti4.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti4.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche3.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| -231.69097Ha || -231.69254Ha || -231.68772Ha ||-231.69266Ha
|}

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 1.''' ''Four conformers of 1,5-hexadiene''</div>

This exercise first involved optimizing a molecule of 1,5-hexadiene with "anti" linkage for the cetral four C atoms at the '''HF/3-21G''' level of theory. The final optimization structure had the total electronic energy of '''-231.69097Ha''' with a '''C<sub>1</sub>''' symmetry. By comparing the energy and point group to that in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1], this structure is confirmed to be ''anti4''. The compositions of energies are also listed in '''Table 2''' as i) the sum of electronic and zero-point energies (potential energy at 0K including the zero-point vibrational energy), ii) the sum of electronic and thermal energies (energy including translational, rotational and vibrational at 298.15K and 1 atm), iii) the sum of electronic and thermal enthalpies (containning an additional correction for RT), and iv) the sum of electronic and thermal free energies (including the entropic contribution to the free energy). For what we are considering, the first two terms are the most useful for further comparisons.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Formula Description'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||E = E<sub>elec</sub> + ZPE||-231.53785
|-
| Sum of electronic and thermal Energies||E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>||-231.53096
|-
| Sum of electronic and thermal Enthalpie||H = E + RT||-231.53002
|-
| Sum of electronic and thermal Free Energie||G = H - TS||-231.56891
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 2.''' ''Frequency calculations of anti4 optimized at '''HF/3-21G''' ([[File:XJB_anti4.log]])''</div>

Next another 1,5-hexadiene molecule with "gauche" linkage for the central four atoms was drawn and optimized at the '''HF/3-21G''' level of theory as before. Energy for this conformation is expected to be higher than that for the ''anti4'' conformation due to the closer position of two alkene ends thus the steric interaction. The final structure had the electronic energy of '''-231.68772Ha''' which was indeed higher than that of the ''anti4'', indicating less favour towards this gauche structure. It had a '''C<sub>2</sub>''' symmetry and was confirmed to be the ''gauche'' conformation.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53439
|-
| Sum of electronic and thermal Energies||-231.52770
|-
| Sum of electronic and thermal Enthalpie||-231.52676
|-
| Sum of electronic and thermal Free Energie||-231.56483
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 3.''' ''Frequency calculations of gauche optimized at '''HF/3-21G''' ([[File:XJB_gauche.log]])''</div>

A lowest energy conformation ''gauche3'' of the reactant molecule was drawn based on and confirmed by [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. The optimization gave a final structure with an energy of '''-231.69266Ha''' with a symmetry of '''C<sub>1</sub>'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53949
|-
| Sum of electronic and thermal Energies||-231.53265
|-
| Sum of electronic and thermal Enthalpie||-231.5317
|-
| Sum of electronic and thermal Free Energie||-231.57064
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 4.''' ''Frequency calculations of gauche3 optimized at '''HF/3-21G''' ([[File:XJB_gauche3.log]])''</div>

The next step required us to draw the C<sub>i</sub> symmetry ''anti2'' conformation of 1,5-hexadiene and optimize it at the '''HF/3-21G''' level of theory. The optimized energy was calculated to be '''-231.69254Ha''' with point group of '''C<sub>i</sub>''', which was consistent with the one given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53954
|-
| Sum of electronic and thermal Energies||-231.53257
|-
| Sum of electronic and thermal Enthalpie||-231.53162
|-
| Sum of electronic and thermal Free Energie||-231.57092
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 5.''' ''Frequency calculations of anti2 optimized at '''HF/3-21G''' ([[File:XJB_anti2.log]])''</div>

The ''anti2'' structure was then reoptimized at '''B3LYP/6-31G*''' which was a improvement over the '''HF/3-21G''' basis set. It adds polarization to atoms and improves the modeling of core electrons thus producing more accurate description of orbitals<ref name="soo">''Nigerian Journal of Chemical Research'', 2007, '''12'''. {{DOI|10.4314/njcr.v12i1.}}</ref>. The reoptimized ''anti2'' structure had the total electronic energy of '''-234.61171Ha''' and the same symmetry C<sub>i</sub>. The comparison of two structures optimized using different levels of theory is summarized in '''Table 6'''. We can see that they have exactly the same dihedral angle of the central four C atoms at 180.00<sup>o</sup> and angles between three of them are very similar as 111.3<sup>o</sup> and 112.7<sup>o</sup> respectively. Thus the the geometry change using '''HF/3-21G''' or '''B3LYP/6-31G*''' is trivial.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Before re-optimization
! scope="col" | After re-optimization
|-
| <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2reoptimized</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2reoptimized.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| ''anti2 optimized at '''HF/3-21G''''' || ''anti2 optimized at '''B3LYP/6-31G*'''''
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 6.''' ''Two optimization of anti2 at '''HF/3-21G''' and '''B3LYP/6-31G*'''([[File:XJB_anti2.log]])''</div>

The frequency calculation was run as well and a list of energies is summarized in '''Table 7'''. A list of all the vibrational frequencies was also generated and no imaginary (negative) frequency was observed and these vibrations are visualized by simulating the infrared spectrum in '''Figure 2'''.

{| align="center"
|- align="center"
|
{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-234.46920
|-
| Sum of electronic and thermal Energies||-234.46186
|-
| Sum of electronic and thermal Enthalpie||-234.46091
|-
| Sum of electronic and thermal Free Energie||-234.50078
|}
|
[[Image:XJB_IR_anti2.jpg|300px]]
|- align="center"
| align="center" | '''Table 7.''' ''Frequency calculations of anti2 optimized at '''B3LYP/6-31G*'''
([[File:XJB_anti2_reoptimized.log]])''
| align="center" | '''Figure 2.''' ''The generated IR spectrum of optimized '''B3LYP/6-31G*''' structure''
|}

===Optimizing the "Chair" and "Boat" Transition Structures===

The study of the chair and boat transition states was carried out using three different approaches. The first one was called the Hessian which involved computing the force constant matrix; the second one was to freeze the reaction coordinate and then optimized the structure once the molecule was fully relaxed; the last one was to use QST2 method to specify the reactants and products then finding the transition structure.

<center>[[Image:XJB_2 ts.jpg|500px|center|thumb|'''Figure 3'''. ''The two transition states of Cope rearrangement'']]</center>

====The Chair Transition State====

In order to set up a transition state optimization, first an allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimized at the '''HF/3-21G''' level of theory. Then this half of the transition structure was duplicated with rotation to make the approximate resembling of the chair transition state with terminal ends of the fragments 2.2 Å apart. The chair transition structure optimization was then done by the first two different approaches.

<center>[[Image:XJB_guess ts.jpg|400px|center|thumb|'''Figure 4'''. ''The guessed chair transition structure of two allyl fragments'']]</center>

For the first optimization approach, the force constants were computed (the Hessian) and the chair transition state was optimized to a TS (Berny) at the '''HF/3-21G''' level of theory. The frequency calculation was carried out at the same time and gave an imaginary frequency at -818 cm<sup>-1</sup>. The electronic energy was calculated to be '''-231.61932Ha''' with C<sub>2h</sub> symmetry point group. The distance between the terminal carbons was 2.0203 Å. An animation of this transition state vibrations was generated to confirm the corresponding Cope rearrangement.

<center>[[File:XJB_chair ts animation.GIF|400px|center|thumb|'''Figure 5.''' ''Animation of the Cope rearrangement via the chair transition state'']]</center>

The chair transition state was then reoptimized using the '''B3LYP/6-31G*''' level of theory and carried out frequency calculations as well. The electronic energy calculated this time was '''-234.55698Ha''' with an imaginary frequency at -566 cm<sup>-1</sup>. We can see that with the geometries quite similar, the energies differ markedly calculated using two levels of theory. The summary and comparison of the calculations optimized at two levels of theory are summarized in '''Table 8'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46670||-234.41493
|-
| Sum of electronic and thermal Energies||-231.46134||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40807
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir1.jpg|200px]]||[[Image:XJB_chairir2.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 8.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the first approach
([[File:XJB_chairts1.1.log]], [[File:XJB_chairts1.2.log]])''</div>

The second approach is the frozen coordinate method using the Redundant Coord Editor to freeze the bond lengths of the terminal ends to 2.2 Å. First the calculation failed because of the recent update of GaussView. After editting the coordinate distance (B 5 1 2.2 F) and the frozen distance (B 5 1 F) the optimization went well this time. The structure was then optimized again using a normal guess Hessian modified including the two differentiating coordinates. This chair transition state was then reoptimized again using the '''B3LYP/6-31G*''' level of theory. The electronic energies were calculated to be the same as in the first approach and frequency calculations were summarized in '''Table 9'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46671||-234.41490
|-
| Sum of electronic and thermal Energies||-231.46135||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40806
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir3.jpg|200px]]||[[Image:XJB_chairir4.jpg|200px]]
|-
| ||[[File:XJB_chairts2.1.log]]||[[File:XJB_chairts2.2.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 9.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the second approach''</div>

As the same energy results given, the bond forming/breaking bond lengths in the chair transition structures in the above two different approaches were compared. The first approach gave the bond lengths of 2.0203 Å while the second approach gave 2.0207 Å. Considering the accuracy limit of bond length estimation in Gaussview, these two bond lengths can be seen as exactly the same. Two methods worked reasonably well because of the good assumption of transition structures. However for more complicated and larger molecule, the Hessian method may fail due to the difficulty to predict the transition geometry (different surface curvature). Thus better way to generate such transition structures is to freeze reaction coordinate.

{| align="center"
|- align="center"
|
[[Image:XJB_chairts bondlength1.jpg|350px|thumb|'''Figure 6.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by Hession'']]
|
[[Image:XJB_chairts bondlength2.jpg|350px|thumb|'''Figure 7.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by frozen coordinate'']]
|- align="center"
|}

====The Boat Transition State====

The boat transition structure was then optimized using the '''QST2''' method. The same numbering for the reactant and product molecules was important in order to specify for this reaction. The renumbering of the molecules is shown as in '''Figure 8''', but the following optimization failed and gave a twisted structure. The reactant and product geometries were then modified further by changing the central C-C-C-C dihedral angle to 0<sup>o</sup> and the inside two C-C-C (i.e. C2-C3-C4 and C3-C4-C5) angles to 100<sup>o</sup>. The job was run successfully and frequency calculations were done with the boat transition state generated. Only one imaginary frequency was detected at -840 cm<sup>-1</sup>. The electronic energy was '''-231.60280Ha''' with C<sub>2v</sub> symmetry. The distances between the terminal carbons of the allyl fragments were examined to be both 2.140 Å which were comparably longer than that of the chair transition structure.

{| align="center"
|- align="center"
|
[[Image:XJB_QST1.jpg|400px|thumb|'''Figure 8.''' ''Numbering of reactant and product'']]
|
[[Image:XJB_QST2.jpg|410px|thumb|'''Figure 9.''' ''Re-numbering of reactant and product'']]
|- align="center"
|}

<center>[[File:XJB_boat ts animation.GIF|400px|center|thumb|'''Figure 10.''' ''Animation of the Cope rearrangement via the boat transition state'']]</center>

The structure was then reoptimized at the higher '''B3LYP/6-31G*''' theory of level and the electronic energy was calculated to be '''-234.54309Ha''' with very similar geometry. The frequency calculation generated an imaginary frequency at -530 cm<sup>-1</sup>. The summary and comparison of two levels' calculations are carried out in the following table.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.45093||-234.40234
|-
| Sum of electronic and thermal Energies||-231.44530||-234.39601
|-
| Sum of electronic and thermal Enthalpie|| -231.44436||-234.39506
|-
| Sum of electronic and thermal Free Energie||-231.47977||-234.43175
|-
| Generated IR spectrum||[[Image:XJB_boatir1.jpg|200px]]||[[Image:XJB_boatir2.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 10.''' ''Frequency calculations of the boat transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by QST2
([[File:XJB_boat ts1.log]], [[File:XJB_boat ts2.log]])''</div>

====Intrinsic Reaction Coordinate====

Next a useful method called '''Intrinct Reaction Coordinate (IRC)''' is applied to the Hessian optimized chair transition structure to follow the mininmum energy path (MEP) from a transition structure down to its local minimum on a potential energy surface. IRC can be used to verify that the transition state you found actually connects the reactants and products you think it does, or to identify any reaction intermediates you haven't thought about, or to generate an animation of the reaction [ref]. The symmetrical reaction coordinate was computed only in the forward direction with force constant always calculated and the number of points along IRC 50. The last structure generated along IRC is given in '''Figure 11''' with the energy calculated as '''-231.69158Ha''' and the dihedral angle and internal angle measured as D = -67.188<sup>o</sup> A = 111.78<sup>o</sup>. This energy does not match to any of the conformations discussed in Appendix 1 thus a minimum geometry has not reached yet.

{| align="center"
|- align="center"
|
<jmol><jmolApplet><title>XJB_IRC.mol</title><color>white</color>
<size>200</size><script>zoom=5</script>
<uploadedFileContents>XJB_IRC.mol</uploadedFileContents>
</jmolApplet></jmol>
|
[[Image:XJB_irc.jpg|500px]]
|- align="center"
| align="center" | '''Figure 11.''' ''The energy minimum structure along IRC''
| align="center" | '''Figure 12.''' ''Total energy along IRC''
|}

Three improving options can be used to reach the minimum energy. The first one is to run a normal Hessian optimization of the last point on IRC; the second one involves using a larger number of points (in this case 100 was used); the last one is to redo the IRC computation specifying to compute force constant at every step. Three methods were all used and the results were summarized in '''Table 11'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Improved IRC method 1
! scope="col" | Improved IRC method 2
! scope="col" | Improved IRC method 3
|-
| [[Image:XJB_IRC1.jpg|300px|center|IRC1]]||[[Image:XJB_IRC2.jpg|300px|center|IRC3]] || [[Image:XJB_IRC3.jpg|300px|center|IRC3]]
|-
| D = -64.17<sup>o</sup> A = 112.04<sup>o</sup> || D = -66.83<sup>o</sup> A = 111.82<sup>o</sup> || D = -64.17<sup>o</sup> A = 112.04<sup>o</sup>
|-
| -231.69167Ha || -231.69150Ha || -231.69167Ha
|-
| [[File:XJB_IRC1.log]]||[[File:XJB_IRC2.log]]||[[File:XJB_IRC3.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 11.''' ''Comparison of three improvement methods of IRC computation upon the chair transition structure''</div>

The first method was very quick but maybe generated relatively low-accuracy result; the second approach involved the controlling of number of points which might result in ending up at the wrong direction and wrong structure; the third method was the most suitable option for this small system but it took longer time than the previous two. From '''Table 11''' it is concluded that method 1 and 3 gave the same lowest energy which was consistent to the energy of the ''gauche2'' conformation given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

====Activation Energies====

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | '''HF/3-21G''' at 0K
! scope="col" | '''HF/3-21G''' at 298.15K
! scope="col" | '''B3LYP/6-31G*''' at 0K
! scope="col" | '''B3LYP/6-31G*''' at 298.15K
! scope="col" | Expt. at 0K
|-
| '''ΔE (chair)'''||45.71||44.70||34.05||33.16||33.5 ± 0.5
|-
| '''ΔE (boat)'''||55.60||54.76||41.96||41.32||44.7 ± 0.5
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 12.''' ''Summary of activation energies for the Cope rearrangement via both transition states in kcal/mol''</div>

The activation energy calculation was done by calculating the difference in energies between the optimized reactant ''anti2'' and the optimized chair and boat transition states. It is observed that chair conformation gives generally lower activation energy thus the more favoured transition state. The structure optimized at '''B3LYP/6-31G*''' level of theory at 0K has closer activation energy to the experimental result. Thus '''B3LYP/6-31G*''' gives a better estimation of transition state than '''HF/3-21G'''.

==The Diels Alder Cycloaddition==

In this exercise, AM1 semi-empirical method is used to compute the Diels Alder cycloaddition which is a concerted pericyclic reaction. The driving force of this reaction is the formation of a new σ bond between the π orbitals of the dieneophile and the diene. The reaction is allowed with HOMO-LUMO interation and a symmetry overlap.

===Cis Butadiene===

<center>[[Image:prototype Diels Alder.jpg|500px|center|thumb|'''Figure 13'''. ''The prototype reaction between butadiene and ethylene'']]</center>

To study a prototype Diels Alder cycloaddition reaction between butadiene and ethylene, first a cis butadiene molecule was built using Gaussview and optimized using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the butadiene were plotted. As seen the HOMO of cis-butadiene is anti-symmetric ('''a''') and the LUMO is symmetric ('''s''') with respect to plane.

{| align="center"
|- align="center"
|
[[Image:XJB_HOMO.jpg|280px|thumb|'''Figure 14.''' ''Anti-symmetric HOMO of cis-butadiene'']]
|
[[Image:XJB_LUMO.jpg|300px|thumb|'''Figure 15.''' ''Symmetric HOMO of cis-butadiene'']]
|- align="center"
|}

===The Transition State of prototype reaction between ethylene and butadiene===

The Hessian method was used to estimate transition state for the reaction between ethylene and butadiene because these two reactants are relatively small and simple molecules. The transition state was drawn by first building a structure as (a) in '''Figure 16''' then removing the -CH<sub>2</sub>-CH<sub>2</sub>- group to form the envelop structure as in (b). The '''clean''' option should not be used and the dashed bond length was guessed to be larger than 2.2 Å otherwise the geometry change would cause a failure in optimization.

<center>[[Image:XJB_dielsguessts.jpg|300px|center|thumb|'''Figure 16'''. ''The guessing transition state for the reaction between butadiene and ethylene'']]</center>

<center>[[File:XJB_butadiene and ethylene ts.GIF|400px|center|thumb|'''Figure 17.''' ''Animation of the butadiene and ethylene cycloaddition transition state'']]</center>

After the Hessian optimization, the transition structure has been successfully determined and an animation was generated in '''Figure 17'''. It has a C<sub>1</sub> symmetry point group and the bond lengths of two partly formed C-C bonds were measured to be 2.12Å. The typical sp<sup>3</sup> and sp<sup>2</sup> C-C bond lengths are 1.54Å and 1.47Å respectively[ref]. The van der Waals radius of the C atom is 1.7 Å[ref]. The measured bond distance of the transition structure is larger than typical bond lengths but still within the van der Waals radius, suggesting the bond forming interation between the carbon atoms.

From the animation ('''Figure 17''') the vibration directions of two molecules are opposite and they are approaching and leaving each other at the same time. This suggests the formation of the two bonds synchronous. The vibrational frequencies were also calculated and an infrared spectrum was simulated ('''Figure 20'''). An imaginary frequency of magnitude -955 cm<sup>-1</sup> was seen and the lowest positive frequency was observed at 147 cm<sup>-1</sup> which corresponded to the reactant vibration but not the bond formation.

{| align="center"
|
[[Image:XJB_HOMOts.jpg|300px|thumb|'''Figure 18'''. HOMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_LUMOts.jpg|300px|thumb|'''Figure 19'''. LUMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_IRts.jpg|300px|thumb|'''Figure 20'''. Generated IR of transition state of reaction between cis-butadiene and ethylene]]
|}

The HOMO and LUMO molecular orbitals of this transition structure were visualized in '''Figure 18''' and '''Figure 19'''. From these two figures it can be observed that the HOMO at the transition structure is anti-symmetric ('''a''') while the LUMO is symmetric ('''s'''). Thus this anti-symmetric transition state HOMO is formed by the HOMO of cis-butadiene and the LUMO of ethylene with the knowledge that they are both anti-symmetrical. This conclusion agrees with the previous talked conditions for allowed reaction.

===The cyclohexa-1,3-diene reaction with maleic anhydride===

<center>[[Image:XJB_mechanism3.jpg|500px|center|thumb|'''Figure 21'''. ''The endo and exo products of the Diels Alder reaction between cyclohexa-1,3-diene and maleic anhydride'']]</center>

The exo and endo transition states were drawn according to '''Figure 21''' with the new C-C bond formation line dashed and bond length estimation as 2.5. Then the same Hessian method was applied to optimize the structures. The electronic energy of the endo structure was calculated to be '''-0.05150Ha''' and '''-0.05042Ha''' for the exo. This reaction is supposed to be kinetically controlled thus the lower energy endo transition state is more favoured.

The bond lengths of the partly formed C-C bond are 2.16 Å and 2.17 Å for endo and exo transition states respectively. Another measured C-C bond length is shown in '''Table 13''' as 2.89 Å for endo (C3-C8 and C2-C7) and 2.95 Å for endo (C7-C10 and C8-C9). The longer atom distance indicates the larger steric repulsion in the exo transition state because of the extra existence of hydrogen atoms on C7 and C8.

Woodward and Hoffmann also proposed a secondary orbital overlap statement as an explanation of the endo stereo-preference in Diels Alder reactions<ref name="soot">''J. Org. Chem.'', 1987, '''52(8)''', pp 1469–1474. {{DOI|10.1021/jo00384a016}}</ref>. This secondary orbital overlap does not involve bond changes but π orbital interactions as the highlighted carbon atoms of the endo transition state in '''Table 13'''. Such overlap stablises the structure and lowers the energy of the endo transition state while there is no such effect existed in the exo transition state.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | endo
! scope="col" | exo
|-
| '''Animation'''||[[File:XJB_endots.GIF|300px]]||[[File:XJB_exots.GIF|300px]]
|-
| '''Sum of electronic and zero-point Energies'''||0.133493||0.134880
|-
| '''Sum of electronic and thermal Energies'''||0.143683||0.144881
|-
| '''Sum of electronic and thermal Enthalpies'''||0.144627||0.145825
|-
| '''Sum of electronic and thermal Free Energies'''||0.097349||0.099117
|-
| '''Bond lengths'''||[[Image:XJB_endomeasure.jpg|300px]]||[[Image:XJB_exomeasure.jpg|300px]]
|-
| '''HOMO'''||[[Image:XJB_endoHOMO.jpg|300px]]||[[Image:XJB_exoHOMO.jpg|285px]]
|-
| '''Secondary orbital overlap effect'''||[[Image:XJB_endosoo.jpg|300px]]||[[Image:XJB_exosoo.jpg|300px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 13.''' ''Summary of comparison of properties of endo and exo transition structures for reaction between cyclohexa-1,3-diene and maleic anhydride''</div>

==Reference==

<references/>

Rep:Mod:XJB1130

2013-12-06T12:56:28Z

Jx1011: /* The cyclohexa-1,3-diene reaction with maleic anhydride */

==The Cope Rearrangement of 1,5-hexadiene==

The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement reaction ('''Figure 1'''). Its mechanism has been studied by a large number of experimental and computational researches, which is recently accepted that the reaction occurs via either a "chair" or a "boat" transition state structure in a concerted manner. The aim of this exercise is to locate the low-energy minimum and the transition structures on the potential energy surface by Gaussian calculation in order to study its mechanism. The reaction pathway and the activation energy can also be calculated by this calculation.

<center>[[Image:XJB_cope rearrangement.jpg|500px|center|thumb|'''Figure 1'''. ''The Cope rearrangement'']]</center>

===Optimizing the Reactants and Products===

Different conformations of 1,5-hexadiene can be optimized using '''Gaussview'''. The optimized structure of four conformers involved in this discussion and their corresponding energies optimized at '''HF/3-21G''' level of theory are concluded in the following table ('''Table 1''').

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | anti4
! scope="col" | anti2
! scope="col" | gauche
! scope="col" | gauche3
|-
| <jmol><jmolApplet><title>XJB_anti4.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti4.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche3.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| -231.69097Ha || -231.69254Ha || -231.68772Ha ||-231.69266Ha
|}

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 1.''' ''Four conformers of 1,5-hexadiene''</div>

This exercise first involved optimizing a molecule of 1,5-hexadiene with "anti" linkage for the cetral four C atoms at the '''HF/3-21G''' level of theory. The final optimization structure had the total electronic energy of '''-231.69097Ha''' with a '''C<sub>1</sub>''' symmetry. By comparing the energy and point group to that in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1], this structure is confirmed to be ''anti4''. The compositions of energies are also listed in '''Table 2''' as i) the sum of electronic and zero-point energies (potential energy at 0K including the zero-point vibrational energy), ii) the sum of electronic and thermal energies (energy including translational, rotational and vibrational at 298.15K and 1 atm), iii) the sum of electronic and thermal enthalpies (containning an additional correction for RT), and iv) the sum of electronic and thermal free energies (including the entropic contribution to the free energy). For what we are considering, the first two terms are the most useful for further comparisons.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Formula Description'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||E = E<sub>elec</sub> + ZPE||-231.53785
|-
| Sum of electronic and thermal Energies||E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>||-231.53096
|-
| Sum of electronic and thermal Enthalpie||H = E + RT||-231.53002
|-
| Sum of electronic and thermal Free Energie||G = H - TS||-231.56891
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 2.''' ''Frequency calculations of anti4 optimized at '''HF/3-21G''' ([[File:XJB_anti4.log]])''</div>

Next another 1,5-hexadiene molecule with "gauche" linkage for the central four atoms was drawn and optimized at the '''HF/3-21G''' level of theory as before. Energy for this conformation is expected to be higher than that for the ''anti4'' conformation due to the closer position of two alkene ends thus the steric interaction. The final structure had the electronic energy of '''-231.68772Ha''' which was indeed higher than that of the ''anti4'', indicating less favour towards this gauche structure. It had a '''C<sub>2</sub>''' symmetry and was confirmed to be the ''gauche'' conformation.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53439
|-
| Sum of electronic and thermal Energies||-231.52770
|-
| Sum of electronic and thermal Enthalpie||-231.52676
|-
| Sum of electronic and thermal Free Energie||-231.56483
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 3.''' ''Frequency calculations of gauche optimized at '''HF/3-21G''' ([[File:XJB_gauche.log]])''</div>

A lowest energy conformation ''gauche3'' of the reactant molecule was drawn based on and confirmed by [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. The optimization gave a final structure with an energy of '''-231.69266Ha''' with a symmetry of '''C<sub>1</sub>'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53949
|-
| Sum of electronic and thermal Energies||-231.53265
|-
| Sum of electronic and thermal Enthalpie||-231.5317
|-
| Sum of electronic and thermal Free Energie||-231.57064
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 4.''' ''Frequency calculations of gauche3 optimized at '''HF/3-21G''' ([[File:XJB_gauche3.log]])''</div>

The next step required us to draw the C<sub>i</sub> symmetry ''anti2'' conformation of 1,5-hexadiene and optimize it at the '''HF/3-21G''' level of theory. The optimized energy was calculated to be '''-231.69254Ha''' with point group of '''C<sub>i</sub>''', which was consistent with the one given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53954
|-
| Sum of electronic and thermal Energies||-231.53257
|-
| Sum of electronic and thermal Enthalpie||-231.53162
|-
| Sum of electronic and thermal Free Energie||-231.57092
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 5.''' ''Frequency calculations of anti2 optimized at '''HF/3-21G''' ([[File:XJB_anti2.log]])''</div>

The ''anti2'' structure was then reoptimized at '''B3LYP/6-31G*''' which was a improvement over the '''HF/3-21G''' basis set. It adds polarization to atoms and improves the modeling of core electrons thus producing more accurate description of orbitals<ref name="soo">''Nigerian Journal of Chemical Research'', 2007, '''12''', pp 1469–1474. {{DOI|10.4314/njcr.v12i1.}}</ref>. The reoptimized ''anti2'' structure had the total electronic energy of '''-234.61171Ha''' and the same symmetry C<sub>i</sub>. The comparison of two structures optimized using different levels of theory is summarized in '''Table 6'''. We can see that they have exactly the same dihedral angle of the central four C atoms at 180.00<sup>o</sup> and angles between three of them are very similar as 111.3<sup>o</sup> and 112.7<sup>o</sup> respectively. Thus the the geometry change using '''HF/3-21G''' or '''B3LYP/6-31G*''' is trivial.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Before re-optimization
! scope="col" | After re-optimization
|-
| <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2reoptimized</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2reoptimized.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| ''anti2 optimized at '''HF/3-21G''''' || ''anti2 optimized at '''B3LYP/6-31G*'''''
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 6.''' ''Two optimization of anti2 at '''HF/3-21G''' and '''B3LYP/6-31G*'''([[File:XJB_anti2.log]])''</div>

The frequency calculation was run as well and a list of energies is summarized in '''Table 7'''. A list of all the vibrational frequencies was also generated and no imaginary (negative) frequency was observed and these vibrations are visualized by simulating the infrared spectrum in '''Figure 2'''.

{| align="center"
|- align="center"
|
{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-234.46920
|-
| Sum of electronic and thermal Energies||-234.46186
|-
| Sum of electronic and thermal Enthalpie||-234.46091
|-
| Sum of electronic and thermal Free Energie||-234.50078
|}
|
[[Image:XJB_IR_anti2.jpg|300px]]
|- align="center"
| align="center" | '''Table 7.''' ''Frequency calculations of anti2 optimized at '''B3LYP/6-31G*'''
([[File:XJB_anti2_reoptimized.log]])''
| align="center" | '''Figure 2.''' ''The generated IR spectrum of optimized '''B3LYP/6-31G*''' structure''
|}

===Optimizing the "Chair" and "Boat" Transition Structures===

The study of the chair and boat transition states is carried out using three different approaches. The first one is called the Hessian which involves computing the force constant matrix; the second one is to freeze the reaction coordinate and then optimize the structure once the molecule is fully relaxed; the last one is to use QST2 method to specify the reactants and products then finding the transition structure.

<center>[[Image:XJB_2 ts.jpg|500px|center|thumb|'''Figure 3'''. ''The two transition states of Cope rearrangement'']]</center>

====The Chair Transition State====

In order to set up a transition state optimization, first an allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimized at the '''HF/3-21G''' level of theory. Then this half of the transition structure was duplicated with rotation to make the approximate resembling of the chair transition state with terminal ends of the fragments 2.2 Å apart. The chair transition structure optimization was then done by the first two different approaches.

<center>[[Image:XJB_guess ts.jpg|400px|center|thumb|'''Figure 4'''. ''The guessed chair transition structure of two allyl fragments'']]</center>

For the first optimization approach, the force constants were computed (the Hessian) and the chair transition state was optimized to a TS (Berny) at the '''HF/3-21G''' level of theory. The frequency calculation was carried out at the same time and gave an imaginary frequency at -818 cm<sup>-1</sup>. The electronic energy was calculated to be '''-231.61932Ha''' with C<sub>2h</sub> symmetry point group. The distance between the terminal carbons was 2.0203 Å. An animation of this transition state vibrations was generated to confirm the corresponding Cope rearrangement.

<center>[[File:XJB_chair ts animation.GIF|400px|center|thumb|'''Figure 5.''' ''Animation of the Cope rearrangement via the chair transition state'']]</center>

The chair transition state was then reoptimized using the '''B3LYP/6-31G*''' level of theory and carried out frequency calculations as well. The electronic energy calculated this time was '''-234.55698Ha''' with an imaginary frequency at -566 cm<sup>-1</sup>. We can see that with the geometries quite similar, the energies differ markedly calculated using two levels of theory. The summary and comparison of the calculations optimized at two levels of theory are summarized in '''Table 8'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46670||-234.41493
|-
| Sum of electronic and thermal Energies||-231.46134||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40807
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir1.jpg|200px]]||[[Image:XJB_chairir2.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 8.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the first approach
([[File:XJB_chairts1.1.log]], [[File:XJB_chairts1.2.log]])''</div>

The second approach is the frozen coordinate method using the Redundant Coord Editor to freeze the bond lengths of the terminal ends to 2.2 Å. First the calculation failed because of the recent update of GaussView. After editting the coordinate distance (B 5 1 2.2 F) and the frozen distance (B 5 1 F) the optimization went well this time. The structure was then optimized again using a normal guess Hessian modified including the two differentiating coordinates. This chair transition state was then reoptimized again using the '''B3LYP/6-31G*''' level of theory. The electronic energies were calculated to be the same as in the first approach and frequency calculations were summarized in '''Table 9'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46671||-234.41490
|-
| Sum of electronic and thermal Energies||-231.46135||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40806
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir3.jpg|200px]]||[[Image:XJB_chairir4.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 9.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the second approach
([[File:XJB_chairts2.1.log]], [[File:XJB_chairts2.2.log]])''</div>

As the same energy results given, the bond forming/breaking bond lengths in the chair transition structures in the above two different approaches were compared. The first approach gave the bond lengths of 2.0203 Å while the second approach gave 2.0207 Å. Considering the accuracy limit of bond length estimation in Gaussview, these two bond lengths can be seen as exactly the same. Two methods worked reasonably well because of the good assumption of transition structures. However for more complicated and larger molecule, the Hessian method may fail due to the difficulty to predict the transition geometry (different surface curvature). Thus better way to generate such transition structures is to freeze reaction coordinate.

{| align="center"
|- align="center"
|
[[Image:XJB_chairts bondlength1.jpg|350px|thumb|'''Figure 6.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by Hession'']]
|
[[Image:XJB_chairts bondlength2.jpg|350px|thumb|'''Figure 7.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by frozen coordinate'']]
|- align="center"
|}

====The Boat Transition State====

The boat transition structure was then optimized using the '''QST2''' method. The same numbering for the reactant and product molecules was important in order to specify for this reaction. The renumbering of the molecules is shown as in '''Figure 8''', but the following optimization failed and gave a twisted structure. The reactant and product geometries were then modified further by changing the central C-C-C-C dihedral angle to 0<sup>o</sup> and the inside two C-C-C (i.e. C2-C3-C4 and C3-C4-C5) angles to 100<sup>o</sup>. The job was run successfully and frequency calculations were done with the boat transition state generated. Only one imaginary frequency was detected also at -840 cm<sup>-1</sup>. The electronic energy was '''-231.60280Ha''' with C<sub>2v</sub> symmetry. The distances between the terminal carbons of the allyl fragments were examined to be both 2.140Å which were comparably longer than that of the chair transition structure.

{| align="center"
|- align="center"
|
[[Image:XJB_QST1.jpg|400px|thumb|'''Figure 8.''' ''Numbering of reactant and product'']]
|
[[Image:XJB_QST2.jpg|410px|thumb|'''Figure 9.''' ''Re-numbering of reactant and product'']]
|- align="center"
|}

<center>[[File:XJB_boat ts animation.GIF|400px|center|thumb|'''Figure 10.''' ''Animation of the Cope rearrangement via the boat transition state'']]</center>

The structure was then reoptimized at the higher '''B3LYP/6-31G*''' theory of level and the electronic energy was calculated to be '''-234.54309Ha''' with very similar geometry. The frequency calculation generated an imaginary frequency at -530 cm<sup>-1</sup>. The summary and comparison of two levels' calculations are carried out in the following table.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.45093||-234.40234
|-
| Sum of electronic and thermal Energies||-231.44530||-234.39601
|-
| Sum of electronic and thermal Enthalpie|| -231.44436||-234.39506
|-
| Sum of electronic and thermal Free Energie||-231.47977||-234.43175
|-
| Generated IR spectrum||[[Image:XJB_boatir1.jpg|200px]]||[[Image:XJB_boatir2.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 10.''' ''Frequency calculations of the boat transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by QST2
([[File:XJB_boat ts1.log]], [[File:XJB_boat ts2.log]])''</div>

====Intrinsic Reaction Coordinate====

Next a useful method called '''Intrinct Reaction Coordinate (IRC)''' is applied to the Hessian optimized chair transition structure to follow the mininmum energy path (MEP) from a transition structure down to its local minimum on a potential energy surface. IRC can be used to verify that the transition state you found actually connects the reactants and products you think it does, or to identify any reaction intermediates you haven't thought about, or to generate an animation of the reaction [ref]. The symmetrical reaction coordinate was computed only in the forward direction with force constant always calculated and the number of points along IRC 50. The last structure generated along IRC is given in '''Figure 11''' with the energy calculated as '''-231.69158Ha''' and the dihedral angle and internal angle measured as D = -67.188<sup>o</sup> A = 111.78<sup>o</sup>. This energy does not match to any of the conformations discussed in Appendix 1 thus a minimum geometry has not reached yet.

{| align="center"
|- align="center"
|
<jmol><jmolApplet><title>XJB_IRC.mol</title><color>white</color>
<size>200</size><script>zoom=5</script>
<uploadedFileContents>XJB_IRC.mol</uploadedFileContents>
</jmolApplet></jmol>
|
[[Image:XJB_irc.jpg|500px]]
|- align="center"
| align="center" | '''Figure 11.''' ''The energy minimum structure along IRC''
| align="center" | '''Figure 12.''' ''Total energy along IRC''
|}

Three improving options can be used to reach the minimum energy. The first one is to run a normal Hessian optimization of the last point on IRC; the second one involves using a larger number of points (in this case 100 was used); the last one is to redo the IRC computation specifying to compute force constant at every step. Three methods were all used and the results were summarized in '''Table 11'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Improved IRC method 1
! scope="col" | Improved IRC method 2
! scope="col" | Improved IRC method 3
|-
| [[Image:XJB_IRC1.jpg|300px|center|IRC1]]||[[Image:XJB_IRC2.jpg|300px|center|IRC3]] || [[Image:XJB_IRC3.jpg|300px|center|IRC3]]
|-
| D = -64.17<sup>o</sup> A = 112.04<sup>o</sup> || D = -66.83<sup>o</sup> A = 111.82<sup>o</sup> || D = -64.17<sup>o</sup> A = 112.04<sup>o</sup>
|-
| -231.69167Ha || -231.69150Ha || -231.69167Ha
|-
| [[File:XJB_IRC1.log]]||[[File:XJB_IRC2.log]]||[[File:XJB_IRC3.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 11.''' ''Comparison of three improvement methods of IRC computation upon the chair transition structure''</div>

The first method was very quick but maybe generated relatively low-accuracy result; the second approach involved the controlling of number of points which might result in ending up at the wrong direction and wrong structure; the third method was the most suitable option for this small system but it took longer time than the previous two. From '''Table 11''' it is concluded that method 1 and 3 gave the same lowest energy which was consistent to the energy of the ''gauche2'' conformation given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

====Activation Energies====

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | '''HF/3-21G''' at 0K
! scope="col" | '''HF/3-21G''' at 298.15K
! scope="col" | '''B3LYP/6-31G*''' at 0K
! scope="col" | '''B3LYP/6-31G*''' at 298.15K
! scope="col" | Expt. at 0K
|-
| '''ΔE (chair)'''||45.71||44.70||34.05||33.16||33.5 ± 0.5
|-
| '''ΔE (boat)'''||55.60||54.76||41.96||41.32||44.7 ± 0.5
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 12.''' ''Summary of activation energies for the Cope rearrangement via both transition states in kcal/mol''</div>

The activation energy calculation was done by calculating the difference in energies between the optimized reactant ''anti2'' and the optimized chair and boat transition states. It is observed that chair conformation gives generally lower activation energy thus the more favoured transition state. The structure optimized at '''B3LYP/6-31G*''' level of theory at 0K has closer activation energy to the experimental result. Thus '''B3LYP/6-31G*''' gives a better estimation of transition state than '''HF/3-21G'''.

==The Diels Alder Cycloaddition==

In this exercise, AM1 semi-empirical method is used to compute the Diels Alder cycloaddition which is a concerted pericyclic reaction. The driving force of this reaction is the formation of a new σ bond between the π orbitals of the dieneophile and the diene. The reaction is allowed with HOMO-LUMO interation and a symmetry overlap.

===Cis Butadiene===

<center>[[Image:prototype Diels Alder.jpg|500px|center|thumb|'''Figure 13'''. ''The prototype reaction between butadiene and ethylene'']]</center>

To study a prototype Diels Alder cycloaddition reaction between butadiene and ethylene, first a cis butadiene molecule was built using Gaussview and optimized using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the butadiene were plotted. As seen the HOMO of cis-butadiene is anti-symmetric ('''a''') and the LUMO is symmetric ('''s''') with respect to plane.

{| align="center"
|- align="center"
|
[[Image:XJB_HOMO.jpg|280px|thumb|'''Figure 14.''' ''Anti-symmetric HOMO of cis-butadiene'']]
|
[[Image:XJB_LUMO.jpg|300px|thumb|'''Figure 15.''' ''Symmetric HOMO of cis-butadiene'']]
|- align="center"
|}

===The Transition State of prototype reaction between ethylene and butadiene===

The Hessian method was used to estimate transition state for the reaction between ethylene and butadiene because these two reactants are relatively small and simple molecules. The transition state was drawn by first building a structure as (a) in '''Figure 16''' then removing the -CH<sub>2</sub>-CH<sub>2</sub>- group to form the envelop structure as in (b). The '''clean''' option should not be used and the dashed bond length was guessed to be larger than 2.2 Å otherwise the geometry change would cause a failure in optimization.

<center>[[Image:XJB_dielsguessts.jpg|300px|center|thumb|'''Figure 16'''. ''The guessing transition state for the reaction between butadiene and ethylene'']]</center>

<center>[[File:XJB_butadiene and ethylene ts.GIF|400px|center|thumb|'''Figure 17.''' ''Animation of the butadiene and ethylene cycloaddition transition state'']]</center>

After the Hessian optimization, the transition structure has been successfully determined and an animation was generated in '''Figure 17'''. It has a C<sub>1</sub> symmetry point group and the bond lengths of two partly formed C-C bonds were measured to be 2.12Å. The typical sp<sup>3</sup> and sp<sup>2</sup> C-C bond lengths are 1.54Å and 1.47Å respectively[ref]. The van der Waals radius of the C atom is 1.7 Å[ref]. The measured bond distance of the transition structure is larger than typical bond lengths but still within the van der Waals radius, suggesting the bond forming interation between the carbon atoms.

From the animation ('''Figure 17''') the vibration directions of two molecules are opposite and they are approaching and leaving each other at the same time. This suggests the formation of the two bonds synchronous. The vibrational frequencies were also calculated and an infrared spectrum was simulated ('''Figure 20'''). An imaginary frequency of magnitude -955 cm<sup>-1</sup> was seen and the lowest positive frequency was observed at 147 cm<sup>-1</sup> which corresponded to the reactant vibration but not the bond formation.

{| align="center"
|
[[Image:XJB_HOMOts.jpg|300px|thumb|'''Figure 18'''. HOMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_LUMOts.jpg|300px|thumb|'''Figure 19'''. LUMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_IRts.jpg|300px|thumb|'''Figure 20'''. Generated IR of transition state of reaction between cis-butadiene and ethylene]]
|}

The HOMO and LUMO molecular orbitals of this transition structure were visualized in '''Figure 18''' and '''Figure 19'''. From these two figures it can be observed that the HOMO at the transition structure is anti-symmetric ('''a''') while the LUMO is symmetric ('''s'''). Thus this anti-symmetric transition state HOMO is formed by the HOMO of cis-butadiene and the LUMO of ethylene with the knowledge that they are both anti-symmetrical. This conclusion agrees with the previous talked conditions for allowed reaction.

===The cyclohexa-1,3-diene reaction with maleic anhydride===

<center>[[Image:XJB_mechanism3.jpg|500px|center|thumb|'''Figure 21'''. ''The endo and exo products of the Diels Alder reaction between cyclohexa-1,3-diene and maleic anhydride'']]</center>

The exo and endo transition states were drawn according to '''Figure 21''' with the new C-C bond formation line dashed and bond length estimation as 2.5. Then the same Hessian method was applied to optimize the structures. The electronic energy of the endo structure was calculated to be '''-0.05150Ha''' and '''-0.05042Ha''' for the exo. This reaction is supposed to be kinetically controlled thus the lower energy endo transition state is more favoured.

The bond lengths of the partly formed C-C bond are 2.16 Å and 2.17 Å for endo and exo transition states respectively. Another measured C-C bond length is shown in '''Table 13''' as 2.89 Å for endo (C3-C8 and C2-C7) and 2.95 Å for endo (C7-C10 and C8-C9). The longer atom distance indicates the larger steric repulsion in the exo transition state because of the extra existence of hydrogen atoms on C7 and C8.

Woodward and Hoffmann also proposed a secondary orbital overlap statement as an explanation of the endo stereo-preference in Diels Alder reactions<ref name="soot">''J. Org. Chem.'', 1987, '''52(8)''', pp 1469–1474. {{DOI|10.1021/jo00384a016}}</ref>. This secondary orbital overlap does not involve bond changes but π orbital interactions as the highlighted carbon atoms of the endo transition state in '''Table 13'''. Such overlap stablises the structure and lowers the energy of the endo transition state while there is no such effect existed in the exo transition state.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | endo
! scope="col" | exo
|-
| '''Animation'''||[[File:XJB_endots.GIF|300px]]||[[File:XJB_exots.GIF|300px]]
|-
| '''Sum of electronic and zero-point Energies'''||0.133493||0.134880
|-
| '''Sum of electronic and thermal Energies'''||0.143683||0.144881
|-
| '''Sum of electronic and thermal Enthalpies'''||0.144627||0.145825
|-
| '''Sum of electronic and thermal Free Energies'''||0.097349||0.099117
|-
| '''Bond lengths'''||[[Image:XJB_endomeasure.jpg|300px]]||[[Image:XJB_exomeasure.jpg|300px]]
|-
| '''HOMO'''||[[Image:XJB_endoHOMO.jpg|300px]]||[[Image:XJB_exoHOMO.jpg|285px]]
|-
| '''Secondary orbital overlap effect'''||[[Image:XJB_endosoo.jpg|300px]]||[[Image:XJB_exosoo.jpg|300px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 13.''' ''Summary of comparison of properties of endo and exo transition structures for reaction between cyclohexa-1,3-diene and maleic anhydride''</div>

==Reference==

<references/>

Rep:Mod:XJB1130

2013-12-06T12:54:30Z

Jx1011:

==The Cope Rearrangement of 1,5-hexadiene==

The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement reaction ('''Figure 1'''). Its mechanism has been studied by a large number of experimental and computational researches, which is recently accepted that the reaction occurs via either a "chair" or a "boat" transition state structure in a concerted manner. The aim of this exercise is to locate the low-energy minimum and the transition structures on the potential energy surface by Gaussian calculation in order to study its mechanism. The reaction pathway and the activation energy can also be calculated by this calculation.

<center>[[Image:XJB_cope rearrangement.jpg|500px|center|thumb|'''Figure 1'''. ''The Cope rearrangement'']]</center>

===Optimizing the Reactants and Products===

Different conformations of 1,5-hexadiene can be optimized using '''Gaussview'''. The optimized structure of four conformers involved in this discussion and their corresponding energies optimized at '''HF/3-21G''' level of theory are concluded in the following table ('''Table 1''').

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | anti4
! scope="col" | anti2
! scope="col" | gauche
! scope="col" | gauche3
|-
| <jmol><jmolApplet><title>XJB_anti4.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti4.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche3.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| -231.69097Ha || -231.69254Ha || -231.68772Ha ||-231.69266Ha
|}

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 1.''' ''Four conformers of 1,5-hexadiene''</div>

This exercise first involved optimizing a molecule of 1,5-hexadiene with "anti" linkage for the cetral four C atoms at the '''HF/3-21G''' level of theory. The final optimization structure had the total electronic energy of '''-231.69097Ha''' with a '''C<sub>1</sub>''' symmetry. By comparing the energy and point group to that in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1], this structure is confirmed to be ''anti4''. The compositions of energies are also listed in '''Table 2''' as i) the sum of electronic and zero-point energies (potential energy at 0K including the zero-point vibrational energy), ii) the sum of electronic and thermal energies (energy including translational, rotational and vibrational at 298.15K and 1 atm), iii) the sum of electronic and thermal enthalpies (containning an additional correction for RT), and iv) the sum of electronic and thermal free energies (including the entropic contribution to the free energy). For what we are considering, the first two terms are the most useful for further comparisons.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Formula Description'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||E = E<sub>elec</sub> + ZPE||-231.53785
|-
| Sum of electronic and thermal Energies||E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>||-231.53096
|-
| Sum of electronic and thermal Enthalpie||H = E + RT||-231.53002
|-
| Sum of electronic and thermal Free Energie||G = H - TS||-231.56891
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 2.''' ''Frequency calculations of anti4 optimized at '''HF/3-21G''' ([[File:XJB_anti4.log]])''</div>

Next another 1,5-hexadiene molecule with "gauche" linkage for the central four atoms was drawn and optimized at the '''HF/3-21G''' level of theory as before. Energy for this conformation is expected to be higher than that for the ''anti4'' conformation due to the closer position of two alkene ends thus the steric interaction. The final structure had the electronic energy of '''-231.68772Ha''' which was indeed higher than that of the ''anti4'', indicating less favour towards this gauche structure. It had a '''C<sub>2</sub>''' symmetry and was confirmed to be the ''gauche'' conformation.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53439
|-
| Sum of electronic and thermal Energies||-231.52770
|-
| Sum of electronic and thermal Enthalpie||-231.52676
|-
| Sum of electronic and thermal Free Energie||-231.56483
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 3.''' ''Frequency calculations of gauche optimized at '''HF/3-21G''' ([[File:XJB_gauche.log]])''</div>

A lowest energy conformation ''gauche3'' of the reactant molecule was drawn based on and confirmed by [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. The optimization gave a final structure with an energy of '''-231.69266Ha''' with a symmetry of '''C<sub>1</sub>'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53949
|-
| Sum of electronic and thermal Energies||-231.53265
|-
| Sum of electronic and thermal Enthalpie||-231.5317
|-
| Sum of electronic and thermal Free Energie||-231.57064
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 4.''' ''Frequency calculations of gauche3 optimized at '''HF/3-21G''' ([[File:XJB_gauche3.log]])''</div>

The next step required us to draw the C<sub>i</sub> symmetry ''anti2'' conformation of 1,5-hexadiene and optimize it at the '''HF/3-21G''' level of theory. The optimized energy was calculated to be '''-231.69254Ha''' with point group of '''C<sub>i</sub>''', which was consistent with the one given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53954
|-
| Sum of electronic and thermal Energies||-231.53257
|-
| Sum of electronic and thermal Enthalpie||-231.53162
|-
| Sum of electronic and thermal Free Energie||-231.57092
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 5.''' ''Frequency calculations of anti2 optimized at '''HF/3-21G''' ([[File:XJB_anti2.log]])''</div>

The ''anti2'' structure was then reoptimized at '''B3LYP/6-31G*''' which was a improvement over the '''HF/3-21G''' basis set. It adds polarization to atoms and improves the modeling of core electrons thus producing more accurate description of orbitals<ref name="soo">''Nigerian Journal of Chemical Research'', 2007, '''12''', pp 1469–1474. {{DOI|10.4314/njcr.v12i1.}}</ref>. The reoptimized ''anti2'' structure had the total electronic energy of '''-234.61171Ha''' and the same symmetry C<sub>i</sub>. The comparison of two structures optimized using different levels of theory is summarized in '''Table 6'''. We can see that they have exactly the same dihedral angle of the central four C atoms at 180.00<sup>o</sup> and angles between three of them are very similar as 111.3<sup>o</sup> and 112.7<sup>o</sup> respectively. Thus the the geometry change using '''HF/3-21G''' or '''B3LYP/6-31G*''' is trivial.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Before re-optimization
! scope="col" | After re-optimization
|-
| <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2reoptimized</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2reoptimized.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| ''anti2 optimized at '''HF/3-21G''''' || ''anti2 optimized at '''B3LYP/6-31G*'''''
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 6.''' ''Two optimization of anti2 at '''HF/3-21G''' and '''B3LYP/6-31G*'''([[File:XJB_anti2.log]])''</div>

The frequency calculation was run as well and a list of energies is summarized in '''Table 7'''. A list of all the vibrational frequencies was also generated and no imaginary (negative) frequency was observed and these vibrations are visualized by simulating the infrared spectrum in '''Figure 2'''.

{| align="center"
|- align="center"
|
{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-234.46920
|-
| Sum of electronic and thermal Energies||-234.46186
|-
| Sum of electronic and thermal Enthalpie||-234.46091
|-
| Sum of electronic and thermal Free Energie||-234.50078
|}
|
[[Image:XJB_IR_anti2.jpg|300px]]
|- align="center"
| align="center" | '''Table 7.''' ''Frequency calculations of anti2 optimized at '''B3LYP/6-31G*'''
([[File:XJB_anti2_reoptimized.log]])''
| align="center" | '''Figure 2.''' ''The generated IR spectrum of optimized '''B3LYP/6-31G*''' structure''
|}

===Optimizing the "Chair" and "Boat" Transition Structures===

The study of the chair and boat transition states is carried out using three different approaches. The first one is called the Hessian which involves computing the force constant matrix; the second one is to freeze the reaction coordinate and then optimize the structure once the molecule is fully relaxed; the last one is to use QST2 method to specify the reactants and products then finding the transition structure.

<center>[[Image:XJB_2 ts.jpg|500px|center|thumb|'''Figure 3'''. ''The two transition states of Cope rearrangement'']]</center>

====The Chair Transition State====

In order to set up a transition state optimization, first an allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimized at the '''HF/3-21G''' level of theory. Then this half of the transition structure was duplicated with rotation to make the approximate resembling of the chair transition state with terminal ends of the fragments 2.2 Å apart. The chair transition structure optimization was then done by the first two different approaches.

<center>[[Image:XJB_guess ts.jpg|400px|center|thumb|'''Figure 4'''. ''The guessed chair transition structure of two allyl fragments'']]</center>

For the first optimization approach, the force constants were computed (the Hessian) and the chair transition state was optimized to a TS (Berny) at the '''HF/3-21G''' level of theory. The frequency calculation was carried out at the same time and gave an imaginary frequency at -818 cm<sup>-1</sup>. The electronic energy was calculated to be '''-231.61932Ha''' with C<sub>2h</sub> symmetry point group. The distance between the terminal carbons was 2.0203 Å. An animation of this transition state vibrations was generated to confirm the corresponding Cope rearrangement.

<center>[[File:XJB_chair ts animation.GIF|400px|center|thumb|'''Figure 5.''' ''Animation of the Cope rearrangement via the chair transition state'']]</center>

The chair transition state was then reoptimized using the '''B3LYP/6-31G*''' level of theory and carried out frequency calculations as well. The electronic energy calculated this time was '''-234.55698Ha''' with an imaginary frequency at -566 cm<sup>-1</sup>. We can see that with the geometries quite similar, the energies differ markedly calculated using two levels of theory. The summary and comparison of the calculations optimized at two levels of theory are summarized in '''Table 8'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46670||-234.41493
|-
| Sum of electronic and thermal Energies||-231.46134||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40807
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir1.jpg|200px]]||[[Image:XJB_chairir2.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 8.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the first approach
([[File:XJB_chairts1.1.log]], [[File:XJB_chairts1.2.log]])''</div>

The second approach is the frozen coordinate method using the Redundant Coord Editor to freeze the bond lengths of the terminal ends to 2.2 Å. First the calculation failed because of the recent update of GaussView. After editting the coordinate distance (B 5 1 2.2 F) and the frozen distance (B 5 1 F) the optimization went well this time. The structure was then optimized again using a normal guess Hessian modified including the two differentiating coordinates. This chair transition state was then reoptimized again using the '''B3LYP/6-31G*''' level of theory. The electronic energies were calculated to be the same as in the first approach and frequency calculations were summarized in '''Table 9'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46671||-234.41490
|-
| Sum of electronic and thermal Energies||-231.46135||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40806
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir3.jpg|200px]]||[[Image:XJB_chairir4.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 9.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the second approach
([[File:XJB_chairts2.1.log]], [[File:XJB_chairts2.2.log]])''</div>

As the same energy results given, the bond forming/breaking bond lengths in the chair transition structures in the above two different approaches were compared. The first approach gave the bond lengths of 2.0203 Å while the second approach gave 2.0207 Å. Considering the accuracy limit of bond length estimation in Gaussview, these two bond lengths can be seen as exactly the same. Two methods worked reasonably well because of the good assumption of transition structures. However for more complicated and larger molecule, the Hessian method may fail due to the difficulty to predict the transition geometry (different surface curvature). Thus better way to generate such transition structures is to freeze reaction coordinate.

{| align="center"
|- align="center"
|
[[Image:XJB_chairts bondlength1.jpg|350px|thumb|'''Figure 6.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by Hession'']]
|
[[Image:XJB_chairts bondlength2.jpg|350px|thumb|'''Figure 7.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by frozen coordinate'']]
|- align="center"
|}

====The Boat Transition State====

The boat transition structure was then optimized using the '''QST2''' method. The same numbering for the reactant and product molecules was important in order to specify for this reaction. The renumbering of the molecules is shown as in '''Figure 8''', but the following optimization failed and gave a twisted structure. The reactant and product geometries were then modified further by changing the central C-C-C-C dihedral angle to 0<sup>o</sup> and the inside two C-C-C (i.e. C2-C3-C4 and C3-C4-C5) angles to 100<sup>o</sup>. The job was run successfully and frequency calculations were done with the boat transition state generated. Only one imaginary frequency was detected also at -840 cm<sup>-1</sup>. The electronic energy was '''-231.60280Ha''' with C<sub>2v</sub> symmetry. The distances between the terminal carbons of the allyl fragments were examined to be both 2.140Å which were comparably longer than that of the chair transition structure.

{| align="center"
|- align="center"
|
[[Image:XJB_QST1.jpg|400px|thumb|'''Figure 8.''' ''Numbering of reactant and product'']]
|
[[Image:XJB_QST2.jpg|410px|thumb|'''Figure 9.''' ''Re-numbering of reactant and product'']]
|- align="center"
|}

<center>[[File:XJB_boat ts animation.GIF|400px|center|thumb|'''Figure 10.''' ''Animation of the Cope rearrangement via the boat transition state'']]</center>

The structure was then reoptimized at the higher '''B3LYP/6-31G*''' theory of level and the electronic energy was calculated to be '''-234.54309Ha''' with very similar geometry. The frequency calculation generated an imaginary frequency at -530 cm<sup>-1</sup>. The summary and comparison of two levels' calculations are carried out in the following table.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.45093||-234.40234
|-
| Sum of electronic and thermal Energies||-231.44530||-234.39601
|-
| Sum of electronic and thermal Enthalpie|| -231.44436||-234.39506
|-
| Sum of electronic and thermal Free Energie||-231.47977||-234.43175
|-
| Generated IR spectrum||[[Image:XJB_boatir1.jpg|200px]]||[[Image:XJB_boatir2.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 10.''' ''Frequency calculations of the boat transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by QST2
([[File:XJB_boat ts1.log]], [[File:XJB_boat ts2.log]])''</div>

====Intrinsic Reaction Coordinate====

Next a useful method called '''Intrinct Reaction Coordinate (IRC)''' is applied to the Hessian optimized chair transition structure to follow the mininmum energy path (MEP) from a transition structure down to its local minimum on a potential energy surface. IRC can be used to verify that the transition state you found actually connects the reactants and products you think it does, or to identify any reaction intermediates you haven't thought about, or to generate an animation of the reaction [ref]. The symmetrical reaction coordinate was computed only in the forward direction with force constant always calculated and the number of points along IRC 50. The last structure generated along IRC is given in '''Figure 11''' with the energy calculated as '''-231.69158Ha''' and the dihedral angle and internal angle measured as D = -67.188<sup>o</sup> A = 111.78<sup>o</sup>. This energy does not match to any of the conformations discussed in Appendix 1 thus a minimum geometry has not reached yet.

{| align="center"
|- align="center"
|
<jmol><jmolApplet><title>XJB_IRC.mol</title><color>white</color>
<size>200</size><script>zoom=5</script>
<uploadedFileContents>XJB_IRC.mol</uploadedFileContents>
</jmolApplet></jmol>
|
[[Image:XJB_irc.jpg|500px]]
|- align="center"
| align="center" | '''Figure 11.''' ''The energy minimum structure along IRC''
| align="center" | '''Figure 12.''' ''Total energy along IRC''
|}

Three improving options can be used to reach the minimum energy. The first one is to run a normal Hessian optimization of the last point on IRC; the second one involves using a larger number of points (in this case 100 was used); the last one is to redo the IRC computation specifying to compute force constant at every step. Three methods were all used and the results were summarized in '''Table 11'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Improved IRC method 1
! scope="col" | Improved IRC method 2
! scope="col" | Improved IRC method 3
|-
| [[Image:XJB_IRC1.jpg|300px|center|IRC1]]||[[Image:XJB_IRC2.jpg|300px|center|IRC3]] || [[Image:XJB_IRC3.jpg|300px|center|IRC3]]
|-
| D = -64.17<sup>o</sup> A = 112.04<sup>o</sup> || D = -66.83<sup>o</sup> A = 111.82<sup>o</sup> || D = -64.17<sup>o</sup> A = 112.04<sup>o</sup>
|-
| -231.69167Ha || -231.69150Ha || -231.69167Ha
|-
| [[File:XJB_IRC1.log]]||[[File:XJB_IRC2.log]]||[[File:XJB_IRC3.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 11.''' ''Comparison of three improvement methods of IRC computation upon the chair transition structure''</div>

The first method was very quick but maybe generated relatively low-accuracy result; the second approach involved the controlling of number of points which might result in ending up at the wrong direction and wrong structure; the third method was the most suitable option for this small system but it took longer time than the previous two. From '''Table 11''' it is concluded that method 1 and 3 gave the same lowest energy which was consistent to the energy of the ''gauche2'' conformation given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

====Activation Energies====

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | '''HF/3-21G''' at 0K
! scope="col" | '''HF/3-21G''' at 298.15K
! scope="col" | '''B3LYP/6-31G*''' at 0K
! scope="col" | '''B3LYP/6-31G*''' at 298.15K
! scope="col" | Expt. at 0K
|-
| '''ΔE (chair)'''||45.71||44.70||34.05||33.16||33.5 ± 0.5
|-
| '''ΔE (boat)'''||55.60||54.76||41.96||41.32||44.7 ± 0.5
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 12.''' ''Summary of activation energies for the Cope rearrangement via both transition states in kcal/mol''</div>

The activation energy calculation was done by calculating the difference in energies between the optimized reactant ''anti2'' and the optimized chair and boat transition states. It is observed that chair conformation gives generally lower activation energy thus the more favoured transition state. The structure optimized at '''B3LYP/6-31G*''' level of theory at 0K has closer activation energy to the experimental result. Thus '''B3LYP/6-31G*''' gives a better estimation of transition state than '''HF/3-21G'''.

==The Diels Alder Cycloaddition==

In this exercise, AM1 semi-empirical method is used to compute the Diels Alder cycloaddition which is a concerted pericyclic reaction. The driving force of this reaction is the formation of a new σ bond between the π orbitals of the dieneophile and the diene. The reaction is allowed with HOMO-LUMO interation and a symmetry overlap.

===Cis Butadiene===

<center>[[Image:prototype Diels Alder.jpg|500px|center|thumb|'''Figure 13'''. ''The prototype reaction between butadiene and ethylene'']]</center>

To study a prototype Diels Alder cycloaddition reaction between butadiene and ethylene, first a cis butadiene molecule was built using Gaussview and optimized using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the butadiene were plotted. As seen the HOMO of cis-butadiene is anti-symmetric ('''a''') and the LUMO is symmetric ('''s''') with respect to plane.

{| align="center"
|- align="center"
|
[[Image:XJB_HOMO.jpg|280px|thumb|'''Figure 14.''' ''Anti-symmetric HOMO of cis-butadiene'']]
|
[[Image:XJB_LUMO.jpg|300px|thumb|'''Figure 15.''' ''Symmetric HOMO of cis-butadiene'']]
|- align="center"
|}

===The Transition State of prototype reaction between ethylene and butadiene===

The Hessian method was used to estimate transition state for the reaction between ethylene and butadiene because these two reactants are relatively small and simple molecules. The transition state was drawn by first building a structure as (a) in '''Figure 16''' then removing the -CH<sub>2</sub>-CH<sub>2</sub>- group to form the envelop structure as in (b). The '''clean''' option should not be used and the dashed bond length was guessed to be larger than 2.2 Å otherwise the geometry change would cause a failure in optimization.

<center>[[Image:XJB_dielsguessts.jpg|300px|center|thumb|'''Figure 16'''. ''The guessing transition state for the reaction between butadiene and ethylene'']]</center>

<center>[[File:XJB_butadiene and ethylene ts.GIF|400px|center|thumb|'''Figure 17.''' ''Animation of the butadiene and ethylene cycloaddition transition state'']]</center>

After the Hessian optimization, the transition structure has been successfully determined and an animation was generated in '''Figure 17'''. It has a C<sub>1</sub> symmetry point group and the bond lengths of two partly formed C-C bonds were measured to be 2.12Å. The typical sp<sup>3</sup> and sp<sup>2</sup> C-C bond lengths are 1.54Å and 1.47Å respectively[ref]. The van der Waals radius of the C atom is 1.7 Å[ref]. The measured bond distance of the transition structure is larger than typical bond lengths but still within the van der Waals radius, suggesting the bond forming interation between the carbon atoms.

From the animation ('''Figure 17''') the vibration directions of two molecules are opposite and they are approaching and leaving each other at the same time. This suggests the formation of the two bonds synchronous. The vibrational frequencies were also calculated and an infrared spectrum was simulated ('''Figure 20'''). An imaginary frequency of magnitude -955 cm<sup>-1</sup> was seen and the lowest positive frequency was observed at 147 cm<sup>-1</sup> which corresponded to the reactant vibration but not the bond formation.

{| align="center"
|
[[Image:XJB_HOMOts.jpg|300px|thumb|'''Figure 18'''. HOMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_LUMOts.jpg|300px|thumb|'''Figure 19'''. LUMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_IRts.jpg|300px|thumb|'''Figure 20'''. Generated IR of transition state of reaction between cis-butadiene and ethylene]]
|}

The HOMO and LUMO molecular orbitals of this transition structure were visualized in '''Figure 18''' and '''Figure 19'''. From these two figures it can be observed that the HOMO at the transition structure is anti-symmetric ('''a''') while the LUMO is symmetric ('''s'''). Thus this anti-symmetric transition state HOMO is formed by the HOMO of cis-butadiene and the LUMO of ethylene with the knowledge that they are both anti-symmetrical. This conclusion agrees with the previous talked conditions for allowed reaction.

===The cyclohexa-1,3-diene reaction with maleic anhydride===

<center>[[Image:XJB_mechanism3.jpg|500px|center|thumb|'''Figure 21'''. ''The endo and exo products of the Diels Alder reaction between cyclohexa-1,3-diene and maleic anhydride'']]</center>

The exo and endo transition states were drawn according to '''Figure 21''' with the new C-C bond formation line dashed and bond length estimation as 2.5. Then the same Hessian method was applied to optimize the structures. The electronic energy of the endo structure was calculated to be '''-0.05150Ha''' and '''-0.05042Ha''' for the exo. This reaction is supposed to be kinetically controlled thus the lower energy endo transition state is more favoured.

The bond lengths of the partly formed C-C bond are 2.16 Å and 2.17 Å for endo and exo transition states respectively. Another measured C-C bond length is shown in '''Table 13''' as 2.89 Å for endo (C3-C8 and C2-C7) and 2.95 Å for endo (C7-C10 and C8-C9). The longer atom distance indicates the larger steric repulsion in the exo transition state because of the extra existence of hydrogen atoms on C7 and C8.

Woodward and Hoffmann also proposed a secondary orbital overlap statement as an explanation of the endo stereo-preference in Diels Alder reactions<ref name="soo">''J. Org. Chem.'', 1987, '''52(8)''', pp 1469–1474. {{DOI|10.1021/jo00384a016}}</ref>. This secondary orbital overlap does not involve bond changes but π orbital interactions as the highlighted carbon atoms of the endo transition state in '''Table 13'''. Such overlap stablises the structure and lowers the energy of the endo transition state while there is no such effect existed in the exo transition state.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | endo
! scope="col" | exo
|-
| '''Animation'''||[[File:XJB_endots.GIF|300px]]||[[File:XJB_exots.GIF|300px]]
|-
| '''Sum of electronic and zero-point Energies'''||0.133493||0.134880
|-
| '''Sum of electronic and thermal Energies'''||0.143683||0.144881
|-
| '''Sum of electronic and thermal Enthalpies'''||0.144627||0.145825
|-
| '''Sum of electronic and thermal Free Energies'''||0.097349||0.099117
|-
| '''Bond lengths'''||[[Image:XJB_endomeasure.jpg|300px]]||[[Image:XJB_exomeasure.jpg|300px]]
|-
| '''HOMO'''||[[Image:XJB_endoHOMO.jpg|300px]]||[[Image:XJB_exoHOMO.jpg|285px]]
|-
| '''Secondary orbital overlap effect'''||[[Image:XJB_endosoo.jpg|300px]]||[[Image:XJB_exosoo.jpg|300px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 13.''' ''Summary of comparison of properties of endo and exo transition structures for reaction between cyclohexa-1,3-diene and maleic anhydride''</div>

==Reference==

<references/>

Rep:Mod:XJB1130

2013-12-06T12:43:56Z

Jx1011: /* The Cope Rearrangement of 1,5-hexadiene */

==The Cope Rearrangement of 1,5-hexadiene==

The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement reaction ('''Figure 1'''). Its mechanism has been studied by a large number of experimental and computational researches, which is recently accepted that the reaction occurs via either a "chair" or a "boat" transition state structure in a concerted manner. The aim of this exercise is to locate the low-energy minimum and the transition structures on the potential energy surface by Gaussian calculation in order to study its mechanism. The reaction pathway and the activation energy can also be calculated by this calculation.

<center>[[Image:XJB_cope rearrangement.jpg|500px|center|thumb|'''Figure 1'''. ''The Cope rearrangement'']]</center>

===Optimizing the Reactants and Products===

Different conformations of 1,5-hexadiene can be optimized using '''Gaussview'''. The optimized structure of four conformers involved in this discussion and their corresponding energies optimized at '''HF/3-21G''' level of theory are concluded in the following table ('''Table 1''').

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | anti4
! scope="col" | anti2
! scope="col" | gauche
! scope="col" | gauche3
|-
| <jmol><jmolApplet><title>XJB_anti4.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti4.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche3.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| -231.69097Ha || -231.69254Ha || -231.68772Ha ||-231.69266Ha
|}

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 1.''' ''Four conformers of 1,5-hexadiene''</div>

This exercise first involved optimizing a molecule of 1,5-hexadiene with "anti" linkage for the cetral four C atoms at the '''HF/3-21G''' level of theory. The final optimization structure had the total electronic energy of '''-231.69097Ha''' with a '''C<sub>1</sub>''' symmetry. By comparing the energy and point group to that in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1], this structure is confirmed to be ''anti4''. The compositions of energies are also listed in '''Table 2''' as i) the sum of electronic and zero-point energies (potential energy at 0K including the zero-point vibrational energy), ii) the sum of electronic and thermal energies (energy including translational, rotational and vibrational at 298.15K and 1 atm), iii) the sum of electronic and thermal enthalpies (containning an additional correction for RT), and iv) the sum of electronic and thermal free energies (including the entropic contribution to the free energy). For what we are considering, the first two terms are the most useful for further comparisons.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Formula Description'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||E = E<sub>elec</sub> + ZPE||-231.53785
|-
| Sum of electronic and thermal Energies||E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>||-231.53096
|-
| Sum of electronic and thermal Enthalpie||H = E + RT||-231.53002
|-
| Sum of electronic and thermal Free Energie||G = H - TS||-231.56891
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 2.''' ''Frequency calculations of anti4 optimized at '''HF/3-21G''' ([[File:XJB_anti4.log]])''</div>

Next another 1,5-hexadiene molecule with "gauche" linkage for the central four atoms was drawn and optimized at the '''HF/3-21G''' level of theory as before. Energy for this conformation is expected to be higher than that for the ''anti4'' conformation due to the closer position of two alkene ends thus the steric interaction. The final structure had the electronic energy of '''-231.68772Ha''' which was indeed higher than that of the ''anti4'', indicating less favour towards this gauche structure. It had a '''C<sub>2</sub>''' symmetry and was confirmed to be the ''gauche'' conformation.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53439
|-
| Sum of electronic and thermal Energies||-231.52770
|-
| Sum of electronic and thermal Enthalpie||-231.52676
|-
| Sum of electronic and thermal Free Energie||-231.56483
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 3.''' ''Frequency calculations of gauche optimized at '''HF/3-21G''' ([[File:XJB_gauche.log]])''</div>

A lowest energy conformation ''gauche3'' of the reactant molecule was drawn based on and confirmed by [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. The optimization gave a final structure with an energy of '''-231.69266Ha''' with a symmetry of '''C<sub>1</sub>'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53949
|-
| Sum of electronic and thermal Energies||-231.53265
|-
| Sum of electronic and thermal Enthalpie||-231.5317
|-
| Sum of electronic and thermal Free Energie||-231.57064
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 4.''' ''Frequency calculations of gauche3 optimized at '''HF/3-21G''' ([[File:XJB_gauche3.log]])''</div>

The next step required us to draw the C<sub>i</sub> symmetry ''anti2'' conformation of 1,5-hexadiene and optimize it at the '''HF/3-21G''' level of theory. The optimized energy was calculated to be '''-231.69254Ha''' with point group of '''C<sub>i</sub>''', which was consistent with the one given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53954
|-
| Sum of electronic and thermal Energies||-231.53257
|-
| Sum of electronic and thermal Enthalpie||-231.53162
|-
| Sum of electronic and thermal Free Energie||-231.57092
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 5.''' ''Frequency calculations of anti2 optimized at '''HF/3-21G''' ([[File:XJB_anti2.log]])''</div>

The ''anti2'' structure was then reoptimized at '''B3LYP/6-31G*''' which was a improvement over the '''HF/3-21G''' basis set. It adds polarization to all atoms and improves the modeling of core electrons thus producing more accurate description of orbitals[ref]. The reoptimized ''anti2'' structure had the total electronic energy of '''-234.61171Ha''' and the same symmetry C<sub>i</sub>. The comparison of two structures optimized using different levels of theory is summarized in '''Table 6'''. We can see that they have exactly the same dihedral angle of the central four C atoms at 180.00<sup>o</sup> and angles between three of them are very similar as 111.3<sup>o</sup> and 112.7<sup>o</sup> respectively. Thus the the geometry change using '''HF/3-21G''' or '''B3LYP/6-31G*''' is trivial.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Before re-optimization
! scope="col" | After re-optimization
|-
| <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2reoptimized</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2reoptimized.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| ''anti2 optimized at '''HF/3-21G''''' || ''anti2 optimized at '''B3LYP/6-31G*'''''
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 6.''' ''Two optimization of anti2 at '''HF/3-21G''' and '''B3LYP/6-31G*'''([[File:XJB_anti2.log]])''</div>

The frequency calculation was run as well and a list of energies is summarized in '''Table 7'''. A list of all the vibrational frequencies was also generated and no imaginary (negative) frequency was observed and these vibrations are visualized by simulating the infrared spectrum in '''Figure 2'''.

{| align="center"
|- align="center"
|
{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-234.46920
|-
| Sum of electronic and thermal Energies||-234.46186
|-
| Sum of electronic and thermal Enthalpie||-234.46091
|-
| Sum of electronic and thermal Free Energie||-234.50078
|}
|
[[Image:XJB_IR_anti2.jpg|300px]]
|- align="center"
| align="center" | '''Table 7.''' ''Frequency calculations of anti2 optimized at '''B3LYP/6-31G*'''
([[File:XJB_anti2_reoptimized.log]])''
| align="center" | '''Figure 2.''' ''The generated IR spectrum of optimized '''B3LYP/6-31G*''' structure''
|}

===Optimizing the "Chair" and "Boat" Transition Structures===

The study of the chair and boat transition states is carried out using three different approaches. The first one is called the Hessian which involves computing the force constant matrix; the second one is to freeze the reaction coordinate and then optimize the structure once the molecule is fully relaxed; the last one is to use QST2 method to specify the reactants and products then finding the transition structure.

<center>[[Image:XJB_2 ts.jpg|500px|center|thumb|'''Figure 3'''. ''The two transition states of Cope rearrangement'']]</center>

====The Chair Transition State====

In order to set up a transition state optimization, first an allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimized at the '''HF/3-21G''' level of theory. Then this half of the transition structure was duplicated with rotation to make the approximate resembling of the chair transition state with terminal ends of the fragments 2.2 Å apart. The chair transition structure optimization was then done by the first two different approaches.

<center>[[Image:XJB_guess ts.jpg|400px|center|thumb|'''Figure 4'''. ''The guessed chair transition structure of two allyl fragments'']]</center>

For the first optimization approach, the force constants were computed (the Hessian) and the chair transition state was optimized to a TS (Berny) at the '''HF/3-21G''' level of theory. The frequency calculation was carried out at the same time and gave an imaginary frequency at -818 cm<sup>-1</sup>. The electronic energy was calculated to be '''-231.61932Ha''' with C<sub>2h</sub> symmetry point group. The distance between the terminal carbons was 2.0203 Å. An animation of this transition state vibrations was generated to confirm the corresponding Cope rearrangement.

<center>[[File:XJB_chair ts animation.GIF|400px|center|thumb|'''Figure 5.''' ''Animation of the Cope rearrangement via the chair transition state'']]</center>

The chair transition state was then reoptimized using the '''B3LYP/6-31G*''' level of theory and carried out frequency calculations as well. The electronic energy calculated this time was '''-234.55698Ha''' with an imaginary frequency at -566 cm<sup>-1</sup>. We can see that with the geometries quite similar, the energies differ markedly calculated using two levels of theory. The summary and comparison of the calculations optimized at two levels of theory are summarized in '''Table 8'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46670||-234.41493
|-
| Sum of electronic and thermal Energies||-231.46134||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40807
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir1.jpg|200px]]||[[Image:XJB_chairir2.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 8.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the first approach
([[File:XJB_chairts1.1.log]], [[File:XJB_chairts1.2.log]])''</div>

The second approach is the frozen coordinate method using the Redundant Coord Editor to freeze the bond lengths of the terminal ends to 2.2 Å. First the calculation failed because of the recent update of GaussView. After editting the coordinate distance (B 5 1 2.2 F) and the frozen distance (B 5 1 F) the optimization went well this time. The structure was then optimized again using a normal guess Hessian modified including the two differentiating coordinates. This chair transition state was then reoptimized again using the '''B3LYP/6-31G*''' level of theory. The electronic energies were calculated to be the same as in the first approach and frequency calculations were summarized in '''Table 9'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46671||-234.41490
|-
| Sum of electronic and thermal Energies||-231.46135||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40806
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir3.jpg|200px]]||[[Image:XJB_chairir4.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 9.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the second approach
([[File:XJB_chairts2.1.log]], [[File:XJB_chairts2.2.log]])''</div>

As the same energy results given, the bond forming/breaking bond lengths in the chair transition structures in the above two different approaches were compared. The first approach gave the bond lengths of 2.0203 Å while the second approach gave 2.0207 Å. Considering the accuracy limit of bond length estimation in Gaussview, these two bond lengths can be seen as exactly the same. Two methods worked reasonably well because of the good assumption of transition structures. However for more complicated and larger molecule, the Hessian method may fail due to the difficulty to predict the transition geometry (different surface curvature). Thus better way to generate such transition structures is to freeze reaction coordinate.

{| align="center"
|- align="center"
|
[[Image:XJB_chairts bondlength1.jpg|350px|thumb|'''Figure 6.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by Hession'']]
|
[[Image:XJB_chairts bondlength2.jpg|350px|thumb|'''Figure 7.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by frozen coordinate'']]
|- align="center"
|}

====The Boat Transition State====

The boat transition structure was then optimized using the '''QST2''' method. The same numbering for the reactant and product molecules was important in order to specify for this reaction. The renumbering of the molecules is shown as in '''Figure 8''', but the following optimization failed and gave a twisted structure. The reactant and product geometries were then modified further by changing the central C-C-C-C dihedral angle to 0<sup>o</sup> and the inside two C-C-C (i.e. C2-C3-C4 and C3-C4-C5) angles to 100<sup>o</sup>. The job was run successfully and frequency calculations were done with the boat transition state generated. Only one imaginary frequency was detected also at -840 cm<sup>-1</sup>. The electronic energy was '''-231.60280Ha''' with C<sub>2v</sub> symmetry. The distances between the terminal carbons of the allyl fragments were examined to be both 2.140Å which were comparably longer than that of the chair transition structure.

{| align="center"
|- align="center"
|
[[Image:XJB_QST1.jpg|400px|thumb|'''Figure 8.''' ''Numbering of reactant and product'']]
|
[[Image:XJB_QST2.jpg|410px|thumb|'''Figure 9.''' ''Re-numbering of reactant and product'']]
|- align="center"
|}

<center>[[File:XJB_boat ts animation.GIF|400px|center|thumb|'''Figure 10.''' ''Animation of the Cope rearrangement via the boat transition state'']]</center>

The structure was then reoptimized at the higher '''B3LYP/6-31G*''' theory of level and the electronic energy was calculated to be '''-234.54309Ha''' with very similar geometry. The frequency calculation generated an imaginary frequency at -530 cm<sup>-1</sup>. The summary and comparison of two levels' calculations are carried out in the following table.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.45093||-234.40234
|-
| Sum of electronic and thermal Energies||-231.44530||-234.39601
|-
| Sum of electronic and thermal Enthalpie|| -231.44436||-234.39506
|-
| Sum of electronic and thermal Free Energie||-231.47977||-234.43175
|-
| Generated IR spectrum||[[Image:XJB_boatir1.jpg|200px]]||[[Image:XJB_boatir2.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 10.''' ''Frequency calculations of the boat transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by QST2
([[File:XJB_boat ts1.log]], [[File:XJB_boat ts2.log]])''</div>

====Intrinsic Reaction Coordinate====

Next a useful method called '''Intrinct Reaction Coordinate (IRC)''' is applied to the Hessian optimized chair transition structure to follow the mininmum energy path (MEP) from a transition structure down to its local minimum on a potential energy surface. IRC can be used to verify that the transition state you found actually connects the reactants and products you think it does, or to identify any reaction intermediates you haven't thought about, or to generate an animation of the reaction [ref]. The symmetrical reaction coordinate was computed only in the forward direction with force constant always calculated and the number of points along IRC 50. The last structure generated along IRC is given in '''Figure 11''' with the energy calculated as '''-231.69158Ha''' and the dihedral angle and internal angle measured as D = -67.188<sup>o</sup> A = 111.78<sup>o</sup>. This energy does not match to any of the conformations discussed in Appendix 1 thus a minimum geometry has not reached yet.

{| align="center"
|- align="center"
|
<jmol><jmolApplet><title>XJB_IRC.mol</title><color>white</color>
<size>200</size><script>zoom=5</script>
<uploadedFileContents>XJB_IRC.mol</uploadedFileContents>
</jmolApplet></jmol>
|
[[Image:XJB_irc.jpg|500px]]
|- align="center"
| align="center" | '''Figure 11.''' ''The energy minimum structure along IRC''
| align="center" | '''Figure 12.''' ''Total energy along IRC''
|}

Three improving options can be used to reach the minimum energy. The first one is to run a normal Hessian optimization of the last point on IRC; the second one involves using a larger number of points (in this case 100 was used); the last one is to redo the IRC computation specifying to compute force constant at every step. Three methods were all used and the results were summarized in '''Table 11'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Improved IRC method 1
! scope="col" | Improved IRC method 2
! scope="col" | Improved IRC method 3
|-
| [[Image:XJB_IRC1.jpg|300px|center|IRC1]]||[[Image:XJB_IRC2.jpg|300px|center|IRC3]] || [[Image:XJB_IRC3.jpg|300px|center|IRC3]]
|-
| D = -64.17<sup>o</sup> A = 112.04<sup>o</sup> || D = -66.83<sup>o</sup> A = 111.82<sup>o</sup> || D = -64.17<sup>o</sup> A = 112.04<sup>o</sup>
|-
| -231.69167Ha || -231.69150Ha || -231.69167Ha
|-
| [[File:XJB_IRC1.log]]||[[File:XJB_IRC2.log]]||[[File:XJB_IRC3.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 11.''' ''Comparison of three improvement methods of IRC computation upon the chair transition structure''</div>

The first method was very quick but maybe generated relatively low-accuracy result; the second approach involved the controlling of number of points which might result in ending up at the wrong direction and wrong structure; the third method was the most suitable option for this small system but it took longer time than the previous two. From '''Table 11''' it is concluded that method 1 and 3 gave the same lowest energy which was consistent to the energy of the ''gauche2'' conformation given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

====Activation Energies====

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | '''HF/3-21G''' at 0K
! scope="col" | '''HF/3-21G''' at 298.15K
! scope="col" | '''B3LYP/6-31G*''' at 0K
! scope="col" | '''B3LYP/6-31G*''' at 298.15K
! scope="col" | Expt. at 0K
|-
| '''ΔE (chair)'''||45.71||44.70||34.05||33.16||33.5 ± 0.5
|-
| '''ΔE (boat)'''||55.60||54.76||41.96||41.32||44.7 ± 0.5
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 12.''' ''Summary of activation energies for the Cope rearrangement via both transition states in kcal/mol''</div>

The activation energy calculation was done by calculating the difference in energies between the optimized reactant ''anti2'' and the optimized chair and boat transition states. It is observed that chair conformation gives generally lower activation energy thus the more favoured transition state. The structure optimized at '''B3LYP/6-31G*''' level of theory at 0K has closer activation energy to the experimental result. Thus '''B3LYP/6-31G*''' gives a better estimation of transition state than '''HF/3-21G'''.

==The Diels Alder Cycloaddition==

In this exercise, AM1 semi-empirical method is used to compute the Diels Alder cycloaddition which is a concerted pericyclic reaction. The driving force of this reaction is the formation of a new σ bond between the π orbitals of the dieneophile and the diene. The reaction is allowed with HOMO-LUMO interation and a symmetry overlap.

===Cis Butadiene===

<center>[[Image:prototype Diels Alder.jpg|500px|center|thumb|'''Figure 13'''. ''The prototype reaction between butadiene and ethylene'']]</center>

To study a prototype Diels Alder cycloaddition reaction between butadiene and ethylene, first a cis butadiene molecule was built using Gaussview and optimized using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the butadiene were plotted. As seen the HOMO of cis-butadiene is anti-symmetric ('''a''') and the LUMO is symmetric ('''s''') with respect to plane.

{| align="center"
|- align="center"
|
[[Image:XJB_HOMO.jpg|280px|thumb|'''Figure 14.''' ''Anti-symmetric HOMO of cis-butadiene'']]
|
[[Image:XJB_LUMO.jpg|300px|thumb|'''Figure 15.''' ''Symmetric HOMO of cis-butadiene'']]
|- align="center"
|}

===The Transition State of prototype reaction between ethylene and butadiene===

The Hessian method was used to estimate transition state for the reaction between ethylene and butadiene because these two reactants are relatively small and simple molecules. The transition state was drawn by first building a structure as (a) in '''Figure 16''' then removing the -CH<sub>2</sub>-CH<sub>2</sub>- group to form the envelop structure as in (b). The '''clean''' option should not be used and the dashed bond length was guessed to be larger than 2.2 Å otherwise the geometry change would cause a failure in optimization.

<center>[[Image:XJB_dielsguessts.jpg|300px|center|thumb|'''Figure 16'''. ''The guessing transition state for the reaction between butadiene and ethylene'']]</center>

<center>[[File:XJB_butadiene and ethylene ts.GIF|400px|center|thumb|'''Figure 17.''' ''Animation of the butadiene and ethylene cycloaddition transition state'']]</center>

After the Hessian optimization, the transition structure has been successfully determined and an animation was generated in '''Figure 17'''. It has a C<sub>1</sub> symmetry point group and the bond lengths of two partly formed C-C bonds were measured to be 2.12Å. The typical sp<sup>3</sup> and sp<sup>2</sup> C-C bond lengths are 1.54Å and 1.47Å respectively[ref]. The van der Waals radius of the C atom is 1.7 Å[ref]. The measured bond distance of the transition structure is larger than typical bond lengths but still within the van der Waals radius, suggesting the bond forming interation between the carbon atoms.

From the animation ('''Figure 17''') the vibration directions of two molecules are opposite and they are approaching and leaving each other at the same time. This suggests the formation of the two bonds synchronous. The vibrational frequencies were also calculated and an infrared spectrum was simulated ('''Figure 20'''). An imaginary frequency of magnitude -955 cm<sup>-1</sup> was seen and the lowest positive frequency was observed at 147 cm<sup>-1</sup> which corresponded to the reactant vibration but not the bond formation.

{| align="center"
|
[[Image:XJB_HOMOts.jpg|300px|thumb|'''Figure 18'''. HOMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_LUMOts.jpg|300px|thumb|'''Figure 19'''. LUMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_IRts.jpg|300px|thumb|'''Figure 20'''. Generated IR of transition state of reaction between cis-butadiene and ethylene]]
|}

The HOMO and LUMO molecular orbitals of this transition structure were visualized in '''Figure 18''' and '''Figure 19'''. From these two figures it can be observed that the HOMO at the transition structure is anti-symmetric ('''a''') while the LUMO is symmetric ('''s'''). Thus this anti-symmetric transition state HOMO is formed by the HOMO of cis-butadiene and the LUMO of ethylene with the knowledge that they are both anti-symmetrical. This conclusion agrees with the previous talked conditions for allowed reaction.

===The cyclohexa-1,3-diene reaction with maleic anhydride===

<center>[[Image:XJB_mechanism3.jpg|500px|center|thumb|'''Figure 21'''. ''The endo and exo products of the Diels Alder reaction between cyclohexa-1,3-diene and maleic anhydride'']]</center>

The exo and endo transition states were drawn according to '''Figure 21''' with the new C-C bond formation line dashed and bond length estimation as 2.5. Then the same Hessian method was applied to optimize the structures. The electronic energy of the endo structure was calculated to be '''-0.05150Ha''' and '''-0.05042Ha''' for the exo. This reaction is supposed to be kinetically controlled thus the lower energy endo transition state is more favoured.

The bond lengths of the partly formed C-C bond are 2.16 Å and 2.17 Å for endo and exo transition states respectively. Another measured C-C bond length is shown in '''Table 13''' as 2.89 Å for endo (C3-C8 and C2-C7) and 2.95 Å for endo (C7-C10 and C8-C9). The longer atom distance indicates the larger steric repulsion in the exo transition state because of the extra existence of hydrogen atoms on C7 and C8.

Woodward and Hoffmann also proposed a secondary orbital overlap statement as an explanation of the endo stereo-preference in Diels Alder reactions<ref name="soo">''J. Org. Chem.'', 1987, '''52(8)''', pp 1469–1474. {{DOI|10.1021/jo00384a016}}</ref>. This secondary orbital overlap does not involve bond changes but π orbital interactions as the highlighted carbon atoms of the endo transition state in '''Table 13'''. Such overlap stablises the structure and lowers the energy of the endo transition state while there is no such effect existed in the exo transition state.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | endo
! scope="col" | exo
|-
| '''Animation'''||[[File:XJB_endots.GIF|300px]]||[[File:XJB_exots.GIF|300px]]
|-
| '''Sum of electronic and zero-point Energies'''||0.133493||0.134880
|-
| '''Sum of electronic and thermal Energies'''||0.143683||0.144881
|-
| '''Sum of electronic and thermal Enthalpies'''||0.144627||0.145825
|-
| '''Sum of electronic and thermal Free Energies'''||0.097349||0.099117
|-
| '''Bond lengths'''||[[Image:XJB_endomeasure.jpg|300px]]||[[Image:XJB_exomeasure.jpg|300px]]
|-
| '''HOMO'''||[[Image:XJB_endoHOMO.jpg|300px]]||[[Image:XJB_exoHOMO.jpg|285px]]
|-
| '''Secondary orbital overlap effect'''||[[Image:XJB_endosoo.jpg|300px]]||[[Image:XJB_exosoo.jpg|300px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 13.''' ''Summary of comparison of properties of endo and exo transition structures for reaction between cyclohexa-1,3-diene and maleic anhydride''</div>

==Reference==

<references/>

Rep:Mod:XJB1130

2013-12-06T12:42:59Z

Jx1011: /* The cyclohexa-1,3-diene reaction with maleic anhydride */

==The Cope Rearrangement of 1,5-hexadiene==

The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement reaction ('''Figure 1'''). Its mechanism has been studied by a large number of experimental and computational researches, which is recently accepted that the reaction occurs via either a "chair" or a "boat" transition state structure in a concerted manner. The aim of this exercise is to locate the low-energy minimum and the transition structures on the potential energy surface by Gaussian calculation in order to study its mechanism. The reaction pathway and the barrier heights can also be calculated by this calculation.

<center>[[Image:XJB_cope rearrangement.jpg|500px|center|thumb|'''Figure 1'''. ''The Cope rearrangement'']]</center>

===Optimizing the Reactants and Products===

Different conformations of 1,5-hexadiene can be optimized using '''Gaussview'''. The optimized structure of four conformers involved in this discussion and their corresponding energies optimized at '''HF/3-21G''' level of theory are concluded in the following table ('''Table 1''').

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | anti4
! scope="col" | anti2
! scope="col" | gauche
! scope="col" | gauche3
|-
| <jmol><jmolApplet><title>XJB_anti4.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti4.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche3.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| -231.69097Ha || -231.69254Ha || -231.68772Ha ||-231.69266Ha
|}

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 1.''' ''Four conformers of 1,5-hexadiene''</div>

This exercise first involved optimizing a molecule of 1,5-hexadiene with "anti" linkage for the cetral four C atoms at the '''HF/3-21G''' level of theory. The final optimization structure had the total electronic energy of '''-231.69097Ha''' with a '''C<sub>1</sub>''' symmetry. By comparing the energy and point group to that in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1], this structure is confirmed to be ''anti4''. The compositions of energies are also listed in '''Table 2''' as i) the sum of electronic and zero-point energies (potential energy at 0K including the zero-point vibrational energy), ii) the sum of electronic and thermal energies (energy including translational, rotational and vibrational at 298.15K and 1 atm), iii) the sum of electronic and thermal enthalpies (containning an additional correction for RT), and iv) the sum of electronic and thermal free energies (including the entropic contribution to the free energy). For what we are considering, the first two terms are the most useful for further comparisons.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Formula Description'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||E = E<sub>elec</sub> + ZPE||-231.53785
|-
| Sum of electronic and thermal Energies||E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>||-231.53096
|-
| Sum of electronic and thermal Enthalpie||H = E + RT||-231.53002
|-
| Sum of electronic and thermal Free Energie||G = H - TS||-231.56891
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 2.''' ''Frequency calculations of anti4 optimized at '''HF/3-21G''' ([[File:XJB_anti4.log]])''</div>

Next another 1,5-hexadiene molecule with "gauche" linkage for the central four atoms was drawn and optimized at the '''HF/3-21G''' level of theory as before. Energy for this conformation is expected to be higher than that for the ''anti4'' conformation due to the closer position of two alkene ends thus the steric interaction. The final structure had the electronic energy of '''-231.68772Ha''' which was indeed higher than that of the ''anti4'', indicating less favour towards this gauche structure. It had a '''C<sub>2</sub>''' symmetry and was confirmed to be the ''gauche'' conformation.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53439
|-
| Sum of electronic and thermal Energies||-231.52770
|-
| Sum of electronic and thermal Enthalpie||-231.52676
|-
| Sum of electronic and thermal Free Energie||-231.56483
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 3.''' ''Frequency calculations of gauche optimized at '''HF/3-21G''' ([[File:XJB_gauche.log]])''</div>

A lowest energy conformation ''gauche3'' of the reactant molecule was drawn based on and confirmed by [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. The optimization gave a final structure with an energy of '''-231.69266Ha''' with a symmetry of '''C<sub>1</sub>'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53949
|-
| Sum of electronic and thermal Energies||-231.53265
|-
| Sum of electronic and thermal Enthalpie||-231.5317
|-
| Sum of electronic and thermal Free Energie||-231.57064
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 4.''' ''Frequency calculations of gauche3 optimized at '''HF/3-21G''' ([[File:XJB_gauche3.log]])''</div>

The next step required us to draw the C<sub>i</sub> symmetry ''anti2'' conformation of 1,5-hexadiene and optimize it at the '''HF/3-21G''' level of theory. The optimized energy was calculated to be '''-231.69254Ha''' with point group of '''C<sub>i</sub>''', which was consistent with the one given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53954
|-
| Sum of electronic and thermal Energies||-231.53257
|-
| Sum of electronic and thermal Enthalpie||-231.53162
|-
| Sum of electronic and thermal Free Energie||-231.57092
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 5.''' ''Frequency calculations of anti2 optimized at '''HF/3-21G''' ([[File:XJB_anti2.log]])''</div>

The ''anti2'' structure was then reoptimized at '''B3LYP/6-31G*''' which was a improvement over the '''HF/3-21G''' basis set. It adds polarization to all atoms and improves the modeling of core electrons thus producing more accurate description of orbitals[ref]. The reoptimized ''anti2'' structure had the total electronic energy of '''-234.61171Ha''' and the same symmetry C<sub>i</sub>. The comparison of two structures optimized using different levels of theory is summarized in '''Table 6'''. We can see that they have exactly the same dihedral angle of the central four C atoms at 180.00<sup>o</sup> and angles between three of them are very similar as 111.3<sup>o</sup> and 112.7<sup>o</sup> respectively. Thus the the geometry change using '''HF/3-21G''' or '''B3LYP/6-31G*''' is trivial.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Before re-optimization
! scope="col" | After re-optimization
|-
| <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2reoptimized</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2reoptimized.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| ''anti2 optimized at '''HF/3-21G''''' || ''anti2 optimized at '''B3LYP/6-31G*'''''
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 6.''' ''Two optimization of anti2 at '''HF/3-21G''' and '''B3LYP/6-31G*'''([[File:XJB_anti2.log]])''</div>

The frequency calculation was run as well and a list of energies is summarized in '''Table 7'''. A list of all the vibrational frequencies was also generated and no imaginary (negative) frequency was observed and these vibrations are visualized by simulating the infrared spectrum in '''Figure 2'''.

{| align="center"
|- align="center"
|
{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-234.46920
|-
| Sum of electronic and thermal Energies||-234.46186
|-
| Sum of electronic and thermal Enthalpie||-234.46091
|-
| Sum of electronic and thermal Free Energie||-234.50078
|}
|
[[Image:XJB_IR_anti2.jpg|300px]]
|- align="center"
| align="center" | '''Table 7.''' ''Frequency calculations of anti2 optimized at '''B3LYP/6-31G*'''
([[File:XJB_anti2_reoptimized.log]])''
| align="center" | '''Figure 2.''' ''The generated IR spectrum of optimized '''B3LYP/6-31G*''' structure''
|}

===Optimizing the "Chair" and "Boat" Transition Structures===

The study of the chair and boat transition states is carried out using three different approaches. The first one is called the Hessian which involves computing the force constant matrix; the second one is to freeze the reaction coordinate and then optimize the structure once the molecule is fully relaxed; the last one is to use QST2 method to specify the reactants and products then finding the transition structure.

<center>[[Image:XJB_2 ts.jpg|500px|center|thumb|'''Figure 3'''. ''The two transition states of Cope rearrangement'']]</center>

====The Chair Transition State====

In order to set up a transition state optimization, first an allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimized at the '''HF/3-21G''' level of theory. Then this half of the transition structure was duplicated with rotation to make the approximate resembling of the chair transition state with terminal ends of the fragments 2.2 Å apart. The chair transition structure optimization was then done by the first two different approaches.

<center>[[Image:XJB_guess ts.jpg|400px|center|thumb|'''Figure 4'''. ''The guessed chair transition structure of two allyl fragments'']]</center>

For the first optimization approach, the force constants were computed (the Hessian) and the chair transition state was optimized to a TS (Berny) at the '''HF/3-21G''' level of theory. The frequency calculation was carried out at the same time and gave an imaginary frequency at -818 cm<sup>-1</sup>. The electronic energy was calculated to be '''-231.61932Ha''' with C<sub>2h</sub> symmetry point group. The distance between the terminal carbons was 2.0203 Å. An animation of this transition state vibrations was generated to confirm the corresponding Cope rearrangement.

<center>[[File:XJB_chair ts animation.GIF|400px|center|thumb|'''Figure 5.''' ''Animation of the Cope rearrangement via the chair transition state'']]</center>

The chair transition state was then reoptimized using the '''B3LYP/6-31G*''' level of theory and carried out frequency calculations as well. The electronic energy calculated this time was '''-234.55698Ha''' with an imaginary frequency at -566 cm<sup>-1</sup>. We can see that with the geometries quite similar, the energies differ markedly calculated using two levels of theory. The summary and comparison of the calculations optimized at two levels of theory are summarized in '''Table 8'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46670||-234.41493
|-
| Sum of electronic and thermal Energies||-231.46134||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40807
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir1.jpg|200px]]||[[Image:XJB_chairir2.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 8.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the first approach
([[File:XJB_chairts1.1.log]], [[File:XJB_chairts1.2.log]])''</div>

The second approach is the frozen coordinate method using the Redundant Coord Editor to freeze the bond lengths of the terminal ends to 2.2 Å. First the calculation failed because of the recent update of GaussView. After editting the coordinate distance (B 5 1 2.2 F) and the frozen distance (B 5 1 F) the optimization went well this time. The structure was then optimized again using a normal guess Hessian modified including the two differentiating coordinates. This chair transition state was then reoptimized again using the '''B3LYP/6-31G*''' level of theory. The electronic energies were calculated to be the same as in the first approach and frequency calculations were summarized in '''Table 9'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46671||-234.41490
|-
| Sum of electronic and thermal Energies||-231.46135||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40806
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir3.jpg|200px]]||[[Image:XJB_chairir4.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 9.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the second approach
([[File:XJB_chairts2.1.log]], [[File:XJB_chairts2.2.log]])''</div>

As the same energy results given, the bond forming/breaking bond lengths in the chair transition structures in the above two different approaches were compared. The first approach gave the bond lengths of 2.0203 Å while the second approach gave 2.0207 Å. Considering the accuracy limit of bond length estimation in Gaussview, these two bond lengths can be seen as exactly the same. Two methods worked reasonably well because of the good assumption of transition structures. However for more complicated and larger molecule, the Hessian method may fail due to the difficulty to predict the transition geometry (different surface curvature). Thus better way to generate such transition structures is to freeze reaction coordinate.

{| align="center"
|- align="center"
|
[[Image:XJB_chairts bondlength1.jpg|350px|thumb|'''Figure 6.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by Hession'']]
|
[[Image:XJB_chairts bondlength2.jpg|350px|thumb|'''Figure 7.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by frozen coordinate'']]
|- align="center"
|}

====The Boat Transition State====

The boat transition structure was then optimized using the '''QST2''' method. The same numbering for the reactant and product molecules was important in order to specify for this reaction. The renumbering of the molecules is shown as in '''Figure 8''', but the following optimization failed and gave a twisted structure. The reactant and product geometries were then modified further by changing the central C-C-C-C dihedral angle to 0<sup>o</sup> and the inside two C-C-C (i.e. C2-C3-C4 and C3-C4-C5) angles to 100<sup>o</sup>. The job was run successfully and frequency calculations were done with the boat transition state generated. Only one imaginary frequency was detected also at -840 cm<sup>-1</sup>. The electronic energy was '''-231.60280Ha''' with C<sub>2v</sub> symmetry. The distances between the terminal carbons of the allyl fragments were examined to be both 2.140Å which were comparably longer than that of the chair transition structure.

{| align="center"
|- align="center"
|
[[Image:XJB_QST1.jpg|400px|thumb|'''Figure 8.''' ''Numbering of reactant and product'']]
|
[[Image:XJB_QST2.jpg|410px|thumb|'''Figure 9.''' ''Re-numbering of reactant and product'']]
|- align="center"
|}

<center>[[File:XJB_boat ts animation.GIF|400px|center|thumb|'''Figure 10.''' ''Animation of the Cope rearrangement via the boat transition state'']]</center>

The structure was then reoptimized at the higher '''B3LYP/6-31G*''' theory of level and the electronic energy was calculated to be '''-234.54309Ha''' with very similar geometry. The frequency calculation generated an imaginary frequency at -530 cm<sup>-1</sup>. The summary and comparison of two levels' calculations are carried out in the following table.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.45093||-234.40234
|-
| Sum of electronic and thermal Energies||-231.44530||-234.39601
|-
| Sum of electronic and thermal Enthalpie|| -231.44436||-234.39506
|-
| Sum of electronic and thermal Free Energie||-231.47977||-234.43175
|-
| Generated IR spectrum||[[Image:XJB_boatir1.jpg|200px]]||[[Image:XJB_boatir2.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 10.''' ''Frequency calculations of the boat transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by QST2
([[File:XJB_boat ts1.log]], [[File:XJB_boat ts2.log]])''</div>

====Intrinsic Reaction Coordinate====

Next a useful method called '''Intrinct Reaction Coordinate (IRC)''' is applied to the Hessian optimized chair transition structure to follow the mininmum energy path (MEP) from a transition structure down to its local minimum on a potential energy surface. IRC can be used to verify that the transition state you found actually connects the reactants and products you think it does, or to identify any reaction intermediates you haven't thought about, or to generate an animation of the reaction [ref]. The symmetrical reaction coordinate was computed only in the forward direction with force constant always calculated and the number of points along IRC 50. The last structure generated along IRC is given in '''Figure 11''' with the energy calculated as '''-231.69158Ha''' and the dihedral angle and internal angle measured as D = -67.188<sup>o</sup> A = 111.78<sup>o</sup>. This energy does not match to any of the conformations discussed in Appendix 1 thus a minimum geometry has not reached yet.

{| align="center"
|- align="center"
|
<jmol><jmolApplet><title>XJB_IRC.mol</title><color>white</color>
<size>200</size><script>zoom=5</script>
<uploadedFileContents>XJB_IRC.mol</uploadedFileContents>
</jmolApplet></jmol>
|
[[Image:XJB_irc.jpg|500px]]
|- align="center"
| align="center" | '''Figure 11.''' ''The energy minimum structure along IRC''
| align="center" | '''Figure 12.''' ''Total energy along IRC''
|}

Three improving options can be used to reach the minimum energy. The first one is to run a normal Hessian optimization of the last point on IRC; the second one involves using a larger number of points (in this case 100 was used); the last one is to redo the IRC computation specifying to compute force constant at every step. Three methods were all used and the results were summarized in '''Table 11'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Improved IRC method 1
! scope="col" | Improved IRC method 2
! scope="col" | Improved IRC method 3
|-
| [[Image:XJB_IRC1.jpg|300px|center|IRC1]]||[[Image:XJB_IRC2.jpg|300px|center|IRC3]] || [[Image:XJB_IRC3.jpg|300px|center|IRC3]]
|-
| D = -64.17<sup>o</sup> A = 112.04<sup>o</sup> || D = -66.83<sup>o</sup> A = 111.82<sup>o</sup> || D = -64.17<sup>o</sup> A = 112.04<sup>o</sup>
|-
| -231.69167Ha || -231.69150Ha || -231.69167Ha
|-
| [[File:XJB_IRC1.log]]||[[File:XJB_IRC2.log]]||[[File:XJB_IRC3.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 11.''' ''Comparison of three improvement methods of IRC computation upon the chair transition structure''</div>

The first method was very quick but maybe generated relatively low-accuracy result; the second approach involved the controlling of number of points which might result in ending up at the wrong direction and wrong structure; the third method was the most suitable option for this small system but it took longer time than the previous two. From '''Table 11''' it is concluded that method 1 and 3 gave the same lowest energy which was consistent to the energy of the ''gauche2'' conformation given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

====Activation Energies====

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | '''HF/3-21G''' at 0K
! scope="col" | '''HF/3-21G''' at 298.15K
! scope="col" | '''B3LYP/6-31G*''' at 0K
! scope="col" | '''B3LYP/6-31G*''' at 298.15K
! scope="col" | Expt. at 0K
|-
| '''ΔE (chair)'''||45.71||44.70||34.05||33.16||33.5 ± 0.5
|-
| '''ΔE (boat)'''||55.60||54.76||41.96||41.32||44.7 ± 0.5
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 12.''' ''Summary of activation energies for the Cope rearrangement via both transition states in kcal/mol''</div>

The activation energy calculation was done by calculating the difference in energies between the optimized reactant ''anti2'' and the optimized chair and boat transition states. It is observed that chair conformation gives generally lower activation energy thus the more favoured transition state. The structure optimized at '''B3LYP/6-31G*''' level of theory at 0K has closer activation energy to the experimental result. Thus '''B3LYP/6-31G*''' gives a better estimation of transition state than '''HF/3-21G'''.

==The Diels Alder Cycloaddition==

In this exercise, AM1 semi-empirical method is used to compute the Diels Alder cycloaddition which is a concerted pericyclic reaction. The driving force of this reaction is the formation of a new σ bond between the π orbitals of the dieneophile and the diene. The reaction is allowed with HOMO-LUMO interation and a symmetry overlap.

===Cis Butadiene===

<center>[[Image:prototype Diels Alder.jpg|500px|center|thumb|'''Figure 13'''. ''The prototype reaction between butadiene and ethylene'']]</center>

To study a prototype Diels Alder cycloaddition reaction between butadiene and ethylene, first a cis butadiene molecule was built using Gaussview and optimized using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the butadiene were plotted. As seen the HOMO of cis-butadiene is anti-symmetric ('''a''') and the LUMO is symmetric ('''s''') with respect to plane.

{| align="center"
|- align="center"
|
[[Image:XJB_HOMO.jpg|280px|thumb|'''Figure 14.''' ''Anti-symmetric HOMO of cis-butadiene'']]
|
[[Image:XJB_LUMO.jpg|300px|thumb|'''Figure 15.''' ''Symmetric HOMO of cis-butadiene'']]
|- align="center"
|}

===The Transition State of prototype reaction between ethylene and butadiene===

The Hessian method was used to estimate transition state for the reaction between ethylene and butadiene because these two reactants are relatively small and simple molecules. The transition state was drawn by first building a structure as (a) in '''Figure 16''' then removing the -CH<sub>2</sub>-CH<sub>2</sub>- group to form the envelop structure as in (b). The '''clean''' option should not be used and the dashed bond length was guessed to be larger than 2.2 Å otherwise the geometry change would cause a failure in optimization.

<center>[[Image:XJB_dielsguessts.jpg|300px|center|thumb|'''Figure 16'''. ''The guessing transition state for the reaction between butadiene and ethylene'']]</center>

<center>[[File:XJB_butadiene and ethylene ts.GIF|400px|center|thumb|'''Figure 17.''' ''Animation of the butadiene and ethylene cycloaddition transition state'']]</center>

After the Hessian optimization, the transition structure has been successfully determined and an animation was generated in '''Figure 17'''. It has a C<sub>1</sub> symmetry point group and the bond lengths of two partly formed C-C bonds were measured to be 2.12Å. The typical sp<sup>3</sup> and sp<sup>2</sup> C-C bond lengths are 1.54Å and 1.47Å respectively[ref]. The van der Waals radius of the C atom is 1.7 Å[ref]. The measured bond distance of the transition structure is larger than typical bond lengths but still within the van der Waals radius, suggesting the bond forming interation between the carbon atoms.

From the animation ('''Figure 17''') the vibration directions of two molecules are opposite and they are approaching and leaving each other at the same time. This suggests the formation of the two bonds synchronous. The vibrational frequencies were also calculated and an infrared spectrum was simulated ('''Figure 20'''). An imaginary frequency of magnitude -955 cm<sup>-1</sup> was seen and the lowest positive frequency was observed at 147 cm<sup>-1</sup> which corresponded to the reactant vibration but not the bond formation.

{| align="center"
|
[[Image:XJB_HOMOts.jpg|300px|thumb|'''Figure 18'''. HOMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_LUMOts.jpg|300px|thumb|'''Figure 19'''. LUMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_IRts.jpg|300px|thumb|'''Figure 20'''. Generated IR of transition state of reaction between cis-butadiene and ethylene]]
|}

The HOMO and LUMO molecular orbitals of this transition structure were visualized in '''Figure 18''' and '''Figure 19'''. From these two figures it can be observed that the HOMO at the transition structure is anti-symmetric ('''a''') while the LUMO is symmetric ('''s'''). Thus this anti-symmetric transition state HOMO is formed by the HOMO of cis-butadiene and the LUMO of ethylene with the knowledge that they are both anti-symmetrical. This conclusion agrees with the previous talked conditions for allowed reaction.

===The cyclohexa-1,3-diene reaction with maleic anhydride===

<center>[[Image:XJB_mechanism3.jpg|500px|center|thumb|'''Figure 21'''. ''The endo and exo products of the Diels Alder reaction between cyclohexa-1,3-diene and maleic anhydride'']]</center>

The exo and endo transition states were drawn according to '''Figure 21''' with the new C-C bond formation line dashed and bond length estimation as 2.5. Then the same Hessian method was applied to optimize the structures. The electronic energy of the endo structure was calculated to be '''-0.05150Ha''' and '''-0.05042Ha''' for the exo. This reaction is supposed to be kinetically controlled thus the lower energy endo transition state is more favoured.

The bond lengths of the partly formed C-C bond are 2.16 Å and 2.17 Å for endo and exo transition states respectively. Another measured C-C bond length is shown in '''Table 13''' as 2.89 Å for endo (C3-C8 and C2-C7) and 2.95 Å for endo (C7-C10 and C8-C9). The longer atom distance indicates the larger steric repulsion in the exo transition state because of the extra existence of hydrogen atoms on C7 and C8.

Woodward and Hoffmann also proposed a secondary orbital overlap statement as an explanation of the endo stereo-preference in Diels Alder reactions<ref name="soo">''J. Org. Chem.'', 1987, '''52(8)''', pp 1469–1474. {{DOI|10.1021/jo00384a016}}</ref>. This secondary orbital overlap does not involve bond changes but π orbital interactions as the highlighted carbon atoms of the endo transition state in '''Table 13'''. Such overlap stablises the structure and lowers the energy of the endo transition state while there is no such effect existed in the exo transition state.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | endo
! scope="col" | exo
|-
| '''Animation'''||[[File:XJB_endots.GIF|300px]]||[[File:XJB_exots.GIF|300px]]
|-
| '''Sum of electronic and zero-point Energies'''||0.133493||0.134880
|-
| '''Sum of electronic and thermal Energies'''||0.143683||0.144881
|-
| '''Sum of electronic and thermal Enthalpies'''||0.144627||0.145825
|-
| '''Sum of electronic and thermal Free Energies'''||0.097349||0.099117
|-
| '''Bond lengths'''||[[Image:XJB_endomeasure.jpg|300px]]||[[Image:XJB_exomeasure.jpg|300px]]
|-
| '''HOMO'''||[[Image:XJB_endoHOMO.jpg|300px]]||[[Image:XJB_exoHOMO.jpg|285px]]
|-
| '''Secondary orbital overlap effect'''||[[Image:XJB_endosoo.jpg|300px]]||[[Image:XJB_exosoo.jpg|300px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 13.''' ''Summary of comparison of properties of endo and exo transition structures for reaction between cyclohexa-1,3-diene and maleic anhydride''</div>

==Reference==

<references/>

File:XJB exosoo.jpg

2013-12-06T12:40:42Z

Jx1011:

Rep:Mod:XJB1130

2013-12-06T00:23:02Z

Jx1011: /* Activation Energies */

==The Cope Rearrangement of 1,5-hexadiene==

The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement reaction ('''Figure 1'''). Its mechanism has been studied by a large number of experimental and computational researches, which is recently accepted that the reaction occurs via either a "chair" or a "boat" transition state structure in a concerted manner. The aim of this exercise is to locate the low-energy minimum and the transition structures on the potential energy surface by Gaussian calculation in order to study its mechanism. The reaction pathway and the barrier heights can also be calculated by this calculation.

<center>[[Image:XJB_cope rearrangement.jpg|500px|center|thumb|'''Figure 1'''. ''The Cope rearrangement'']]</center>

===Optimizing the Reactants and Products===

Different conformations of 1,5-hexadiene can be optimized using '''Gaussview'''. The optimized structure of four conformers involved in this discussion and their corresponding energies optimized at '''HF/3-21G''' level of theory are concluded in the following table ('''Table 1''').

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | anti4
! scope="col" | anti2
! scope="col" | gauche
! scope="col" | gauche3
|-
| <jmol><jmolApplet><title>XJB_anti4.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti4.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche3.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| -231.69097Ha || -231.69254Ha || -231.68772Ha ||-231.69266Ha
|}

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 1.''' ''Four conformers of 1,5-hexadiene''</div>

This exercise first involved optimizing a molecule of 1,5-hexadiene with "anti" linkage for the cetral four C atoms at the '''HF/3-21G''' level of theory. The final optimization structure had the total electronic energy of '''-231.69097Ha''' with a '''C<sub>1</sub>''' symmetry. By comparing the energy and point group to that in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1], this structure is confirmed to be ''anti4''. The compositions of energies are also listed in '''Table 2''' as i) the sum of electronic and zero-point energies (potential energy at 0K including the zero-point vibrational energy), ii) the sum of electronic and thermal energies (energy including translational, rotational and vibrational at 298.15K and 1 atm), iii) the sum of electronic and thermal enthalpies (containning an additional correction for RT), and iv) the sum of electronic and thermal free energies (including the entropic contribution to the free energy). For what we are considering, the first two terms are the most useful for further comparisons.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Formula Description'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||E = E<sub>elec</sub> + ZPE||-231.53785
|-
| Sum of electronic and thermal Energies||E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>||-231.53096
|-
| Sum of electronic and thermal Enthalpie||H = E + RT||-231.53002
|-
| Sum of electronic and thermal Free Energie||G = H - TS||-231.56891
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 2.''' ''Frequency calculations of anti4 optimized at '''HF/3-21G''' ([[File:XJB_anti4.log]])''</div>

Next another 1,5-hexadiene molecule with "gauche" linkage for the central four atoms was drawn and optimized at the '''HF/3-21G''' level of theory as before. Energy for this conformation is expected to be higher than that for the ''anti4'' conformation due to the closer position of two alkene ends thus the steric interaction. The final structure had the electronic energy of '''-231.68772Ha''' which was indeed higher than that of the ''anti4'', indicating less favour towards this gauche structure. It had a '''C<sub>2</sub>''' symmetry and was confirmed to be the ''gauche'' conformation.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53439
|-
| Sum of electronic and thermal Energies||-231.52770
|-
| Sum of electronic and thermal Enthalpie||-231.52676
|-
| Sum of electronic and thermal Free Energie||-231.56483
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 3.''' ''Frequency calculations of gauche optimized at '''HF/3-21G''' ([[File:XJB_gauche.log]])''</div>

A lowest energy conformation ''gauche3'' of the reactant molecule was drawn based on and confirmed by [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. The optimization gave a final structure with an energy of '''-231.69266Ha''' with a symmetry of '''C<sub>1</sub>'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53949
|-
| Sum of electronic and thermal Energies||-231.53265
|-
| Sum of electronic and thermal Enthalpie||-231.5317
|-
| Sum of electronic and thermal Free Energie||-231.57064
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 4.''' ''Frequency calculations of gauche3 optimized at '''HF/3-21G''' ([[File:XJB_gauche3.log]])''</div>

The next step required us to draw the C<sub>i</sub> symmetry ''anti2'' conformation of 1,5-hexadiene and optimize it at the '''HF/3-21G''' level of theory. The optimized energy was calculated to be '''-231.69254Ha''' with point group of '''C<sub>i</sub>''', which was consistent with the one given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53954
|-
| Sum of electronic and thermal Energies||-231.53257
|-
| Sum of electronic and thermal Enthalpie||-231.53162
|-
| Sum of electronic and thermal Free Energie||-231.57092
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 5.''' ''Frequency calculations of anti2 optimized at '''HF/3-21G''' ([[File:XJB_anti2.log]])''</div>

The ''anti2'' structure was then reoptimized at '''B3LYP/6-31G*''' which was a improvement over the '''HF/3-21G''' basis set. It adds polarization to all atoms and improves the modeling of core electrons thus producing more accurate description of orbitals[ref]. The reoptimized ''anti2'' structure had the total electronic energy of '''-234.61171Ha''' and the same symmetry C<sub>i</sub>. The comparison of two structures optimized using different levels of theory is summarized in '''Table 6'''. We can see that they have exactly the same dihedral angle of the central four C atoms at 180.00<sup>o</sup> and angles between three of them are very similar as 111.3<sup>o</sup> and 112.7<sup>o</sup> respectively. Thus the the geometry change using '''HF/3-21G''' or '''B3LYP/6-31G*''' is trivial.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Before re-optimization
! scope="col" | After re-optimization
|-
| <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2reoptimized</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2reoptimized.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| ''anti2 optimized at '''HF/3-21G''''' || ''anti2 optimized at '''B3LYP/6-31G*'''''
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 6.''' ''Two optimization of anti2 at '''HF/3-21G''' and '''B3LYP/6-31G*'''([[File:XJB_anti2.log]])''</div>

The frequency calculation was run as well and a list of energies is summarized in '''Table 7'''. A list of all the vibrational frequencies was also generated and no imaginary (negative) frequency was observed and these vibrations are visualized by simulating the infrared spectrum in '''Figure 2'''.

{| align="center"
|- align="center"
|
{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-234.46920
|-
| Sum of electronic and thermal Energies||-234.46186
|-
| Sum of electronic and thermal Enthalpie||-234.46091
|-
| Sum of electronic and thermal Free Energie||-234.50078
|}
|
[[Image:XJB_IR_anti2.jpg|300px]]
|- align="center"
| align="center" | '''Table 7.''' ''Frequency calculations of anti2 optimized at '''B3LYP/6-31G*'''
([[File:XJB_anti2_reoptimized.log]])''
| align="center" | '''Figure 2.''' ''The generated IR spectrum of optimized '''B3LYP/6-31G*''' structure''
|}

===Optimizing the "Chair" and "Boat" Transition Structures===

The study of the chair and boat transition states is carried out using three different approaches. The first one is called the Hessian which involves computing the force constant matrix; the second one is to freeze the reaction coordinate and then optimize the structure once the molecule is fully relaxed; the last one is to use QST2 method to specify the reactants and products then finding the transition structure.

<center>[[Image:XJB_2 ts.jpg|500px|center|thumb|'''Figure 3'''. ''The two transition states of Cope rearrangement'']]</center>

====The Chair Transition State====

In order to set up a transition state optimization, first an allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimized at the '''HF/3-21G''' level of theory. Then this half of the transition structure was duplicated with rotation to make the approximate resembling of the chair transition state with terminal ends of the fragments 2.2 Å apart. The chair transition structure optimization was then done by the first two different approaches.

<center>[[Image:XJB_guess ts.jpg|400px|center|thumb|'''Figure 4'''. ''The guessed chair transition structure of two allyl fragments'']]</center>

For the first optimization approach, the force constants were computed (the Hessian) and the chair transition state was optimized to a TS (Berny) at the '''HF/3-21G''' level of theory. The frequency calculation was carried out at the same time and gave an imaginary frequency at -818 cm<sup>-1</sup>. The electronic energy was calculated to be '''-231.61932Ha''' with C<sub>2h</sub> symmetry point group. The distance between the terminal carbons was 2.0203 Å. An animation of this transition state vibrations was generated to confirm the corresponding Cope rearrangement.

<center>[[File:XJB_chair ts animation.GIF|400px|center|thumb|'''Figure 5.''' ''Animation of the Cope rearrangement via the chair transition state'']]</center>

The chair transition state was then reoptimized using the '''B3LYP/6-31G*''' level of theory and carried out frequency calculations as well. The electronic energy calculated this time was '''-234.55698Ha''' with an imaginary frequency at -566 cm<sup>-1</sup>. We can see that with the geometries quite similar, the energies differ markedly calculated using two levels of theory. The summary and comparison of the calculations optimized at two levels of theory are summarized in '''Table 8'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46670||-234.41493
|-
| Sum of electronic and thermal Energies||-231.46134||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40807
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir1.jpg|200px]]||[[Image:XJB_chairir2.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 8.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the first approach
([[File:XJB_chairts1.1.log]], [[File:XJB_chairts1.2.log]])''</div>

The second approach is the frozen coordinate method using the Redundant Coord Editor to freeze the bond lengths of the terminal ends to 2.2 Å. First the calculation failed because of the recent update of GaussView. After editting the coordinate distance (B 5 1 2.2 F) and the frozen distance (B 5 1 F) the optimization went well this time. The structure was then optimized again using a normal guess Hessian modified including the two differentiating coordinates. This chair transition state was then reoptimized again using the '''B3LYP/6-31G*''' level of theory. The electronic energies were calculated to be the same as in the first approach and frequency calculations were summarized in '''Table 9'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46671||-234.41490
|-
| Sum of electronic and thermal Energies||-231.46135||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40806
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir3.jpg|200px]]||[[Image:XJB_chairir4.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 9.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the second approach
([[File:XJB_chairts2.1.log]], [[File:XJB_chairts2.2.log]])''</div>

As the same energy results given, the bond forming/breaking bond lengths in the chair transition structures in the above two different approaches were compared. The first approach gave the bond lengths of 2.0203 Å while the second approach gave 2.0207 Å. Considering the accuracy limit of bond length estimation in Gaussview, these two bond lengths can be seen as exactly the same. Two methods worked reasonably well because of the good assumption of transition structures. However for more complicated and larger molecule, the Hessian method may fail due to the difficulty to predict the transition geometry (different surface curvature). Thus better way to generate such transition structures is to freeze reaction coordinate.

{| align="center"
|- align="center"
|
[[Image:XJB_chairts bondlength1.jpg|350px|thumb|'''Figure 6.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by Hession'']]
|
[[Image:XJB_chairts bondlength2.jpg|350px|thumb|'''Figure 7.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by frozen coordinate'']]
|- align="center"
|}

====The Boat Transition State====

The boat transition structure was then optimized using the '''QST2''' method. The same numbering for the reactant and product molecules was important in order to specify for this reaction. The renumbering of the molecules is shown as in '''Figure 8''', but the following optimization failed and gave a twisted structure. The reactant and product geometries were then modified further by changing the central C-C-C-C dihedral angle to 0<sup>o</sup> and the inside two C-C-C (i.e. C2-C3-C4 and C3-C4-C5) angles to 100<sup>o</sup>. The job was run successfully and frequency calculations were done with the boat transition state generated. Only one imaginary frequency was detected also at -840 cm<sup>-1</sup>. The electronic energy was '''-231.60280Ha''' with C<sub>2v</sub> symmetry. The distances between the terminal carbons of the allyl fragments were examined to be both 2.140Å which were comparably longer than that of the chair transition structure.

{| align="center"
|- align="center"
|
[[Image:XJB_QST1.jpg|400px|thumb|'''Figure 8.''' ''Numbering of reactant and product'']]
|
[[Image:XJB_QST2.jpg|410px|thumb|'''Figure 9.''' ''Re-numbering of reactant and product'']]
|- align="center"
|}

<center>[[File:XJB_boat ts animation.GIF|400px|center|thumb|'''Figure 10.''' ''Animation of the Cope rearrangement via the boat transition state'']]</center>

The structure was then reoptimized at the higher '''B3LYP/6-31G*''' theory of level and the electronic energy was calculated to be '''-234.54309Ha''' with very similar geometry. The frequency calculation generated an imaginary frequency at -530 cm<sup>-1</sup>. The summary and comparison of two levels' calculations are carried out in the following table.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.45093||-234.40234
|-
| Sum of electronic and thermal Energies||-231.44530||-234.39601
|-
| Sum of electronic and thermal Enthalpie|| -231.44436||-234.39506
|-
| Sum of electronic and thermal Free Energie||-231.47977||-234.43175
|-
| Generated IR spectrum||[[Image:XJB_boatir1.jpg|200px]]||[[Image:XJB_boatir2.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 10.''' ''Frequency calculations of the boat transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by QST2
([[File:XJB_boat ts1.log]], [[File:XJB_boat ts2.log]])''</div>

====Intrinsic Reaction Coordinate====

Next a useful method called '''Intrinct Reaction Coordinate (IRC)''' is applied to the Hessian optimized chair transition structure to follow the mininmum energy path (MEP) from a transition structure down to its local minimum on a potential energy surface. IRC can be used to verify that the transition state you found actually connects the reactants and products you think it does, or to identify any reaction intermediates you haven't thought about, or to generate an animation of the reaction [ref]. The symmetrical reaction coordinate was computed only in the forward direction with force constant always calculated and the number of points along IRC 50. The last structure generated along IRC is given in '''Figure 11''' with the energy calculated as '''-231.69158Ha''' and the dihedral angle and internal angle measured as D = -67.188<sup>o</sup> A = 111.78<sup>o</sup>. This energy does not match to any of the conformations discussed in Appendix 1 thus a minimum geometry has not reached yet.

{| align="center"
|- align="center"
|
<jmol><jmolApplet><title>XJB_IRC.mol</title><color>white</color>
<size>200</size><script>zoom=5</script>
<uploadedFileContents>XJB_IRC.mol</uploadedFileContents>
</jmolApplet></jmol>
|
[[Image:XJB_irc.jpg|500px]]
|- align="center"
| align="center" | '''Figure 11.''' ''The energy minimum structure along IRC''
| align="center" | '''Figure 12.''' ''Total energy along IRC''
|}

Three improving options can be used to reach the minimum energy. The first one is to run a normal Hessian optimization of the last point on IRC; the second one involves using a larger number of points (in this case 100 was used); the last one is to redo the IRC computation specifying to compute force constant at every step. Three methods were all used and the results were summarized in '''Table 11'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Improved IRC method 1
! scope="col" | Improved IRC method 2
! scope="col" | Improved IRC method 3
|-
| [[Image:XJB_IRC1.jpg|300px|center|IRC1]]||[[Image:XJB_IRC2.jpg|300px|center|IRC3]] || [[Image:XJB_IRC3.jpg|300px|center|IRC3]]
|-
| D = -64.17<sup>o</sup> A = 112.04<sup>o</sup> || D = -66.83<sup>o</sup> A = 111.82<sup>o</sup> || D = -64.17<sup>o</sup> A = 112.04<sup>o</sup>
|-
| -231.69167Ha || -231.69150Ha || -231.69167Ha
|-
| [[File:XJB_IRC1.log]]||[[File:XJB_IRC2.log]]||[[File:XJB_IRC3.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 11.''' ''Comparison of three improvement methods of IRC computation upon the chair transition structure''</div>

The first method was very quick but maybe generated relatively low-accuracy result; the second approach involved the controlling of number of points which might result in ending up at the wrong direction and wrong structure; the third method was the most suitable option for this small system but it took longer time than the previous two. From '''Table 11''' it is concluded that method 1 and 3 gave the same lowest energy which was consistent to the energy of the ''gauche2'' conformation given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

====Activation Energies====

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | '''HF/3-21G''' at 0K
! scope="col" | '''HF/3-21G''' at 298.15K
! scope="col" | '''B3LYP/6-31G*''' at 0K
! scope="col" | '''B3LYP/6-31G*''' at 298.15K
! scope="col" | Expt. at 0K
|-
| '''ΔE (chair)'''||45.71||44.70||34.05||33.16||33.5 ± 0.5
|-
| '''ΔE (boat)'''||55.60||54.76||41.96||41.32||44.7 ± 0.5
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 12.''' ''Summary of activation energies for the Cope rearrangement via both transition states in kcal/mol''</div>

The activation energy calculation was done by calculating the difference in energies between the optimized reactant ''anti2'' and the optimized chair and boat transition states. It is observed that chair conformation gives generally lower activation energy thus the more favoured transition state. The structure optimized at '''B3LYP/6-31G*''' level of theory at 0K has closer activation energy to the experimental result. Thus '''B3LYP/6-31G*''' gives a better estimation of transition state than '''HF/3-21G'''.

==The Diels Alder Cycloaddition==

In this exercise, AM1 semi-empirical method is used to compute the Diels Alder cycloaddition which is a concerted pericyclic reaction. The driving force of this reaction is the formation of a new σ bond between the π orbitals of the dieneophile and the diene. The reaction is allowed with HOMO-LUMO interation and a symmetry overlap.

===Cis Butadiene===

<center>[[Image:prototype Diels Alder.jpg|500px|center|thumb|'''Figure 13'''. ''The prototype reaction between butadiene and ethylene'']]</center>

To study a prototype Diels Alder cycloaddition reaction between butadiene and ethylene, first a cis butadiene molecule was built using Gaussview and optimized using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the butadiene were plotted. As seen the HOMO of cis-butadiene is anti-symmetric ('''a''') and the LUMO is symmetric ('''s''') with respect to plane.

{| align="center"
|- align="center"
|
[[Image:XJB_HOMO.jpg|280px|thumb|'''Figure 14.''' ''Anti-symmetric HOMO of cis-butadiene'']]
|
[[Image:XJB_LUMO.jpg|300px|thumb|'''Figure 15.''' ''Symmetric HOMO of cis-butadiene'']]
|- align="center"
|}

===The Transition State of prototype reaction between ethylene and butadiene===

The Hessian method was used to estimate transition state for the reaction between ethylene and butadiene because these two reactants are relatively small and simple molecules. The transition state was drawn by first building a structure as (a) in '''Figure 16''' then removing the -CH<sub>2</sub>-CH<sub>2</sub>- group to form the envelop structure as in (b). The '''clean''' option should not be used and the dashed bond length was guessed to be larger than 2.2 Å otherwise the geometry change would cause a failure in optimization.

<center>[[Image:XJB_dielsguessts.jpg|300px|center|thumb|'''Figure 16'''. ''The guessing transition state for the reaction between butadiene and ethylene'']]</center>

<center>[[File:XJB_butadiene and ethylene ts.GIF|400px|center|thumb|'''Figure 17.''' ''Animation of the butadiene and ethylene cycloaddition transition state'']]</center>

After the Hessian optimization, the transition structure has been successfully determined and an animation was generated in '''Figure 17'''. It has a C<sub>1</sub> symmetry point group and the bond lengths of two partly formed C-C bonds were measured to be 2.12Å. The typical sp<sup>3</sup> and sp<sup>2</sup> C-C bond lengths are 1.54Å and 1.47Å respectively[ref]. The van der Waals radius of the C atom is 1.7 Å[ref]. The measured bond distance of the transition structure is larger than typical bond lengths but still within the van der Waals radius, suggesting the bond forming interation between the carbon atoms.

From the animation ('''Figure 17''') the vibration directions of two molecules are opposite and they are approaching and leaving each other at the same time. This suggests the formation of the two bonds synchronous. The vibrational frequencies were also calculated and an infrared spectrum was simulated ('''Figure 20'''). An imaginary frequency of magnitude -955 cm<sup>-1</sup> was seen and the lowest positive frequency was observed at 147 cm<sup>-1</sup> which corresponded to the reactant vibration but not the bond formation.

{| align="center"
|
[[Image:XJB_HOMOts.jpg|300px|thumb|'''Figure 18'''. HOMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_LUMOts.jpg|300px|thumb|'''Figure 19'''. LUMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_IRts.jpg|300px|thumb|'''Figure 20'''. Generated IR of transition state of reaction between cis-butadiene and ethylene]]
|}

The HOMO and LUMO molecular orbitals of this transition structure were visualized in '''Figure 18''' and '''Figure 19'''. From these two figures it can be observed that the HOMO at the transition structure is anti-symmetric ('''a''') while the LUMO is symmetric ('''s'''). Thus this anti-symmetric transition state HOMO is formed by the HOMO of cis-butadiene and the LUMO of ethylene with the knowledge that they are both anti-symmetrical. This conclusion agrees with the previous talked conditions for allowed reaction.

===The cyclohexa-1,3-diene reaction with maleic anhydride===

<center>[[Image:XJB_mechanism3.jpg|500px|center|thumb|'''Figure 21'''. ''The endo and exo products of the Diels Alder reaction between cyclohexa-1,3-diene and maleic anhydride'']]</center>

The exo and endo transition states were drawn according to '''Figure 21''' with the new C-C bond formation line dashed and bond length estimation as 2.5. Then the same Hessian method was applied to optimize the structures. The electronic energy of the endo structure was calculated to be '''-0.05150Ha''' and '''-0.05042Ha''' for the exo. This reaction is supposed to be kinetically controlled thus the lower energy endo transition state is more favoured.

The bond lengths of the partly formed C-C bond are 2.16 Å and 2.17 Å for endo and exo transition states respectively. Another measured C-C bond length is shown in '''Table 13''' as 2.89 Å for endo (C3-C8 and C2-C7) and 2.95 Å for endo (C7-C10 and C8-C9). The longer atom distance indicates the larger steric repulsion in the exo transition state because of the extra existence of hydrogen atoms on C7 and C8.

Woodward and Hoffmann also proposed a secondary orbital overlap statement as an explanation of the endo stereo-preference in Diels Alder reactions<ref name="soo">''J. Org. Chem.'', 1987, '''52(8)''', pp 1469–1474. {{DOI|10.1021/jo00384a016}}</ref>. This secondary orbital overlap does not involve bond changes but π orbital interactions as the highlighted carbon atoms of the endo transition state in '''Table 13'''. Such overlap stablises the structure and lowers the energy of the endo transition state while there is no such effect existed in the exo transition state.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | endo
! scope="col" | exo
|-
| '''Animation'''||[[File:XJB_endots.GIF|300px]]||[[File:XJB_exots.GIF|300px]]
|-
| '''Sum of electronic and zero-point Energies'''||0.133493||0.134880
|-
| '''Sum of electronic and thermal Energies'''||0.143683||0.144881
|-
| '''Sum of electronic and thermal Enthalpies'''||0.144627||0.145825
|-
| '''Sum of electronic and thermal Free Energies'''||0.097349||0.099117
|-
| '''Bond lengths'''||[[Image:XJB_endomeasure.jpg|300px]]||[[Image:XJB_exomeasure.jpg|300px]]
|-
| '''HOMO'''||[[Image:XJB_endoHOMO.jpg|270px]]||[[Image:XJB_exoHOMO.jpg|255px]]
|-
| '''Secondary orbital overlap effect'''||[[Image:XJB_endosoo.jpg|300px]]||[[Image:XJB_exosoo.jpg|300px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 13.''' ''Summary of comparison of properties of endo and exo transition structures for reaction between cyclohexa-1,3-diene and maleic anhydride''</div>

==Reference==

<references/>

Rep:Mod:XJB1130

2013-12-06T00:17:26Z

Jx1011: /* The cyclohexa-1,3-diene reaction with maleic anhydride */

==The Cope Rearrangement of 1,5-hexadiene==

The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement reaction ('''Figure 1'''). Its mechanism has been studied by a large number of experimental and computational researches, which is recently accepted that the reaction occurs via either a "chair" or a "boat" transition state structure in a concerted manner. The aim of this exercise is to locate the low-energy minimum and the transition structures on the potential energy surface by Gaussian calculation in order to study its mechanism. The reaction pathway and the barrier heights can also be calculated by this calculation.

<center>[[Image:XJB_cope rearrangement.jpg|500px|center|thumb|'''Figure 1'''. ''The Cope rearrangement'']]</center>

===Optimizing the Reactants and Products===

Different conformations of 1,5-hexadiene can be optimized using '''Gaussview'''. The optimized structure of four conformers involved in this discussion and their corresponding energies optimized at '''HF/3-21G''' level of theory are concluded in the following table ('''Table 1''').

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | anti4
! scope="col" | anti2
! scope="col" | gauche
! scope="col" | gauche3
|-
| <jmol><jmolApplet><title>XJB_anti4.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti4.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche3.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| -231.69097Ha || -231.69254Ha || -231.68772Ha ||-231.69266Ha
|}

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 1.''' ''Four conformers of 1,5-hexadiene''</div>

This exercise first involved optimizing a molecule of 1,5-hexadiene with "anti" linkage for the cetral four C atoms at the '''HF/3-21G''' level of theory. The final optimization structure had the total electronic energy of '''-231.69097Ha''' with a '''C<sub>1</sub>''' symmetry. By comparing the energy and point group to that in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1], this structure is confirmed to be ''anti4''. The compositions of energies are also listed in '''Table 2''' as i) the sum of electronic and zero-point energies (potential energy at 0K including the zero-point vibrational energy), ii) the sum of electronic and thermal energies (energy including translational, rotational and vibrational at 298.15K and 1 atm), iii) the sum of electronic and thermal enthalpies (containning an additional correction for RT), and iv) the sum of electronic and thermal free energies (including the entropic contribution to the free energy). For what we are considering, the first two terms are the most useful for further comparisons.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Formula Description'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||E = E<sub>elec</sub> + ZPE||-231.53785
|-
| Sum of electronic and thermal Energies||E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>||-231.53096
|-
| Sum of electronic and thermal Enthalpie||H = E + RT||-231.53002
|-
| Sum of electronic and thermal Free Energie||G = H - TS||-231.56891
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 2.''' ''Frequency calculations of anti4 optimized at '''HF/3-21G''' ([[File:XJB_anti4.log]])''</div>

Next another 1,5-hexadiene molecule with "gauche" linkage for the central four atoms was drawn and optimized at the '''HF/3-21G''' level of theory as before. Energy for this conformation is expected to be higher than that for the ''anti4'' conformation due to the closer position of two alkene ends thus the steric interaction. The final structure had the electronic energy of '''-231.68772Ha''' which was indeed higher than that of the ''anti4'', indicating less favour towards this gauche structure. It had a '''C<sub>2</sub>''' symmetry and was confirmed to be the ''gauche'' conformation.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53439
|-
| Sum of electronic and thermal Energies||-231.52770
|-
| Sum of electronic and thermal Enthalpie||-231.52676
|-
| Sum of electronic and thermal Free Energie||-231.56483
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 3.''' ''Frequency calculations of gauche optimized at '''HF/3-21G''' ([[File:XJB_gauche.log]])''</div>

A lowest energy conformation ''gauche3'' of the reactant molecule was drawn based on and confirmed by [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. The optimization gave a final structure with an energy of '''-231.69266Ha''' with a symmetry of '''C<sub>1</sub>'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53949
|-
| Sum of electronic and thermal Energies||-231.53265
|-
| Sum of electronic and thermal Enthalpie||-231.5317
|-
| Sum of electronic and thermal Free Energie||-231.57064
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 4.''' ''Frequency calculations of gauche3 optimized at '''HF/3-21G''' ([[File:XJB_gauche3.log]])''</div>

The next step required us to draw the C<sub>i</sub> symmetry ''anti2'' conformation of 1,5-hexadiene and optimize it at the '''HF/3-21G''' level of theory. The optimized energy was calculated to be '''-231.69254Ha''' with point group of '''C<sub>i</sub>''', which was consistent with the one given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53954
|-
| Sum of electronic and thermal Energies||-231.53257
|-
| Sum of electronic and thermal Enthalpie||-231.53162
|-
| Sum of electronic and thermal Free Energie||-231.57092
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 5.''' ''Frequency calculations of anti2 optimized at '''HF/3-21G''' ([[File:XJB_anti2.log]])''</div>

The ''anti2'' structure was then reoptimized at '''B3LYP/6-31G*''' which was a improvement over the '''HF/3-21G''' basis set. It adds polarization to all atoms and improves the modeling of core electrons thus producing more accurate description of orbitals[ref]. The reoptimized ''anti2'' structure had the total electronic energy of '''-234.61171Ha''' and the same symmetry C<sub>i</sub>. The comparison of two structures optimized using different levels of theory is summarized in '''Table 6'''. We can see that they have exactly the same dihedral angle of the central four C atoms at 180.00<sup>o</sup> and angles between three of them are very similar as 111.3<sup>o</sup> and 112.7<sup>o</sup> respectively. Thus the the geometry change using '''HF/3-21G''' or '''B3LYP/6-31G*''' is trivial.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Before re-optimization
! scope="col" | After re-optimization
|-
| <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2reoptimized</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2reoptimized.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| ''anti2 optimized at '''HF/3-21G''''' || ''anti2 optimized at '''B3LYP/6-31G*'''''
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 6.''' ''Two optimization of anti2 at '''HF/3-21G''' and '''B3LYP/6-31G*'''([[File:XJB_anti2.log]])''</div>

The frequency calculation was run as well and a list of energies is summarized in '''Table 7'''. A list of all the vibrational frequencies was also generated and no imaginary (negative) frequency was observed and these vibrations are visualized by simulating the infrared spectrum in '''Figure 2'''.

{| align="center"
|- align="center"
|
{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-234.46920
|-
| Sum of electronic and thermal Energies||-234.46186
|-
| Sum of electronic and thermal Enthalpie||-234.46091
|-
| Sum of electronic and thermal Free Energie||-234.50078
|}
|
[[Image:XJB_IR_anti2.jpg|300px]]
|- align="center"
| align="center" | '''Table 7.''' ''Frequency calculations of anti2 optimized at '''B3LYP/6-31G*'''
([[File:XJB_anti2_reoptimized.log]])''
| align="center" | '''Figure 2.''' ''The generated IR spectrum of optimized '''B3LYP/6-31G*''' structure''
|}

===Optimizing the "Chair" and "Boat" Transition Structures===

The study of the chair and boat transition states is carried out using three different approaches. The first one is called the Hessian which involves computing the force constant matrix; the second one is to freeze the reaction coordinate and then optimize the structure once the molecule is fully relaxed; the last one is to use QST2 method to specify the reactants and products then finding the transition structure.

<center>[[Image:XJB_2 ts.jpg|500px|center|thumb|'''Figure 3'''. ''The two transition states of Cope rearrangement'']]</center>

====The Chair Transition State====

In order to set up a transition state optimization, first an allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimized at the '''HF/3-21G''' level of theory. Then this half of the transition structure was duplicated with rotation to make the approximate resembling of the chair transition state with terminal ends of the fragments 2.2 Å apart. The chair transition structure optimization was then done by the first two different approaches.

<center>[[Image:XJB_guess ts.jpg|400px|center|thumb|'''Figure 4'''. ''The guessed chair transition structure of two allyl fragments'']]</center>

For the first optimization approach, the force constants were computed (the Hessian) and the chair transition state was optimized to a TS (Berny) at the '''HF/3-21G''' level of theory. The frequency calculation was carried out at the same time and gave an imaginary frequency at -818 cm<sup>-1</sup>. The electronic energy was calculated to be '''-231.61932Ha''' with C<sub>2h</sub> symmetry point group. The distance between the terminal carbons was 2.0203 Å. An animation of this transition state vibrations was generated to confirm the corresponding Cope rearrangement.

<center>[[File:XJB_chair ts animation.GIF|400px|center|thumb|'''Figure 5.''' ''Animation of the Cope rearrangement via the chair transition state'']]</center>

The chair transition state was then reoptimized using the '''B3LYP/6-31G*''' level of theory and carried out frequency calculations as well. The electronic energy calculated this time was '''-234.55698Ha''' with an imaginary frequency at -566 cm<sup>-1</sup>. We can see that with the geometries quite similar, the energies differ markedly calculated using two levels of theory. The summary and comparison of the calculations optimized at two levels of theory are summarized in '''Table 8'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46670||-234.41493
|-
| Sum of electronic and thermal Energies||-231.46134||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40807
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir1.jpg|200px]]||[[Image:XJB_chairir2.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 8.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the first approach
([[File:XJB_chairts1.1.log]], [[File:XJB_chairts1.2.log]])''</div>

The second approach is the frozen coordinate method using the Redundant Coord Editor to freeze the bond lengths of the terminal ends to 2.2 Å. First the calculation failed because of the recent update of GaussView. After editting the coordinate distance (B 5 1 2.2 F) and the frozen distance (B 5 1 F) the optimization went well this time. The structure was then optimized again using a normal guess Hessian modified including the two differentiating coordinates. This chair transition state was then reoptimized again using the '''B3LYP/6-31G*''' level of theory. The electronic energies were calculated to be the same as in the first approach and frequency calculations were summarized in '''Table 9'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46671||-234.41490
|-
| Sum of electronic and thermal Energies||-231.46135||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40806
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir3.jpg|200px]]||[[Image:XJB_chairir4.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 9.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the second approach
([[File:XJB_chairts2.1.log]], [[File:XJB_chairts2.2.log]])''</div>

As the same energy results given, the bond forming/breaking bond lengths in the chair transition structures in the above two different approaches were compared. The first approach gave the bond lengths of 2.0203 Å while the second approach gave 2.0207 Å. Considering the accuracy limit of bond length estimation in Gaussview, these two bond lengths can be seen as exactly the same. Two methods worked reasonably well because of the good assumption of transition structures. However for more complicated and larger molecule, the Hessian method may fail due to the difficulty to predict the transition geometry (different surface curvature). Thus better way to generate such transition structures is to freeze reaction coordinate.

{| align="center"
|- align="center"
|
[[Image:XJB_chairts bondlength1.jpg|350px|thumb|'''Figure 6.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by Hession'']]
|
[[Image:XJB_chairts bondlength2.jpg|350px|thumb|'''Figure 7.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by frozen coordinate'']]
|- align="center"
|}

====The Boat Transition State====

The boat transition structure was then optimized using the '''QST2''' method. The same numbering for the reactant and product molecules was important in order to specify for this reaction. The renumbering of the molecules is shown as in '''Figure 8''', but the following optimization failed and gave a twisted structure. The reactant and product geometries were then modified further by changing the central C-C-C-C dihedral angle to 0<sup>o</sup> and the inside two C-C-C (i.e. C2-C3-C4 and C3-C4-C5) angles to 100<sup>o</sup>. The job was run successfully and frequency calculations were done with the boat transition state generated. Only one imaginary frequency was detected also at -840 cm<sup>-1</sup>. The electronic energy was '''-231.60280Ha''' with C<sub>2v</sub> symmetry. The distances between the terminal carbons of the allyl fragments were examined to be both 2.140Å which were comparably longer than that of the chair transition structure.

{| align="center"
|- align="center"
|
[[Image:XJB_QST1.jpg|400px|thumb|'''Figure 8.''' ''Numbering of reactant and product'']]
|
[[Image:XJB_QST2.jpg|410px|thumb|'''Figure 9.''' ''Re-numbering of reactant and product'']]
|- align="center"
|}

<center>[[File:XJB_boat ts animation.GIF|400px|center|thumb|'''Figure 10.''' ''Animation of the Cope rearrangement via the boat transition state'']]</center>

The structure was then reoptimized at the higher '''B3LYP/6-31G*''' theory of level and the electronic energy was calculated to be '''-234.54309Ha''' with very similar geometry. The frequency calculation generated an imaginary frequency at -530 cm<sup>-1</sup>. The summary and comparison of two levels' calculations are carried out in the following table.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.45093||-234.40234
|-
| Sum of electronic and thermal Energies||-231.44530||-234.39601
|-
| Sum of electronic and thermal Enthalpie|| -231.44436||-234.39506
|-
| Sum of electronic and thermal Free Energie||-231.47977||-234.43175
|-
| Generated IR spectrum||[[Image:XJB_boatir1.jpg|200px]]||[[Image:XJB_boatir2.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 10.''' ''Frequency calculations of the boat transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by QST2
([[File:XJB_boat ts1.log]], [[File:XJB_boat ts2.log]])''</div>

====Intrinsic Reaction Coordinate====

Next a useful method called '''Intrinct Reaction Coordinate (IRC)''' is applied to the Hessian optimized chair transition structure to follow the mininmum energy path (MEP) from a transition structure down to its local minimum on a potential energy surface. IRC can be used to verify that the transition state you found actually connects the reactants and products you think it does, or to identify any reaction intermediates you haven't thought about, or to generate an animation of the reaction [ref]. The symmetrical reaction coordinate was computed only in the forward direction with force constant always calculated and the number of points along IRC 50. The last structure generated along IRC is given in '''Figure 11''' with the energy calculated as '''-231.69158Ha''' and the dihedral angle and internal angle measured as D = -67.188<sup>o</sup> A = 111.78<sup>o</sup>. This energy does not match to any of the conformations discussed in Appendix 1 thus a minimum geometry has not reached yet.

{| align="center"
|- align="center"
|
<jmol><jmolApplet><title>XJB_IRC.mol</title><color>white</color>
<size>200</size><script>zoom=5</script>
<uploadedFileContents>XJB_IRC.mol</uploadedFileContents>
</jmolApplet></jmol>
|
[[Image:XJB_irc.jpg|500px]]
|- align="center"
| align="center" | '''Figure 11.''' ''The energy minimum structure along IRC''
| align="center" | '''Figure 12.''' ''Total energy along IRC''
|}

Three improving options can be used to reach the minimum energy. The first one is to run a normal Hessian optimization of the last point on IRC; the second one involves using a larger number of points (in this case 100 was used); the last one is to redo the IRC computation specifying to compute force constant at every step. Three methods were all used and the results were summarized in '''Table 11'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Improved IRC method 1
! scope="col" | Improved IRC method 2
! scope="col" | Improved IRC method 3
|-
| [[Image:XJB_IRC1.jpg|300px|center|IRC1]]||[[Image:XJB_IRC2.jpg|300px|center|IRC3]] || [[Image:XJB_IRC3.jpg|300px|center|IRC3]]
|-
| D = -64.17<sup>o</sup> A = 112.04<sup>o</sup> || D = -66.83<sup>o</sup> A = 111.82<sup>o</sup> || D = -64.17<sup>o</sup> A = 112.04<sup>o</sup>
|-
| -231.69167Ha || -231.69150Ha || -231.69167Ha
|-
| [[File:XJB_IRC1.log]]||[[File:XJB_IRC2.log]]||[[File:XJB_IRC3.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 11.''' ''Comparison of three improvement methods of IRC computation upon the chair transition structure''</div>

The first method was very quick but maybe generated relatively low-accuracy result; the second approach involved the controlling of number of points which might result in ending up at the wrong direction and wrong structure; the third method was the most suitable option for this small system but it took longer time than the previous two. From '''Table 11''' it is concluded that method 1 and 3 gave the same lowest energy which was consistent to the energy of the ''gauche2'' conformation given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

====Activation Energies====

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | HF/3-21G at 0K
! scope="col" | HF/3-21G at 298.15K
! scope="col" | B3LYP/6-31G* at 0K
! scope="col" | B3LYP/6-31G* at 298.15K
! scope="col" | Expt. at 0K
|-
| '''ΔE (chair)'''||45.71||44.70||34.05||33.16||33.5 ± 0.5
|-
| '''ΔE (boat)'''||55.60||54.76||41.96||41.32||44.7 ± 0.5
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 12.''' ''Summary of activation energies for the Cope rearrangement via both transition states in kcal/mol''</div>

==The Diels Alder Cycloaddition==

In this exercise, AM1 semi-empirical method is used to compute the Diels Alder cycloaddition which is a concerted pericyclic reaction. The driving force of this reaction is the formation of a new σ bond between the π orbitals of the dieneophile and the diene. The reaction is allowed with HOMO-LUMO interation and a symmetry overlap.

===Cis Butadiene===

<center>[[Image:prototype Diels Alder.jpg|500px|center|thumb|'''Figure 13'''. ''The prototype reaction between butadiene and ethylene'']]</center>

To study a prototype Diels Alder cycloaddition reaction between butadiene and ethylene, first a cis butadiene molecule was built using Gaussview and optimized using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the butadiene were plotted. As seen the HOMO of cis-butadiene is anti-symmetric ('''a''') and the LUMO is symmetric ('''s''') with respect to plane.

{| align="center"
|- align="center"
|
[[Image:XJB_HOMO.jpg|280px|thumb|'''Figure 14.''' ''Anti-symmetric HOMO of cis-butadiene'']]
|
[[Image:XJB_LUMO.jpg|300px|thumb|'''Figure 15.''' ''Symmetric HOMO of cis-butadiene'']]
|- align="center"
|}

===The Transition State of prototype reaction between ethylene and butadiene===

The Hessian method was used to estimate transition state for the reaction between ethylene and butadiene because these two reactants are relatively small and simple molecules. The transition state was drawn by first building a structure as (a) in '''Figure 16''' then removing the -CH<sub>2</sub>-CH<sub>2</sub>- group to form the envelop structure as in (b). The '''clean''' option should not be used and the dashed bond length was guessed to be larger than 2.2 Å otherwise the geometry change would cause a failure in optimization.

<center>[[Image:XJB_dielsguessts.jpg|300px|center|thumb|'''Figure 16'''. ''The guessing transition state for the reaction between butadiene and ethylene'']]</center>

<center>[[File:XJB_butadiene and ethylene ts.GIF|400px|center|thumb|'''Figure 17.''' ''Animation of the butadiene and ethylene cycloaddition transition state'']]</center>

After the Hessian optimization, the transition structure has been successfully determined and an animation was generated in '''Figure 17'''. It has a C<sub>1</sub> symmetry point group and the bond lengths of two partly formed C-C bonds were measured to be 2.12Å. The typical sp<sup>3</sup> and sp<sup>2</sup> C-C bond lengths are 1.54Å and 1.47Å respectively[ref]. The van der Waals radius of the C atom is 1.7 Å[ref]. The measured bond distance of the transition structure is larger than typical bond lengths but still within the van der Waals radius, suggesting the bond forming interation between the carbon atoms.

From the animation ('''Figure 17''') the vibration directions of two molecules are opposite and they are approaching and leaving each other at the same time. This suggests the formation of the two bonds synchronous. The vibrational frequencies were also calculated and an infrared spectrum was simulated ('''Figure 20'''). An imaginary frequency of magnitude -955 cm<sup>-1</sup> was seen and the lowest positive frequency was observed at 147 cm<sup>-1</sup> which corresponded to the reactant vibration but not the bond formation.

{| align="center"
|
[[Image:XJB_HOMOts.jpg|300px|thumb|'''Figure 18'''. HOMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_LUMOts.jpg|300px|thumb|'''Figure 19'''. LUMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_IRts.jpg|300px|thumb|'''Figure 20'''. Generated IR of transition state of reaction between cis-butadiene and ethylene]]
|}

The HOMO and LUMO molecular orbitals of this transition structure were visualized in '''Figure 18''' and '''Figure 19'''. From these two figures it can be observed that the HOMO at the transition structure is anti-symmetric ('''a''') while the LUMO is symmetric ('''s'''). Thus this anti-symmetric transition state HOMO is formed by the HOMO of cis-butadiene and the LUMO of ethylene with the knowledge that they are both anti-symmetrical. This conclusion agrees with the previous talked conditions for allowed reaction.

===The cyclohexa-1,3-diene reaction with maleic anhydride===

<center>[[Image:XJB_mechanism3.jpg|500px|center|thumb|'''Figure 21'''. ''The endo and exo products of the Diels Alder reaction between cyclohexa-1,3-diene and maleic anhydride'']]</center>

The exo and endo transition states were drawn according to '''Figure 21''' with the new C-C bond formation line dashed and bond length estimation as 2.5. Then the same Hessian method was applied to optimize the structures. The electronic energy of the endo structure was calculated to be '''-0.05150Ha''' and '''-0.05042Ha''' for the exo. This reaction is supposed to be kinetically controlled thus the lower energy endo transition state is more favoured.

The bond lengths of the partly formed C-C bond are 2.16 Å and 2.17 Å for endo and exo transition states respectively. Another measured C-C bond length is shown in '''Table 13''' as 2.89 Å for endo (C3-C8 and C2-C7) and 2.95 Å for endo (C7-C10 and C8-C9). The longer atom distance indicates the larger steric repulsion in the exo transition state because of the extra existence of hydrogen atoms on C7 and C8.

Woodward and Hoffmann also proposed a secondary orbital overlap statement as an explanation of the endo stereo-preference in Diels Alder reactions<ref name="soo">''J. Org. Chem.'', 1987, '''52(8)''', pp 1469–1474. {{DOI|10.1021/jo00384a016}}</ref>. This secondary orbital overlap does not involve bond changes but π orbital interactions as the highlighted carbon atoms of the endo transition state in '''Table 13'''. Such overlap stablises the structure and lowers the energy of the endo transition state while there is no such effect existed in the exo transition state.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | endo
! scope="col" | exo
|-
| '''Animation'''||[[File:XJB_endots.GIF|300px]]||[[File:XJB_exots.GIF|300px]]
|-
| '''Sum of electronic and zero-point Energies'''||0.133493||0.134880
|-
| '''Sum of electronic and thermal Energies'''||0.143683||0.144881
|-
| '''Sum of electronic and thermal Enthalpies'''||0.144627||0.145825
|-
| '''Sum of electronic and thermal Free Energies'''||0.097349||0.099117
|-
| '''Bond lengths'''||[[Image:XJB_endomeasure.jpg|300px]]||[[Image:XJB_exomeasure.jpg|300px]]
|-
| '''HOMO'''||[[Image:XJB_endoHOMO.jpg|270px]]||[[Image:XJB_exoHOMO.jpg|255px]]
|-
| '''Secondary orbital overlap effect'''||[[Image:XJB_endosoo.jpg|300px]]||[[Image:XJB_exosoo.jpg|300px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 13.''' ''Summary of comparison of properties of endo and exo transition structures for reaction between cyclohexa-1,3-diene and maleic anhydride''</div>

==Reference==

<references/>

Rep:Mod:XJB1130

2013-12-06T00:15:45Z

Jx1011: /* Intrinsic Reaction Coordinate */

==The Cope Rearrangement of 1,5-hexadiene==

The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement reaction ('''Figure 1'''). Its mechanism has been studied by a large number of experimental and computational researches, which is recently accepted that the reaction occurs via either a "chair" or a "boat" transition state structure in a concerted manner. The aim of this exercise is to locate the low-energy minimum and the transition structures on the potential energy surface by Gaussian calculation in order to study its mechanism. The reaction pathway and the barrier heights can also be calculated by this calculation.

<center>[[Image:XJB_cope rearrangement.jpg|500px|center|thumb|'''Figure 1'''. ''The Cope rearrangement'']]</center>

===Optimizing the Reactants and Products===

Different conformations of 1,5-hexadiene can be optimized using '''Gaussview'''. The optimized structure of four conformers involved in this discussion and their corresponding energies optimized at '''HF/3-21G''' level of theory are concluded in the following table ('''Table 1''').

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | anti4
! scope="col" | anti2
! scope="col" | gauche
! scope="col" | gauche3
|-
| <jmol><jmolApplet><title>XJB_anti4.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti4.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche3.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| -231.69097Ha || -231.69254Ha || -231.68772Ha ||-231.69266Ha
|}

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 1.''' ''Four conformers of 1,5-hexadiene''</div>

This exercise first involved optimizing a molecule of 1,5-hexadiene with "anti" linkage for the cetral four C atoms at the '''HF/3-21G''' level of theory. The final optimization structure had the total electronic energy of '''-231.69097Ha''' with a '''C<sub>1</sub>''' symmetry. By comparing the energy and point group to that in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1], this structure is confirmed to be ''anti4''. The compositions of energies are also listed in '''Table 2''' as i) the sum of electronic and zero-point energies (potential energy at 0K including the zero-point vibrational energy), ii) the sum of electronic and thermal energies (energy including translational, rotational and vibrational at 298.15K and 1 atm), iii) the sum of electronic and thermal enthalpies (containning an additional correction for RT), and iv) the sum of electronic and thermal free energies (including the entropic contribution to the free energy). For what we are considering, the first two terms are the most useful for further comparisons.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Formula Description'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||E = E<sub>elec</sub> + ZPE||-231.53785
|-
| Sum of electronic and thermal Energies||E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>||-231.53096
|-
| Sum of electronic and thermal Enthalpie||H = E + RT||-231.53002
|-
| Sum of electronic and thermal Free Energie||G = H - TS||-231.56891
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 2.''' ''Frequency calculations of anti4 optimized at '''HF/3-21G''' ([[File:XJB_anti4.log]])''</div>

Next another 1,5-hexadiene molecule with "gauche" linkage for the central four atoms was drawn and optimized at the '''HF/3-21G''' level of theory as before. Energy for this conformation is expected to be higher than that for the ''anti4'' conformation due to the closer position of two alkene ends thus the steric interaction. The final structure had the electronic energy of '''-231.68772Ha''' which was indeed higher than that of the ''anti4'', indicating less favour towards this gauche structure. It had a '''C<sub>2</sub>''' symmetry and was confirmed to be the ''gauche'' conformation.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53439
|-
| Sum of electronic and thermal Energies||-231.52770
|-
| Sum of electronic and thermal Enthalpie||-231.52676
|-
| Sum of electronic and thermal Free Energie||-231.56483
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 3.''' ''Frequency calculations of gauche optimized at '''HF/3-21G''' ([[File:XJB_gauche.log]])''</div>

A lowest energy conformation ''gauche3'' of the reactant molecule was drawn based on and confirmed by [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. The optimization gave a final structure with an energy of '''-231.69266Ha''' with a symmetry of '''C<sub>1</sub>'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53949
|-
| Sum of electronic and thermal Energies||-231.53265
|-
| Sum of electronic and thermal Enthalpie||-231.5317
|-
| Sum of electronic and thermal Free Energie||-231.57064
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 4.''' ''Frequency calculations of gauche3 optimized at '''HF/3-21G''' ([[File:XJB_gauche3.log]])''</div>

The next step required us to draw the C<sub>i</sub> symmetry ''anti2'' conformation of 1,5-hexadiene and optimize it at the '''HF/3-21G''' level of theory. The optimized energy was calculated to be '''-231.69254Ha''' with point group of '''C<sub>i</sub>''', which was consistent with the one given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53954
|-
| Sum of electronic and thermal Energies||-231.53257
|-
| Sum of electronic and thermal Enthalpie||-231.53162
|-
| Sum of electronic and thermal Free Energie||-231.57092
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 5.''' ''Frequency calculations of anti2 optimized at '''HF/3-21G''' ([[File:XJB_anti2.log]])''</div>

The ''anti2'' structure was then reoptimized at '''B3LYP/6-31G*''' which was a improvement over the '''HF/3-21G''' basis set. It adds polarization to all atoms and improves the modeling of core electrons thus producing more accurate description of orbitals[ref]. The reoptimized ''anti2'' structure had the total electronic energy of '''-234.61171Ha''' and the same symmetry C<sub>i</sub>. The comparison of two structures optimized using different levels of theory is summarized in '''Table 6'''. We can see that they have exactly the same dihedral angle of the central four C atoms at 180.00<sup>o</sup> and angles between three of them are very similar as 111.3<sup>o</sup> and 112.7<sup>o</sup> respectively. Thus the the geometry change using '''HF/3-21G''' or '''B3LYP/6-31G*''' is trivial.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Before re-optimization
! scope="col" | After re-optimization
|-
| <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2reoptimized</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2reoptimized.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| ''anti2 optimized at '''HF/3-21G''''' || ''anti2 optimized at '''B3LYP/6-31G*'''''
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 6.''' ''Two optimization of anti2 at '''HF/3-21G''' and '''B3LYP/6-31G*'''([[File:XJB_anti2.log]])''</div>

The frequency calculation was run as well and a list of energies is summarized in '''Table 7'''. A list of all the vibrational frequencies was also generated and no imaginary (negative) frequency was observed and these vibrations are visualized by simulating the infrared spectrum in '''Figure 2'''.

{| align="center"
|- align="center"
|
{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-234.46920
|-
| Sum of electronic and thermal Energies||-234.46186
|-
| Sum of electronic and thermal Enthalpie||-234.46091
|-
| Sum of electronic and thermal Free Energie||-234.50078
|}
|
[[Image:XJB_IR_anti2.jpg|300px]]
|- align="center"
| align="center" | '''Table 7.''' ''Frequency calculations of anti2 optimized at '''B3LYP/6-31G*'''
([[File:XJB_anti2_reoptimized.log]])''
| align="center" | '''Figure 2.''' ''The generated IR spectrum of optimized '''B3LYP/6-31G*''' structure''
|}

===Optimizing the "Chair" and "Boat" Transition Structures===

The study of the chair and boat transition states is carried out using three different approaches. The first one is called the Hessian which involves computing the force constant matrix; the second one is to freeze the reaction coordinate and then optimize the structure once the molecule is fully relaxed; the last one is to use QST2 method to specify the reactants and products then finding the transition structure.

<center>[[Image:XJB_2 ts.jpg|500px|center|thumb|'''Figure 3'''. ''The two transition states of Cope rearrangement'']]</center>

====The Chair Transition State====

In order to set up a transition state optimization, first an allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimized at the '''HF/3-21G''' level of theory. Then this half of the transition structure was duplicated with rotation to make the approximate resembling of the chair transition state with terminal ends of the fragments 2.2 Å apart. The chair transition structure optimization was then done by the first two different approaches.

<center>[[Image:XJB_guess ts.jpg|400px|center|thumb|'''Figure 4'''. ''The guessed chair transition structure of two allyl fragments'']]</center>

For the first optimization approach, the force constants were computed (the Hessian) and the chair transition state was optimized to a TS (Berny) at the '''HF/3-21G''' level of theory. The frequency calculation was carried out at the same time and gave an imaginary frequency at -818 cm<sup>-1</sup>. The electronic energy was calculated to be '''-231.61932Ha''' with C<sub>2h</sub> symmetry point group. The distance between the terminal carbons was 2.0203 Å. An animation of this transition state vibrations was generated to confirm the corresponding Cope rearrangement.

<center>[[File:XJB_chair ts animation.GIF|400px|center|thumb|'''Figure 5.''' ''Animation of the Cope rearrangement via the chair transition state'']]</center>

The chair transition state was then reoptimized using the '''B3LYP/6-31G*''' level of theory and carried out frequency calculations as well. The electronic energy calculated this time was '''-234.55698Ha''' with an imaginary frequency at -566 cm<sup>-1</sup>. We can see that with the geometries quite similar, the energies differ markedly calculated using two levels of theory. The summary and comparison of the calculations optimized at two levels of theory are summarized in '''Table 8'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46670||-234.41493
|-
| Sum of electronic and thermal Energies||-231.46134||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40807
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir1.jpg|200px]]||[[Image:XJB_chairir2.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 8.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the first approach
([[File:XJB_chairts1.1.log]], [[File:XJB_chairts1.2.log]])''</div>

The second approach is the frozen coordinate method using the Redundant Coord Editor to freeze the bond lengths of the terminal ends to 2.2 Å. First the calculation failed because of the recent update of GaussView. After editting the coordinate distance (B 5 1 2.2 F) and the frozen distance (B 5 1 F) the optimization went well this time. The structure was then optimized again using a normal guess Hessian modified including the two differentiating coordinates. This chair transition state was then reoptimized again using the '''B3LYP/6-31G*''' level of theory. The electronic energies were calculated to be the same as in the first approach and frequency calculations were summarized in '''Table 9'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46671||-234.41490
|-
| Sum of electronic and thermal Energies||-231.46135||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40806
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir3.jpg|200px]]||[[Image:XJB_chairir4.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 9.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the second approach
([[File:XJB_chairts2.1.log]], [[File:XJB_chairts2.2.log]])''</div>

As the same energy results given, the bond forming/breaking bond lengths in the chair transition structures in the above two different approaches were compared. The first approach gave the bond lengths of 2.0203 Å while the second approach gave 2.0207 Å. Considering the accuracy limit of bond length estimation in Gaussview, these two bond lengths can be seen as exactly the same. Two methods worked reasonably well because of the good assumption of transition structures. However for more complicated and larger molecule, the Hessian method may fail due to the difficulty to predict the transition geometry (different surface curvature). Thus better way to generate such transition structures is to freeze reaction coordinate.

{| align="center"
|- align="center"
|
[[Image:XJB_chairts bondlength1.jpg|350px|thumb|'''Figure 6.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by Hession'']]
|
[[Image:XJB_chairts bondlength2.jpg|350px|thumb|'''Figure 7.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by frozen coordinate'']]
|- align="center"
|}

====The Boat Transition State====

The boat transition structure was then optimized using the '''QST2''' method. The same numbering for the reactant and product molecules was important in order to specify for this reaction. The renumbering of the molecules is shown as in '''Figure 8''', but the following optimization failed and gave a twisted structure. The reactant and product geometries were then modified further by changing the central C-C-C-C dihedral angle to 0<sup>o</sup> and the inside two C-C-C (i.e. C2-C3-C4 and C3-C4-C5) angles to 100<sup>o</sup>. The job was run successfully and frequency calculations were done with the boat transition state generated. Only one imaginary frequency was detected also at -840 cm<sup>-1</sup>. The electronic energy was '''-231.60280Ha''' with C<sub>2v</sub> symmetry. The distances between the terminal carbons of the allyl fragments were examined to be both 2.140Å which were comparably longer than that of the chair transition structure.

{| align="center"
|- align="center"
|
[[Image:XJB_QST1.jpg|400px|thumb|'''Figure 8.''' ''Numbering of reactant and product'']]
|
[[Image:XJB_QST2.jpg|410px|thumb|'''Figure 9.''' ''Re-numbering of reactant and product'']]
|- align="center"
|}

<center>[[File:XJB_boat ts animation.GIF|400px|center|thumb|'''Figure 10.''' ''Animation of the Cope rearrangement via the boat transition state'']]</center>

The structure was then reoptimized at the higher '''B3LYP/6-31G*''' theory of level and the electronic energy was calculated to be '''-234.54309Ha''' with very similar geometry. The frequency calculation generated an imaginary frequency at -530 cm<sup>-1</sup>. The summary and comparison of two levels' calculations are carried out in the following table.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.45093||-234.40234
|-
| Sum of electronic and thermal Energies||-231.44530||-234.39601
|-
| Sum of electronic and thermal Enthalpie|| -231.44436||-234.39506
|-
| Sum of electronic and thermal Free Energie||-231.47977||-234.43175
|-
| Generated IR spectrum||[[Image:XJB_boatir1.jpg|200px]]||[[Image:XJB_boatir2.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 10.''' ''Frequency calculations of the boat transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by QST2
([[File:XJB_boat ts1.log]], [[File:XJB_boat ts2.log]])''</div>

====Intrinsic Reaction Coordinate====

Next a useful method called '''Intrinct Reaction Coordinate (IRC)''' is applied to the Hessian optimized chair transition structure to follow the mininmum energy path (MEP) from a transition structure down to its local minimum on a potential energy surface. IRC can be used to verify that the transition state you found actually connects the reactants and products you think it does, or to identify any reaction intermediates you haven't thought about, or to generate an animation of the reaction [ref]. The symmetrical reaction coordinate was computed only in the forward direction with force constant always calculated and the number of points along IRC 50. The last structure generated along IRC is given in '''Figure 11''' with the energy calculated as '''-231.69158Ha''' and the dihedral angle and internal angle measured as D = -67.188<sup>o</sup> A = 111.78<sup>o</sup>. This energy does not match to any of the conformations discussed in Appendix 1 thus a minimum geometry has not reached yet.

{| align="center"
|- align="center"
|
<jmol><jmolApplet><title>XJB_IRC.mol</title><color>white</color>
<size>200</size><script>zoom=5</script>
<uploadedFileContents>XJB_IRC.mol</uploadedFileContents>
</jmolApplet></jmol>
|
[[Image:XJB_irc.jpg|500px]]
|- align="center"
| align="center" | '''Figure 11.''' ''The energy minimum structure along IRC''
| align="center" | '''Figure 12.''' ''Total energy along IRC''
|}

Three improving options can be used to reach the minimum energy. The first one is to run a normal Hessian optimization of the last point on IRC; the second one involves using a larger number of points (in this case 100 was used); the last one is to redo the IRC computation specifying to compute force constant at every step. Three methods were all used and the results were summarized in '''Table 11'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Improved IRC method 1
! scope="col" | Improved IRC method 2
! scope="col" | Improved IRC method 3
|-
| [[Image:XJB_IRC1.jpg|300px|center|IRC1]]||[[Image:XJB_IRC2.jpg|300px|center|IRC3]] || [[Image:XJB_IRC3.jpg|300px|center|IRC3]]
|-
| D = -64.17<sup>o</sup> A = 112.04<sup>o</sup> || D = -66.83<sup>o</sup> A = 111.82<sup>o</sup> || D = -64.17<sup>o</sup> A = 112.04<sup>o</sup>
|-
| -231.69167Ha || -231.69150Ha || -231.69167Ha
|-
| [[File:XJB_IRC1.log]]||[[File:XJB_IRC2.log]]||[[File:XJB_IRC3.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 11.''' ''Comparison of three improvement methods of IRC computation upon the chair transition structure''</div>

The first method was very quick but maybe generated relatively low-accuracy result; the second approach involved the controlling of number of points which might result in ending up at the wrong direction and wrong structure; the third method was the most suitable option for this small system but it took longer time than the previous two. From '''Table 11''' it is concluded that method 1 and 3 gave the same lowest energy which was consistent to the energy of the ''gauche2'' conformation given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

====Activation Energies====

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | HF/3-21G at 0K
! scope="col" | HF/3-21G at 298.15K
! scope="col" | B3LYP/6-31G* at 0K
! scope="col" | B3LYP/6-31G* at 298.15K
! scope="col" | Expt. at 0K
|-
| '''ΔE (chair)'''||45.71||44.70||34.05||33.16||33.5 ± 0.5
|-
| '''ΔE (boat)'''||55.60||54.76||41.96||41.32||44.7 ± 0.5
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 12.''' ''Summary of activation energies for the Cope rearrangement via both transition states in kcal/mol''</div>

==The Diels Alder Cycloaddition==

In this exercise, AM1 semi-empirical method is used to compute the Diels Alder cycloaddition which is a concerted pericyclic reaction. The driving force of this reaction is the formation of a new σ bond between the π orbitals of the dieneophile and the diene. The reaction is allowed with HOMO-LUMO interation and a symmetry overlap.

===Cis Butadiene===

<center>[[Image:prototype Diels Alder.jpg|500px|center|thumb|'''Figure 13'''. ''The prototype reaction between butadiene and ethylene'']]</center>

To study a prototype Diels Alder cycloaddition reaction between butadiene and ethylene, first a cis butadiene molecule was built using Gaussview and optimized using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the butadiene were plotted. As seen the HOMO of cis-butadiene is anti-symmetric ('''a''') and the LUMO is symmetric ('''s''') with respect to plane.

{| align="center"
|- align="center"
|
[[Image:XJB_HOMO.jpg|280px|thumb|'''Figure 14.''' ''Anti-symmetric HOMO of cis-butadiene'']]
|
[[Image:XJB_LUMO.jpg|300px|thumb|'''Figure 15.''' ''Symmetric HOMO of cis-butadiene'']]
|- align="center"
|}

===The Transition State of prototype reaction between ethylene and butadiene===

The Hessian method was used to estimate transition state for the reaction between ethylene and butadiene because these two reactants are relatively small and simple molecules. The transition state was drawn by first building a structure as (a) in '''Figure 16''' then removing the -CH<sub>2</sub>-CH<sub>2</sub>- group to form the envelop structure as in (b). The '''clean''' option should not be used and the dashed bond length was guessed to be larger than 2.2 Å otherwise the geometry change would cause a failure in optimization.

<center>[[Image:XJB_dielsguessts.jpg|300px|center|thumb|'''Figure 16'''. ''The guessing transition state for the reaction between butadiene and ethylene'']]</center>

<center>[[File:XJB_butadiene and ethylene ts.GIF|400px|center|thumb|'''Figure 17.''' ''Animation of the butadiene and ethylene cycloaddition transition state'']]</center>

After the Hessian optimization, the transition structure has been successfully determined and an animation was generated in '''Figure 17'''. It has a C<sub>1</sub> symmetry point group and the bond lengths of two partly formed C-C bonds were measured to be 2.12Å. The typical sp<sup>3</sup> and sp<sup>2</sup> C-C bond lengths are 1.54Å and 1.47Å respectively[ref]. The van der Waals radius of the C atom is 1.7 Å[ref]. The measured bond distance of the transition structure is larger than typical bond lengths but still within the van der Waals radius, suggesting the bond forming interation between the carbon atoms.

From the animation ('''Figure 17''') the vibration directions of two molecules are opposite and they are approaching and leaving each other at the same time. This suggests the formation of the two bonds synchronous. The vibrational frequencies were also calculated and an infrared spectrum was simulated ('''Figure 20'''). An imaginary frequency of magnitude -955 cm<sup>-1</sup> was seen and the lowest positive frequency was observed at 147 cm<sup>-1</sup> which corresponded to the reactant vibration but not the bond formation.

{| align="center"
|
[[Image:XJB_HOMOts.jpg|300px|thumb|'''Figure 18'''. HOMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_LUMOts.jpg|300px|thumb|'''Figure 19'''. LUMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_IRts.jpg|300px|thumb|'''Figure 20'''. Generated IR of transition state of reaction between cis-butadiene and ethylene]]
|}

The HOMO and LUMO molecular orbitals of this transition structure were visualized in '''Figure 18''' and '''Figure 19'''. From these two figures it can be observed that the HOMO at the transition structure is anti-symmetric ('''a''') while the LUMO is symmetric ('''s'''). Thus this anti-symmetric transition state HOMO is formed by the HOMO of cis-butadiene and the LUMO of ethylene with the knowledge that they are both anti-symmetrical. This conclusion agrees with the previous talked conditions for allowed reaction.

===The cyclohexa-1,3-diene reaction with maleic anhydride===

<center>[[Image:XJB_mechanism3.jpg|500px|center|thumb|'''Figure 21'''. ''The endo and exo products of the Diels Alder reaction between cyclohexa-1,3-diene and maleic anhydride'']]</center>

The exo and endo transition states were drawn according to '''Figure 21''' with the new C-C bond formation line dashed and bond length estimation as 2.5. Then the same Hessian method was applied to optimize the structures. The electronic energy of the endo structure was calculated to be '''-0.05150Ha''' and '''-0.05042Ha''' for the exo. This reaction is supposed to be kinetically controlled thus the lower energy endo transition state is more favoured.

The bond lengths of the partly formed C-C bond are 2.16 Å and 2.17 Å for endo and exo transition states respectively. Another measured C-C bond length is shown in '''Table 13''' as 2.89 Å for endo (C3-C8 and C2-C7) and 2.95 Å for endo (C7-C10 and C8-C9). The longer atom distance indicates the larger steric repulsion in the exo transition state because of the extra existence of hydrogen atoms on C7 and C8.

Woodward and Hoffmann also proposed a secondary orbital overlap statement as an explanation of the endo stereo-preference in Diels Alder reactions<ref name="soo">''J. Org. Chem.'', 1987, '''52(8)''', pp 1469–1474{{DOI|10.1021/jo00384a016}}</ref>. This secondary orbital overlap does not involve bond changes but π orbital interactions as the highlighted carbon atoms of the endo transition state in '''Table 13'''. Such overlap stablises the structure and lowers the energy of the endo transition state while there is no such effect existed in the exo transition state.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | endo
! scope="col" | exo
|-
| '''Animation'''||[[File:XJB_endots.GIF|300px]]||[[File:XJB_exots.GIF|300px]]
|-
| '''Sum of electronic and zero-point Energies'''||0.133493||0.134880
|-
| '''Sum of electronic and thermal Energies'''||0.143683||0.144881
|-
| '''Sum of electronic and thermal Enthalpies'''||0.144627||0.145825
|-
| '''Sum of electronic and thermal Free Energies'''||0.097349||0.099117
|-
| '''Bond lengths'''||[[Image:XJB_endomeasure.jpg|300px]]||[[Image:XJB_exomeasure.jpg|300px]]
|-
| '''HOMO'''||[[Image:XJB_endoHOMO.jpg|270px]]||[[Image:XJB_exoHOMO.jpg|255px]]
|-
| '''Secondary orbital overlap effect'''||[[Image:XJB_endosoo.jpg|300px]]||[[Image:XJB_exosoo.jpg|300px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 13.''' ''Summary of comparison of properties of endo and exo transition structures for reaction between cyclohexa-1,3-diene and maleic anhydride''</div>

==Reference==

<references/>

Rep:Mod:XJB1130

2013-12-06T00:06:16Z

Jx1011:

==The Cope Rearrangement of 1,5-hexadiene==

The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement reaction ('''Figure 1'''). Its mechanism has been studied by a large number of experimental and computational researches, which is recently accepted that the reaction occurs via either a "chair" or a "boat" transition state structure in a concerted manner. The aim of this exercise is to locate the low-energy minimum and the transition structures on the potential energy surface by Gaussian calculation in order to study its mechanism. The reaction pathway and the barrier heights can also be calculated by this calculation.

<center>[[Image:XJB_cope rearrangement.jpg|500px|center|thumb|'''Figure 1'''. ''The Cope rearrangement'']]</center>

===Optimizing the Reactants and Products===

Different conformations of 1,5-hexadiene can be optimized using '''Gaussview'''. The optimized structure of four conformers involved in this discussion and their corresponding energies optimized at '''HF/3-21G''' level of theory are concluded in the following table ('''Table 1''').

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | anti4
! scope="col" | anti2
! scope="col" | gauche
! scope="col" | gauche3
|-
| <jmol><jmolApplet><title>XJB_anti4.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti4.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche3.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| -231.69097Ha || -231.69254Ha || -231.68772Ha ||-231.69266Ha
|}

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 1.''' ''Four conformers of 1,5-hexadiene''</div>

This exercise first involved optimizing a molecule of 1,5-hexadiene with "anti" linkage for the cetral four C atoms at the '''HF/3-21G''' level of theory. The final optimization structure had the total electronic energy of '''-231.69097Ha''' with a '''C<sub>1</sub>''' symmetry. By comparing the energy and point group to that in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1], this structure is confirmed to be ''anti4''. The compositions of energies are also listed in '''Table 2''' as i) the sum of electronic and zero-point energies (potential energy at 0K including the zero-point vibrational energy), ii) the sum of electronic and thermal energies (energy including translational, rotational and vibrational at 298.15K and 1 atm), iii) the sum of electronic and thermal enthalpies (containning an additional correction for RT), and iv) the sum of electronic and thermal free energies (including the entropic contribution to the free energy). For what we are considering, the first two terms are the most useful for further comparisons.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Formula Description'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||E = E<sub>elec</sub> + ZPE||-231.53785
|-
| Sum of electronic and thermal Energies||E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>||-231.53096
|-
| Sum of electronic and thermal Enthalpie||H = E + RT||-231.53002
|-
| Sum of electronic and thermal Free Energie||G = H - TS||-231.56891
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 2.''' ''Frequency calculations of anti4 optimized at '''HF/3-21G''' ([[File:XJB_anti4.log]])''</div>

Next another 1,5-hexadiene molecule with "gauche" linkage for the central four atoms was drawn and optimized at the '''HF/3-21G''' level of theory as before. Energy for this conformation is expected to be higher than that for the ''anti4'' conformation due to the closer position of two alkene ends thus the steric interaction. The final structure had the electronic energy of '''-231.68772Ha''' which was indeed higher than that of the ''anti4'', indicating less favour towards this gauche structure. It had a '''C<sub>2</sub>''' symmetry and was confirmed to be the ''gauche'' conformation.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53439
|-
| Sum of electronic and thermal Energies||-231.52770
|-
| Sum of electronic and thermal Enthalpie||-231.52676
|-
| Sum of electronic and thermal Free Energie||-231.56483
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 3.''' ''Frequency calculations of gauche optimized at '''HF/3-21G''' ([[File:XJB_gauche.log]])''</div>

A lowest energy conformation ''gauche3'' of the reactant molecule was drawn based on and confirmed by [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. The optimization gave a final structure with an energy of '''-231.69266Ha''' with a symmetry of '''C<sub>1</sub>'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53949
|-
| Sum of electronic and thermal Energies||-231.53265
|-
| Sum of electronic and thermal Enthalpie||-231.5317
|-
| Sum of electronic and thermal Free Energie||-231.57064
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 4.''' ''Frequency calculations of gauche3 optimized at '''HF/3-21G''' ([[File:XJB_gauche3.log]])''</div>

The next step required us to draw the C<sub>i</sub> symmetry ''anti2'' conformation of 1,5-hexadiene and optimize it at the '''HF/3-21G''' level of theory. The optimized energy was calculated to be '''-231.69254Ha''' with point group of '''C<sub>i</sub>''', which was consistent with the one given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53954
|-
| Sum of electronic and thermal Energies||-231.53257
|-
| Sum of electronic and thermal Enthalpie||-231.53162
|-
| Sum of electronic and thermal Free Energie||-231.57092
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 5.''' ''Frequency calculations of anti2 optimized at '''HF/3-21G''' ([[File:XJB_anti2.log]])''</div>

The ''anti2'' structure was then reoptimized at '''B3LYP/6-31G*''' which was a improvement over the '''HF/3-21G''' basis set. It adds polarization to all atoms and improves the modeling of core electrons thus producing more accurate description of orbitals[ref]. The reoptimized ''anti2'' structure had the total electronic energy of '''-234.61171Ha''' and the same symmetry C<sub>i</sub>. The comparison of two structures optimized using different levels of theory is summarized in '''Table 6'''. We can see that they have exactly the same dihedral angle of the central four C atoms at 180.00<sup>o</sup> and angles between three of them are very similar as 111.3<sup>o</sup> and 112.7<sup>o</sup> respectively. Thus the the geometry change using '''HF/3-21G''' or '''B3LYP/6-31G*''' is trivial.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Before re-optimization
! scope="col" | After re-optimization
|-
| <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2reoptimized</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2reoptimized.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| ''anti2 optimized at '''HF/3-21G''''' || ''anti2 optimized at '''B3LYP/6-31G*'''''
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 6.''' ''Two optimization of anti2 at '''HF/3-21G''' and '''B3LYP/6-31G*'''([[File:XJB_anti2.log]])''</div>

The frequency calculation was run as well and a list of energies is summarized in '''Table 7'''. A list of all the vibrational frequencies was also generated and no imaginary (negative) frequency was observed and these vibrations are visualized by simulating the infrared spectrum in '''Figure 2'''.

{| align="center"
|- align="center"
|
{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-234.46920
|-
| Sum of electronic and thermal Energies||-234.46186
|-
| Sum of electronic and thermal Enthalpie||-234.46091
|-
| Sum of electronic and thermal Free Energie||-234.50078
|}
|
[[Image:XJB_IR_anti2.jpg|300px]]
|- align="center"
| align="center" | '''Table 7.''' ''Frequency calculations of anti2 optimized at '''B3LYP/6-31G*'''
([[File:XJB_anti2_reoptimized.log]])''
| align="center" | '''Figure 2.''' ''The generated IR spectrum of optimized '''B3LYP/6-31G*''' structure''
|}

===Optimizing the "Chair" and "Boat" Transition Structures===

The study of the chair and boat transition states is carried out using three different approaches. The first one is called the Hessian which involves computing the force constant matrix; the second one is to freeze the reaction coordinate and then optimize the structure once the molecule is fully relaxed; the last one is to use QST2 method to specify the reactants and products then finding the transition structure.

<center>[[Image:XJB_2 ts.jpg|500px|center|thumb|'''Figure 3'''. ''The two transition states of Cope rearrangement'']]</center>

====The Chair Transition State====

In order to set up a transition state optimization, first an allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimized at the '''HF/3-21G''' level of theory. Then this half of the transition structure was duplicated with rotation to make the approximate resembling of the chair transition state with terminal ends of the fragments 2.2 Å apart. The chair transition structure optimization was then done by the first two different approaches.

<center>[[Image:XJB_guess ts.jpg|400px|center|thumb|'''Figure 4'''. ''The guessed chair transition structure of two allyl fragments'']]</center>

For the first optimization approach, the force constants were computed (the Hessian) and the chair transition state was optimized to a TS (Berny) at the '''HF/3-21G''' level of theory. The frequency calculation was carried out at the same time and gave an imaginary frequency at -818 cm<sup>-1</sup>. The electronic energy was calculated to be '''-231.61932Ha''' with C<sub>2h</sub> symmetry point group. The distance between the terminal carbons was 2.0203 Å. An animation of this transition state vibrations was generated to confirm the corresponding Cope rearrangement.

<center>[[File:XJB_chair ts animation.GIF|400px|center|thumb|'''Figure 5.''' ''Animation of the Cope rearrangement via the chair transition state'']]</center>

The chair transition state was then reoptimized using the '''B3LYP/6-31G*''' level of theory and carried out frequency calculations as well. The electronic energy calculated this time was '''-234.55698Ha''' with an imaginary frequency at -566 cm<sup>-1</sup>. We can see that with the geometries quite similar, the energies differ markedly calculated using two levels of theory. The summary and comparison of the calculations optimized at two levels of theory are summarized in '''Table 8'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46670||-234.41493
|-
| Sum of electronic and thermal Energies||-231.46134||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40807
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir1.jpg|200px]]||[[Image:XJB_chairir2.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 8.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the first approach
([[File:XJB_chairts1.1.log]], [[File:XJB_chairts1.2.log]])''</div>

The second approach is the frozen coordinate method using the Redundant Coord Editor to freeze the bond lengths of the terminal ends to 2.2 Å. First the calculation failed because of the recent update of GaussView. After editting the coordinate distance (B 5 1 2.2 F) and the frozen distance (B 5 1 F) the optimization went well this time. The structure was then optimized again using a normal guess Hessian modified including the two differentiating coordinates. This chair transition state was then reoptimized again using the '''B3LYP/6-31G*''' level of theory. The electronic energies were calculated to be the same as in the first approach and frequency calculations were summarized in '''Table 9'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46671||-234.41490
|-
| Sum of electronic and thermal Energies||-231.46135||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40806
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir3.jpg|200px]]||[[Image:XJB_chairir4.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 9.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the second approach
([[File:XJB_chairts2.1.log]], [[File:XJB_chairts2.2.log]])''</div>

As the same energy results given, the bond forming/breaking bond lengths in the chair transition structures in the above two different approaches were compared. The first approach gave the bond lengths of 2.0203 Å while the second approach gave 2.0207 Å. Considering the accuracy limit of bond length estimation in Gaussview, these two bond lengths can be seen as exactly the same. Two methods worked reasonably well because of the good assumption of transition structures. However for more complicated and larger molecule, the Hessian method may fail due to the difficulty to predict the transition geometry (different surface curvature). Thus better way to generate such transition structures is to freeze reaction coordinate.

{| align="center"
|- align="center"
|
[[Image:XJB_chairts bondlength1.jpg|350px|thumb|'''Figure 6.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by Hession'']]
|
[[Image:XJB_chairts bondlength2.jpg|350px|thumb|'''Figure 7.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by frozen coordinate'']]
|- align="center"
|}

====The Boat Transition State====

The boat transition structure was then optimized using the '''QST2''' method. The same numbering for the reactant and product molecules was important in order to specify for this reaction. The renumbering of the molecules is shown as in '''Figure 8''', but the following optimization failed and gave a twisted structure. The reactant and product geometries were then modified further by changing the central C-C-C-C dihedral angle to 0<sup>o</sup> and the inside two C-C-C (i.e. C2-C3-C4 and C3-C4-C5) angles to 100<sup>o</sup>. The job was run successfully and frequency calculations were done with the boat transition state generated. Only one imaginary frequency was detected also at -840 cm<sup>-1</sup>. The electronic energy was '''-231.60280Ha''' with C<sub>2v</sub> symmetry. The distances between the terminal carbons of the allyl fragments were examined to be both 2.140Å which were comparably longer than that of the chair transition structure.

{| align="center"
|- align="center"
|
[[Image:XJB_QST1.jpg|400px|thumb|'''Figure 8.''' ''Numbering of reactant and product'']]
|
[[Image:XJB_QST2.jpg|410px|thumb|'''Figure 9.''' ''Re-numbering of reactant and product'']]
|- align="center"
|}

<center>[[File:XJB_boat ts animation.GIF|400px|center|thumb|'''Figure 10.''' ''Animation of the Cope rearrangement via the boat transition state'']]</center>

The structure was then reoptimized at the higher '''B3LYP/6-31G*''' theory of level and the electronic energy was calculated to be '''-234.54309Ha''' with very similar geometry. The frequency calculation generated an imaginary frequency at -530 cm<sup>-1</sup>. The summary and comparison of two levels' calculations are carried out in the following table.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.45093||-234.40234
|-
| Sum of electronic and thermal Energies||-231.44530||-234.39601
|-
| Sum of electronic and thermal Enthalpie|| -231.44436||-234.39506
|-
| Sum of electronic and thermal Free Energie||-231.47977||-234.43175
|-
| Generated IR spectrum||[[Image:XJB_boatir1.jpg|200px]]||[[Image:XJB_boatir2.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 10.''' ''Frequency calculations of the boat transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by QST2
([[File:XJB_boat ts1.log]], [[File:XJB_boat ts2.log]])''</div>

====Intrinsic Reaction Coordinate====

Next a useful method called '''Intrinct Reaction Coordinate (IRC)''' is applied to the Hessian optimized chair transition structure to follow the mininmum energy path (MEP) from a transition structure down to its local minimum on a potential energy surface. IRC can be used to verify that the transition state you found actually connects the reactants and products you think it does, or to identify any reaction intermediates you haven't thought about, or to generate an animation of the reaction [ref]. The symmetrical reaction coordinate was computed only in the forward direction with force constant always calculated and the number of points along IRC 50. The last structure generated along IRC is given in '''Figure 11''' with the energy calculated as '''-231.69158Ha''' and the dihedral angle and internal angle measured as D = -67.188<sup>o</sup> A = 111.78<sup>o</sup>. This energy does not match to any of the conformations discussed in Appendix 1 thus a minimum geometry has not reached yet.

{| align="center"
|- align="center"
|
<jmol><jmolApplet><title>XJB_IRC.mol</title><color>white</color>
<size>200</size><script>zoom=5</script>
<uploadedFileContents>XJB_IRC.mol</uploadedFileContents>
</jmolApplet></jmol>
|
[[Image:XJB_irc.jpg|500px]]
|- align="center"
| align="center" | '''Figure 11.''' ''The energy minimum structure along IRC''
| align="center" | '''Figure 12.''' ''Total energy along IRC''
|}

Three improving options can be used to reach the minimum energy. The first one is to run a normal Hessian optimization of the last point on IRC; the second one involves using a larger number of points (in this case 100 was used); the last one is to redo the IRC computation specifying to compute force constant at every step. Three methods were all used and the results were summarized in '''Table 11'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Improved IRC method 1
! scope="col" | Improved IRC method 2
! scope="col" | Improved IRC method 3
|-
| [[Image:XJB_IRC1.jpg|300px|center|IRC1]]||[[Image:XJB_IRC2.jpg|300px|center|IRC3]] || [[Image:XJB_IRC3.jpg|300px|center|IRC3]]
|-
| D = -64.17<sup>o</sup> A = 112.04<sup>o</sup> || D = -66.83<sup>o</sup> A = 111.82<sup>o</sup> || D = -64.17<sup>o</sup> A = 112.04<sup>o</sup>
|-
| -231.69167Ha || -231.69150Ha || -231.69167Ha
|-
| [[File:XJB_IRC1.log]]||[[File:XJB_IRC2.log]]||[[File:XJB_IRC3.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 11.''' ''Comparison of three improvement methods of IRC computation upon the chair transition structure''</div>

It is concluded that method 1 and 3 gave the same the lowest energy which was consistent to the energy given in Appendix 1.

====Activation Energies====

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | HF/3-21G at 0K
! scope="col" | HF/3-21G at 298.15K
! scope="col" | B3LYP/6-31G* at 0K
! scope="col" | B3LYP/6-31G* at 298.15K
! scope="col" | Expt. at 0K
|-
| '''ΔE (chair)'''||45.71||44.70||34.05||33.16||33.5 ± 0.5
|-
| '''ΔE (boat)'''||55.60||54.76||41.96||41.32||44.7 ± 0.5
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 12.''' ''Summary of activation energies for the Cope rearrangement via both transition states in kcal/mol''</div>

==The Diels Alder Cycloaddition==

In this exercise, AM1 semi-empirical method is used to compute the Diels Alder cycloaddition which is a concerted pericyclic reaction. The driving force of this reaction is the formation of a new σ bond between the π orbitals of the dieneophile and the diene. The reaction is allowed with HOMO-LUMO interation and a symmetry overlap.

===Cis Butadiene===

<center>[[Image:prototype Diels Alder.jpg|500px|center|thumb|'''Figure 13'''. ''The prototype reaction between butadiene and ethylene'']]</center>

To study a prototype Diels Alder cycloaddition reaction between butadiene and ethylene, first a cis butadiene molecule was built using Gaussview and optimized using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the butadiene were plotted. As seen the HOMO of cis-butadiene is anti-symmetric ('''a''') and the LUMO is symmetric ('''s''') with respect to plane.

{| align="center"
|- align="center"
|
[[Image:XJB_HOMO.jpg|280px|thumb|'''Figure 14.''' ''Anti-symmetric HOMO of cis-butadiene'']]
|
[[Image:XJB_LUMO.jpg|300px|thumb|'''Figure 15.''' ''Symmetric HOMO of cis-butadiene'']]
|- align="center"
|}

===The Transition State of prototype reaction between ethylene and butadiene===

The Hessian method was used to estimate transition state for the reaction between ethylene and butadiene because these two reactants are relatively small and simple molecules. The transition state was drawn by first building a structure as (a) in '''Figure 16''' then removing the -CH<sub>2</sub>-CH<sub>2</sub>- group to form the envelop structure as in (b). The '''clean''' option should not be used and the dashed bond length was guessed to be larger than 2.2 Å otherwise the geometry change would cause a failure in optimization.

<center>[[Image:XJB_dielsguessts.jpg|300px|center|thumb|'''Figure 16'''. ''The guessing transition state for the reaction between butadiene and ethylene'']]</center>

<center>[[File:XJB_butadiene and ethylene ts.GIF|400px|center|thumb|'''Figure 17.''' ''Animation of the butadiene and ethylene cycloaddition transition state'']]</center>

After the Hessian optimization, the transition structure has been successfully determined and an animation was generated in '''Figure 17'''. It has a C<sub>1</sub> symmetry point group and the bond lengths of two partly formed C-C bonds were measured to be 2.12Å. The typical sp<sup>3</sup> and sp<sup>2</sup> C-C bond lengths are 1.54Å and 1.47Å respectively[ref]. The van der Waals radius of the C atom is 1.7 Å[ref]. The measured bond distance of the transition structure is larger than typical bond lengths but still within the van der Waals radius, suggesting the bond forming interation between the carbon atoms.

From the animation ('''Figure 17''') the vibration directions of two molecules are opposite and they are approaching and leaving each other at the same time. This suggests the formation of the two bonds synchronous. The vibrational frequencies were also calculated and an infrared spectrum was simulated ('''Figure 20'''). An imaginary frequency of magnitude -955 cm<sup>-1</sup> was seen and the lowest positive frequency was observed at 147 cm<sup>-1</sup> which corresponded to the reactant vibration but not the bond formation.

{| align="center"
|
[[Image:XJB_HOMOts.jpg|300px|thumb|'''Figure 18'''. HOMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_LUMOts.jpg|300px|thumb|'''Figure 19'''. LUMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_IRts.jpg|300px|thumb|'''Figure 20'''. Generated IR of transition state of reaction between cis-butadiene and ethylene]]
|}

The HOMO and LUMO molecular orbitals of this transition structure were visualized in '''Figure 18''' and '''Figure 19'''. From these two figures it can be observed that the HOMO at the transition structure is anti-symmetric ('''a''') while the LUMO is symmetric ('''s'''). Thus this anti-symmetric transition state HOMO is formed by the HOMO of cis-butadiene and the LUMO of ethylene with the knowledge that they are both anti-symmetrical. This conclusion agrees with the previous talked conditions for allowed reaction.

===The cyclohexa-1,3-diene reaction with maleic anhydride===

<center>[[Image:XJB_mechanism3.jpg|500px|center|thumb|'''Figure 21'''. ''The endo and exo products of the Diels Alder reaction between cyclohexa-1,3-diene and maleic anhydride'']]</center>

The exo and endo transition states were drawn according to '''Figure 21''' with the new C-C bond formation line dashed and bond length estimation as 2.5. Then the same Hessian method was applied to optimize the structures. The electronic energy of the endo structure was calculated to be '''-0.05150Ha''' and '''-0.05042Ha''' for the exo. This reaction is supposed to be kinetically controlled thus the lower energy endo transition state is more favoured.

The bond lengths of the partly formed C-C bond are 2.16 Å and 2.17 Å for endo and exo transition states respectively. Another measured C-C bond length is shown in '''Table 13''' as 2.89 Å for endo (C3-C8 and C2-C7) and 2.95 Å for endo (C7-C10 and C8-C9). The longer atom distance indicates the larger steric repulsion in the exo transition state because of the extra existence of hydrogen atoms on C7 and C8.

Woodward and Hoffmann also proposed a secondary orbital overlap statement as an explanation of the endo stereo-preference in Diels Alder reactions<ref name="soo">''J. Org. Chem.'', 1987, '''52(8)''', pp 1469–1474{{DOI|10.1021/jo00384a016}}</ref>. This secondary orbital overlap does not involve bond changes but π orbital interactions as the highlighted carbon atoms of the endo transition state in '''Table 13'''. Such overlap stablises the structure and lowers the energy of the endo transition state while there is no such effect existed in the exo transition state.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | endo
! scope="col" | exo
|-
| '''Animation'''||[[File:XJB_endots.GIF|300px]]||[[File:XJB_exots.GIF|300px]]
|-
| '''Sum of electronic and zero-point Energies'''||0.133493||0.134880
|-
| '''Sum of electronic and thermal Energies'''||0.143683||0.144881
|-
| '''Sum of electronic and thermal Enthalpies'''||0.144627||0.145825
|-
| '''Sum of electronic and thermal Free Energies'''||0.097349||0.099117
|-
| '''Bond lengths'''||[[Image:XJB_endomeasure.jpg|300px]]||[[Image:XJB_exomeasure.jpg|300px]]
|-
| '''HOMO'''||[[Image:XJB_endoHOMO.jpg|270px]]||[[Image:XJB_exoHOMO.jpg|255px]]
|-
| '''Secondary orbital overlap effect'''||[[Image:XJB_endosoo.jpg|300px]]||[[Image:XJB_exosoo.jpg|300px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 13.''' ''Summary of comparison of properties of endo and exo transition structures for reaction between cyclohexa-1,3-diene and maleic anhydride''</div>

==Reference==

<references/>

Rep:Mod:XJB1130

2013-12-06T00:05:35Z

Jx1011:

==The Cope Rearrangement of 1,5-hexadiene==

The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement reaction ('''Figure 1'''). Its mechanism has been studied by a large number of experimental and computational researches, which is recently accepted that the reaction occurs via either a "chair" or a "boat" transition state structure in a concerted manner. The aim of this exercise is to locate the low-energy minimum and the transition structures on the potential energy surface by Gaussian calculation in order to study its mechanism. The reaction pathway and the barrier heights can also be calculated by this calculation.

<center>[[Image:XJB_cope rearrangement.jpg|500px|center|thumb|'''Figure 1'''. ''The Cope rearrangement'']]</center>

===Optimizing the Reactants and Products===

Different conformations of 1,5-hexadiene can be optimized using '''Gaussview'''. The optimized structure of four conformers involved in this discussion and their corresponding energies optimized at '''HF/3-21G''' level of theory are concluded in the following table ('''Table 1''').

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | anti4
! scope="col" | anti2
! scope="col" | gauche
! scope="col" | gauche3
|-
| <jmol><jmolApplet><title>XJB_anti4.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti4.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche3.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| -231.69097Ha || -231.69254Ha || -231.68772Ha ||-231.69266Ha
|}

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 1.''' ''Four conformers of 1,5-hexadiene''</div>

This exercise first involved optimizing a molecule of 1,5-hexadiene with "anti" linkage for the cetral four C atoms at the '''HF/3-21G''' level of theory. The final optimization structure had the total electronic energy of '''-231.69097Ha''' with a '''C<sub>1</sub>''' symmetry. By comparing the energy and point group to that in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1], this structure is confirmed to be ''anti4''. The compositions of energies are also listed in '''Table 2''' as i) the sum of electronic and zero-point energies (potential energy at 0K including the zero-point vibrational energy), ii) the sum of electronic and thermal energies (energy including translational, rotational and vibrational at 298.15K and 1 atm), iii) the sum of electronic and thermal enthalpies (containning an additional correction for RT), and iv) the sum of electronic and thermal free energies (including the entropic contribution to the free energy). For what we are considering, the first two terms are the most useful for further comparisons.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Formula Description'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||E = E<sub>elec</sub> + ZPE||-231.53785
|-
| Sum of electronic and thermal Energies||E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>||-231.53096
|-
| Sum of electronic and thermal Enthalpie||H = E + RT||-231.53002
|-
| Sum of electronic and thermal Free Energie||G = H - TS||-231.56891
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 2.''' ''Frequency calculations of anti4 optimized at '''HF/3-21G''' ([[File:XJB_anti4.log]])''</div>

Next another 1,5-hexadiene molecule with "gauche" linkage for the central four atoms was drawn and optimized at the '''HF/3-21G''' level of theory as before. Energy for this conformation is expected to be higher than that for the ''anti4'' conformation due to the closer position of two alkene ends thus the steric interaction. The final structure had the electronic energy of '''-231.68772Ha''' which was indeed higher than that of the ''anti4'', indicating less favour towards this gauche structure. It had a '''C<sub>2</sub>''' symmetry and was confirmed to be the ''gauche'' conformation.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53439
|-
| Sum of electronic and thermal Energies||-231.52770
|-
| Sum of electronic and thermal Enthalpie||-231.52676
|-
| Sum of electronic and thermal Free Energie||-231.56483
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 3.''' ''Frequency calculations of gauche optimized at '''HF/3-21G''' ([[File:XJB_gauche.log]])''</div>

A lowest energy conformation ''gauche3'' of the reactant molecule was drawn based on and confirmed by [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. The optimization gave a final structure with an energy of '''-231.69266Ha''' with a symmetry of '''C<sub>1</sub>'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53949
|-
| Sum of electronic and thermal Energies||-231.53265
|-
| Sum of electronic and thermal Enthalpie||-231.5317
|-
| Sum of electronic and thermal Free Energie||-231.57064
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 4.''' ''Frequency calculations of gauche3 optimized at '''HF/3-21G''' ([[File:XJB_gauche3.log]])''</div>

The next step required us to draw the C<sub>i</sub> symmetry ''anti2'' conformation of 1,5-hexadiene and optimize it at the '''HF/3-21G''' level of theory. The optimized energy was calculated to be '''-231.69254Ha''' with point group of '''C<sub>i</sub>''', which was consistent with the one given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53954
|-
| Sum of electronic and thermal Energies||-231.53257
|-
| Sum of electronic and thermal Enthalpie||-231.53162
|-
| Sum of electronic and thermal Free Energie||-231.57092
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 5.''' ''Frequency calculations of anti2 optimized at '''HF/3-21G''' ([[File:XJB_anti2.log]])''</div>

The ''anti2'' structure was then reoptimized at '''B3LYP/6-31G*''' which was a improvement over the '''HF/3-21G''' basis set. It adds polarization to all atoms and improves the modeling of core electrons thus producing more accurate description of orbitals[ref]. The reoptimized ''anti2'' structure had the total electronic energy of '''-234.61171Ha''' and the same symmetry C<sub>i</sub>. The comparison of two structures optimized using different levels of theory is summarized in '''Table 6'''. We can see that they have exactly the same dihedral angle of the central four C atoms at 180.00<sup>o</sup> and angles between three of them are very similar as 111.3<sup>o</sup> and 112.7<sup>o</sup> respectively. Thus the the geometry change using '''HF/3-21G''' or '''B3LYP/6-31G*''' is trivial.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Before re-optimization
! scope="col" | After re-optimization
|-
| <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2reoptimized</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2reoptimized.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| ''anti2 optimized at '''HF/3-21G''''' || ''anti2 optimized at '''B3LYP/6-31G*'''''
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 6.''' ''Two optimization of anti2 at '''HF/3-21G''' and '''B3LYP/6-31G*'''([[File:XJB_anti2.log]])''</div>

The frequency calculation was run as well and a list of energies is summarized in '''Table 7'''. A list of all the vibrational frequencies was also generated and no imaginary (negative) frequency was observed and these vibrations are visualized by simulating the infrared spectrum in '''Figure 2'''.

{| align="center"
|- align="center"
|
{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-234.46920
|-
| Sum of electronic and thermal Energies||-234.46186
|-
| Sum of electronic and thermal Enthalpie||-234.46091
|-
| Sum of electronic and thermal Free Energie||-234.50078
|}
|
[[Image:XJB_IR_anti2.jpg|300px]]
|- align="center"
| align="center" | '''Table 7.''' ''Frequency calculations of anti2 optimized at '''B3LYP/6-31G*'''
([[File:XJB_anti2_reoptimized.log]])''
| align="center" | '''Figure 2.''' ''The generated IR spectrum of optimized '''B3LYP/6-31G*''' structure''
|}

===Optimizing the "Chair" and "Boat" Transition Structures===

The study of the chair and boat transition states is carried out using three different approaches. The first one is called the Hessian which involves computing the force constant matrix; the second one is to freeze the reaction coordinate and then optimize the structure once the molecule is fully relaxed; the last one is to use QST2 method to specify the reactants and products then finding the transition structure.

<center>[[Image:XJB_2 ts.jpg|500px|center|thumb|'''Figure 3'''. ''The two transition states of Cope rearrangement'']]</center>

====The Chair Transition State====

In order to set up a transition state optimization, first an allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimized at the '''HF/3-21G''' level of theory. Then this half of the transition structure was duplicated with rotation to make the approximate resembling of the chair transition state with terminal ends of the fragments 2.2 Å apart. The chair transition structure optimization was then done by the first two different approaches.

<center>[[Image:XJB_guess ts.jpg|400px|center|thumb|'''Figure 4'''. ''The guessed chair transition structure of two allyl fragments'']]</center>

For the first optimization approach, the force constants were computed (the Hessian) and the chair transition state was optimized to a TS (Berny) at the '''HF/3-21G''' level of theory. The frequency calculation was carried out at the same time and gave an imaginary frequency at -818 cm<sup>-1</sup>. The electronic energy was calculated to be '''-231.61932Ha''' with C<sub>2h</sub> symmetry point group. The distance between the terminal carbons was 2.0203 Å. An animation of this transition state vibrations was generated to confirm the corresponding Cope rearrangement.

<center>[[File:XJB_chair ts animation.GIF|400px|center|thumb|'''Figure 5.''' ''Animation of the Cope rearrangement via the chair transition state'']]</center>

The chair transition state was then reoptimized using the '''B3LYP/6-31G*''' level of theory and carried out frequency calculations as well. The electronic energy calculated this time was '''-234.55698Ha''' with an imaginary frequency at -566 cm<sup>-1</sup>. We can see that with the geometries quite similar, the energies differ markedly calculated using two levels of theory. The summary and comparison of the calculations optimized at two levels of theory are summarized in '''Table 8'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46670||-234.41493
|-
| Sum of electronic and thermal Energies||-231.46134||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40807
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir1.jpg|200px]]||[[Image:XJB_chairir2.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 8.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the first approach
([[File:XJB_chairts1.1.log]], [[File:XJB_chairts1.2.log]])''</div>

The second approach is the frozen coordinate method using the Redundant Coord Editor to freeze the bond lengths of the terminal ends to 2.2 Å. First the calculation failed because of the recent update of GaussView. After editting the coordinate distance (B 5 1 2.2 F) and the frozen distance (B 5 1 F) the optimization went well this time. The structure was then optimized again using a normal guess Hessian modified including the two differentiating coordinates. This chair transition state was then reoptimized again using the '''B3LYP/6-31G*''' level of theory. The electronic energies were calculated to be the same as in the first approach and frequency calculations were summarized in '''Table 9'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46671||-234.41490
|-
| Sum of electronic and thermal Energies||-231.46135||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40806
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir3.jpg|200px]]||[[Image:XJB_chairir4.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 9.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the second approach
([[File:XJB_chairts2.1.log]], [[File:XJB_chairts2.2.log]])''</div>

As the same energy results given, the bond forming/breaking bond lengths in the chair transition structures in the above two different approaches were compared. The first approach gave the bond lengths of 2.0203 Å while the second approach gave 2.0207 Å. Considering the accuracy limit of bond length estimation in Gaussview, these two bond lengths can be seen as exactly the same. Two methods worked reasonably well because of the good assumption of transition structures. However for more complicated and larger molecule, the Hessian method may fail due to the difficulty to predict the transition geometry (different surface curvature). Thus better way to generate such transition structures is to freeze reaction coordinate.

{| align="center"
|- align="center"
|
[[Image:XJB_chairts bondlength1.jpg|350px|thumb|'''Figure 6.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by Hession'']]
|
[[Image:XJB_chairts bondlength2.jpg|350px|thumb|'''Figure 7.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by frozen coordinate'']]
|- align="center"
|}

====The Boat Transition State====

The boat transition structure was then optimized using the '''QST2''' method. The same numbering for the reactant and product molecules was important in order to specify for this reaction. The renumbering of the molecules is shown as in '''Figure 8''', but the following optimization failed and gave a twisted structure. The reactant and product geometries were then modified further by changing the central C-C-C-C dihedral angle to 0<sup>o</sup> and the inside two C-C-C (i.e. C2-C3-C4 and C3-C4-C5) angles to 100<sup>o</sup>. The job was run successfully and frequency calculations were done with the boat transition state generated. Only one imaginary frequency was detected also at -840 cm<sup>-1</sup>. The electronic energy was '''-231.60280Ha''' with C<sub>2v</sub> symmetry. The distances between the terminal carbons of the allyl fragments were examined to be both 2.140Å which were comparably longer than that of the chair transition structure.

{| align="center"
|- align="center"
|
[[Image:XJB_QST1.jpg|400px|thumb|'''Figure 8.''' ''Numbering of reactant and product'']]
|
[[Image:XJB_QST2.jpg|410px|thumb|'''Figure 9.''' ''Re-numbering of reactant and product'']]
|- align="center"
|}

<center>[[File:XJB_boat ts animation.GIF|400px|center|thumb|'''Figure 10.''' ''Animation of the Cope rearrangement via the boat transition state'']]</center>

The structure was then reoptimized at the higher '''B3LYP/6-31G*''' theory of level and the electronic energy was calculated to be '''-234.54309Ha''' with very similar geometry. The frequency calculation generated an imaginary frequency at -530 cm<sup>-1</sup>. The summary and comparison of two levels' calculations are carried out in the following table.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.45093||-234.40234
|-
| Sum of electronic and thermal Energies||-231.44530||-234.39601
|-
| Sum of electronic and thermal Enthalpie|| -231.44436||-234.39506
|-
| Sum of electronic and thermal Free Energie||-231.47977||-234.43175
|-
| Generated IR spectrum||[[Image:XJB_boatir1.jpg|200px]]||[[Image:XJB_boatir2.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 10.''' ''Frequency calculations of the boat transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by QST2
([[File:XJB_boat ts1.log]], [[File:XJB_boat ts2.log]])''</div>

====Intrinsic Reaction Coordinate====

Next a useful method called '''Intrinct Reaction Coordinate (IRC)''' is applied to the Hessian optimized chair transition structure to follow the mininmum energy path (MEP) from a transition structure down to its local minimum on a potential energy surface. IRC can be used to verify that the transition state you found actually connects the reactants and products you think it does, or to identify any reaction intermediates you haven't thought about, or to generate an animation of the reaction [ref]. The symmetrical reaction coordinate was computed only in the forward direction with force constant always calculated and the number of points along IRC 50. The last structure generated along IRC is given in '''Figure 11''' with the energy calculated as '''-231.69158Ha''' and the dihedral angle and internal angle measured as D = -67.188<sup>o</sup> A = 111.78<sup>o</sup>. This energy does not match to any of the conformations discussed in Appendix 1 thus a minimum geometry has not reached yet.

{| align="center"
|- align="center"
|
<jmol><jmolApplet><title>XJB_IRC.mol</title><color>white</color>
<size>200</size><script>zoom=5</script>
<uploadedFileContents>XJB_IRC.mol</uploadedFileContents>
</jmolApplet></jmol>
|
[[Image:XJB_irc.jpg|500px]]
|- align="center"
| align="center" | '''Figure 11.''' ''The energy minimum structure along IRC''
| align="center" | '''Figure 12.''' ''Total energy along IRC''
|}

Three improving options can be used to reach the minimum energy. The first one is to run a normal Hessian optimization of the last point on IRC; the second one involves using a larger number of points (in this case 100 was used); the last one is to redo the IRC computation specifying to compute force constant at every step. Three methods were all used and the results were summarized in '''Table 11'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Improved IRC method 1
! scope="col" | Improved IRC method 2
! scope="col" | Improved IRC method 3
|-
| [[Image:XJB_IRC1.jpg|300px|center|IRC1]]||[[Image:XJB_IRC2.jpg|300px|center|IRC3]] || [[Image:XJB_IRC3.jpg|300px|center|IRC3]]
|-
| D = -64.17<sup>o</sup> A = 112.04<sup>o</sup> || D = -66.83<sup>o</sup> A = 111.82<sup>o</sup> || D = -64.17<sup>o</sup> A = 112.04<sup>o</sup>
|-
| -231.69167Ha || -231.69150Ha || -231.69167Ha
|-
| [[File:XJB_IRC1.log]]||[[File:XJB_IRC2.log]]||[[File:XJB_IRC3.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 11.''' ''Comparison of three improvement methods of IRC computation upon the chair transition structure''</div>

It is concluded that method 1 and 3 gave the same the lowest energy which was consistent to the energy given in Appendix 1.

====Activation Energies====

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | HF/3-21G at 0K
! scope="col" | HF/3-21G at 298.15K
! scope="col" | B3LYP/6-31G* at 0K
! scope="col" | B3LYP/6-31G* at 298.15K
! scope="col" | Expt. at 0K
|-
| '''ΔE (chair)'''||45.71||44.70||34.05||33.16||33.5 ± 0.5
|-
| '''ΔE (boat)'''||55.60||54.76||41.96||41.32||44.7 ± 0.5
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 12.''' ''Summary of activation energies for the Cope rearrangement via both transition states in kcal/mol''</div>

==The Diels Alder Cycloaddition==

In this exercise, AM1 semi-empirical method is used to compute the Diels Alder cycloaddition which is a concerted pericyclic reaction. The driving force of this reaction is the formation of a new σ bond between the π orbitals of the dieneophile and the diene. The reaction is allowed with HOMO-LUMO interation and a symmetry overlap.

===Cis Butadiene===

<center>[[Image:prototype Diels Alder.jpg|500px|center|thumb|'''Figure 13'''. ''The prototype reaction between butadiene and ethylene'']]</center>

To study a prototype Diels Alder cycloaddition reaction between butadiene and ethylene, first a cis butadiene molecule was built using Gaussview and optimized using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the butadiene were plotted. As seen the HOMO of cis-butadiene is anti-symmetric ('''a''') and the LUMO is symmetric ('''s''') with respect to plane.

{| align="center"
|- align="center"
|
[[Image:XJB_HOMO.jpg|280px|thumb|'''Figure 14.''' ''Anti-symmetric HOMO of cis-butadiene'']]
|
[[Image:XJB_LUMO.jpg|300px|thumb|'''Figure 15.''' ''Symmetric HOMO of cis-butadiene'']]
|- align="center"
|}

===The Transition State of prototype reaction between ethylene and butadiene===

The Hessian method was used to estimate transition state for the reaction between ethylene and butadiene because these two reactants are relatively small and simple molecules. The transition state was drawn by first building a structure as (a) in '''Figure 16''' then removing the -CH<sub>2</sub>-CH<sub>2</sub>- group to form the envelop structure as in (b). The '''clean''' option should not be used and the dashed bond length was guessed to be larger than 2.2 Å otherwise the geometry change would cause a failure in optimization.

<center>[[Image:XJB_dielsguessts.jpg|300px|center|thumb|'''Figure 16'''. ''The guessing transition state for the reaction between butadiene and ethylene'']]</center>

<center>[[File:XJB_butadiene and ethylene ts.GIF|400px|center|thumb|'''Figure 17.''' ''Animation of the butadiene and ethylene cycloaddition transition state'']]</center>

After the Hessian optimization, the transition structure has been successfully determined and an animation was generated in '''Figure 17'''. It has a C<sub>1</sub> symmetry point group and the bond lengths of two partly formed C-C bonds were measured to be 2.12Å. The typical sp<sup>3</sup> and sp<sup>2</sup> C-C bond lengths are 1.54Å and 1.47Å respectively[ref]. The van der Waals radius of the C atom is 1.7 Å[ref]. The measured bond distance of the transition structure is larger than typical bond lengths but still within the van der Waals radius, suggesting the bond forming interation between the carbon atoms.

From the animation ('''Figure 17''') the vibration directions of two molecules are opposite and they are approaching and leaving each other at the same time. This suggests the formation of the two bonds synchronous. The vibrational frequencies were also calculated and an infrared spectrum was simulated ('''Figure 20'''). An imaginary frequency of magnitude -955 cm<sup>-1</sup> was seen and the lowest positive frequency was observed at 147 cm<sup>-1</sup> which corresponded to the reactant vibration but not the bond formation.

{| align="center"
|
[[Image:XJB_HOMOts.jpg|300px|thumb|'''Figure 18'''. HOMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_LUMOts.jpg|300px|thumb|'''Figure 19'''. LUMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_IRts.jpg|300px|thumb|'''Figure 20'''. Generated IR of transition state of reaction between cis-butadiene and ethylene]]
|}

The HOMO and LUMO molecular orbitals of this transition structure were visualized in '''Figure 18''' and '''Figure 19'''. From these two figures it can be observed that the HOMO at the transition structure is anti-symmetric ('''a''') while the LUMO is symmetric ('''s'''). Thus this anti-symmetric transition state HOMO is formed by the HOMO of cis-butadiene and the LUMO of ethylene with the knowledge that they are both anti-symmetrical. This conclusion agrees with the previous talked conditions for allowed reaction.

===The cyclohexa-1,3-diene reaction with maleic anhydride===

<center>[[Image:XJB_mechanism3.jpg|500px|center|thumb|'''Figure 21'''. ''The endo and exo products of the Diels Alder reaction between cyclohexa-1,3-diene and maleic anhydride'']]</center>

The exo and endo transition states were drawn according to '''Figure 21''' with the new C-C bond formation line dashed and bond length estimation as 2.5. Then the same Hessian method was applied to optimize the structures. The electronic energy of the endo structure was calculated to be '''-0.05150Ha''' and '''-0.05042Ha''' for the exo. This reaction is supposed to be kinetically controlled thus the lower energy endo transition state is more favoured.

The bond lengths of the partly formed C-C bond are 2.16 Å and 2.17 Å for endo and exo transition states respectively. Another measured C-C bond length is shown in '''Table 13''' as 2.89 Å for endo (C3-C8 and C2-C7) and 2.95 Å for endo (C7-C10 and C8-C9). The longer atom distance indicates the larger steric repulsion in the exo transition state because of the extra existence of hydrogen atoms on C7 and C8.

Woodward and Hoffmann also proposed a secondary orbital overlap statement as an explanation of the endo stereo-preference in Diels Alder reactions<ref name="soo">''J. Org. Chem.'', 1987, '''52(8)''', pp 1469–1474{{DOI|10.1021/jo00384a016}}</ref>. This secondary orbital overlap does not involve bond changes but π orbital interactions as the highlighted carbon atoms of the endo transition state in '''Table 13'''. Such overlap stablises the structure and lowers the energy of the endo transition state while there is no such effect existed in the exo transition state.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | endo
! scope="col" | exo
|-
| '''Animation'''||[[File:XJB_endots.GIF|300px]]||[[File:XJB_exots.GIF|300px]]
|-
| '''Sum of electronic and zero-point Energies'''||0.133493||0.134880
|-
| '''Sum of electronic and thermal Energies'''||0.143683||0.144881
|-
| '''Sum of electronic and thermal Enthalpies'''||0.144627||0.145825
|-
| '''Sum of electronic and thermal Free Energies'''||0.097349||0.099117
|-
| '''Bond lengths'''||[[Image:XJB_endomeasure.jpg|300px]]||[[Image:XJB_exomeasure.jpg|300px]]
|-
| '''HOMO'''||[[Image:XJB_endoHOMO.jpg|270px]]||[[Image:XJB_exoHOMO.jpg|255px]]
|-
| '''Secondary orbital overlap effect'''||[[Image:XJB_endosoo.jpg|300px]]||[[Image:XJB_exosoo.jpg|300px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 13.''' ''Summary of comparison of properties of endo and exo transition structures for reaction between cyclohexa-1,3-diene and maleic anhydride''</div>

==Reference==

<references/>

Rep:Mod:XJB1130

2013-12-06T00:03:50Z

Jx1011: /* The cyclohexa-1,3-diene reaction with maleic anhydride */

==The Cope Rearrangement of 1,5-hexadiene==

The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement reaction ('''Figure 1'''). Its mechanism has been studied by a large number of experimental and computational researches, which is recently accepted that the reaction occurs via either a "chair" or a "boat" transition state structure in a concerted manner. The aim of this exercise is to locate the low-energy minimum and the transition structures on the potential energy surface by Gaussian calculation in order to study its mechanism. The reaction pathway and the barrier heights can also be calculated by this calculation.

<center>[[Image:XJB_cope rearrangement.jpg|500px|center|thumb|'''Figure 1'''. ''The Cope rearrangement'']]</center>

===Optimizing the Reactants and Products===

Different conformations of 1,5-hexadiene can be optimized using '''Gaussview'''. The optimized structure of four conformers involved in this discussion and their corresponding energies optimized at '''HF/3-21G''' level of theory are concluded in the following table ('''Table 1''').

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | anti4
! scope="col" | anti2
! scope="col" | gauche
! scope="col" | gauche3
|-
| <jmol><jmolApplet><title>XJB_anti4.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti4.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche3.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| -231.69097Ha || -231.69254Ha || -231.68772Ha ||-231.69266Ha
|}

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 1.''' ''Four conformers of 1,5-hexadiene''</div>

This exercise first involved optimizing a molecule of 1,5-hexadiene with "anti" linkage for the cetral four C atoms at the '''HF/3-21G''' level of theory. The final optimization structure had the total electronic energy of '''-231.69097Ha''' with a '''C<sub>1</sub>''' symmetry. By comparing the energy and point group to that in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1], this structure is confirmed to be ''anti4''. The compositions of energies are also listed in '''Table 2''' as i) the sum of electronic and zero-point energies (potential energy at 0K including the zero-point vibrational energy), ii) the sum of electronic and thermal energies (energy including translational, rotational and vibrational at 298.15K and 1 atm), iii) the sum of electronic and thermal enthalpies (containning an additional correction for RT), and iv) the sum of electronic and thermal free energies (including the entropic contribution to the free energy). For what we are considering, the first two terms are the most useful for further comparisons.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Formula Description'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||E = E<sub>elec</sub> + ZPE||-231.53785
|-
| Sum of electronic and thermal Energies||E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>||-231.53096
|-
| Sum of electronic and thermal Enthalpie||H = E + RT||-231.53002
|-
| Sum of electronic and thermal Free Energie||G = H - TS||-231.56891
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 2.''' ''Frequency calculations of anti4 optimized at '''HF/3-21G''' ([[File:XJB_anti4.log]])''</div>

Next another 1,5-hexadiene molecule with "gauche" linkage for the central four atoms was drawn and optimized at the '''HF/3-21G''' level of theory as before. Energy for this conformation is expected to be higher than that for the ''anti4'' conformation due to the closer position of two alkene ends thus the steric interaction. The final structure had the electronic energy of '''-231.68772Ha''' which was indeed higher than that of the ''anti4'', indicating less favour towards this gauche structure. It had a '''C<sub>2</sub>''' symmetry and was confirmed to be the ''gauche'' conformation.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53439
|-
| Sum of electronic and thermal Energies||-231.52770
|-
| Sum of electronic and thermal Enthalpie||-231.52676
|-
| Sum of electronic and thermal Free Energie||-231.56483
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 3.''' ''Frequency calculations of gauche optimized at '''HF/3-21G''' ([[File:XJB_gauche.log]])''</div>

A lowest energy conformation ''gauche3'' of the reactant molecule was drawn based on and confirmed by [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. The optimization gave a final structure with an energy of '''-231.69266Ha''' with a symmetry of '''C<sub>1</sub>'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53949
|-
| Sum of electronic and thermal Energies||-231.53265
|-
| Sum of electronic and thermal Enthalpie||-231.5317
|-
| Sum of electronic and thermal Free Energie||-231.57064
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 4.''' ''Frequency calculations of gauche3 optimized at '''HF/3-21G''' ([[File:XJB_gauche3.log]])''</div>

The next step required us to draw the C<sub>i</sub> symmetry ''anti2'' conformation of 1,5-hexadiene and optimize it at the '''HF/3-21G''' level of theory. The optimized energy was calculated to be '''-231.69254Ha''' with point group of '''C<sub>i</sub>''', which was consistent with the one given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53954
|-
| Sum of electronic and thermal Energies||-231.53257
|-
| Sum of electronic and thermal Enthalpie||-231.53162
|-
| Sum of electronic and thermal Free Energie||-231.57092
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 5.''' ''Frequency calculations of anti2 optimized at '''HF/3-21G''' ([[File:XJB_anti2.log]])''</div>

The ''anti2'' structure was then reoptimized at '''B3LYP/6-31G*''' which was a improvement over the '''HF/3-21G''' basis set. It adds polarization to all atoms and improves the modeling of core electrons thus producing more accurate description of orbitals[ref]. The reoptimized ''anti2'' structure had the total electronic energy of '''-234.61171Ha''' and the same symmetry C<sub>i</sub>. The comparison of two structures optimized using different levels of theory is summarized in '''Table 6'''. We can see that they have exactly the same dihedral angle of the central four C atoms at 180.00<sup>o</sup> and angles between three of them are very similar as 111.3<sup>o</sup> and 112.7<sup>o</sup> respectively. Thus the the geometry change using '''HF/3-21G''' or '''B3LYP/6-31G*''' is trivial.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Before re-optimization
! scope="col" | After re-optimization
|-
| <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2reoptimized</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2reoptimized.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| ''anti2 optimized at '''HF/3-21G''''' || ''anti2 optimized at '''B3LYP/6-31G*'''''
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 6.''' ''Two optimization of anti2 at '''HF/3-21G''' and '''B3LYP/6-31G*'''([[File:XJB_anti2.log]])''</div>

The frequency calculation was run as well and a list of energies is summarized in '''Table 7'''. A list of all the vibrational frequencies was also generated and no imaginary (negative) frequency was observed and these vibrations are visualized by simulating the infrared spectrum in '''Figure 2'''.

{| align="center"
|- align="center"
|
{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-234.46920
|-
| Sum of electronic and thermal Energies||-234.46186
|-
| Sum of electronic and thermal Enthalpie||-234.46091
|-
| Sum of electronic and thermal Free Energie||-234.50078
|}
|
[[Image:XJB_IR_anti2.jpg|300px]]
|- align="center"
| align="center" | '''Table 7.''' ''Frequency calculations of anti2 optimized at '''B3LYP/6-31G*'''
([[File:XJB_anti2_reoptimized.log]])''
| align="center" | '''Figure 2.''' ''The generated IR spectrum of optimized '''B3LYP/6-31G*''' structure''
|}

===Optimizing the "Chair" and "Boat" Transition Structures===

The study of the chair and boat transition states is carried out using three different approaches. The first one is called the Hessian which involves computing the force constant matrix; the second one is to freeze the reaction coordinate and then optimize the structure once the molecule is fully relaxed; the last one is to use QST2 method to specify the reactants and products then finding the transition structure.

<center>[[Image:XJB_2 ts.jpg|500px|center|thumb|'''Figure 3'''. ''The two transition states of Cope rearrangement'']]</center>

====The Chair Transition State====

In order to set up a transition state optimization, first an allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimized at the '''HF/3-21G''' level of theory. Then this half of the transition structure was duplicated with rotation to make the approximate resembling of the chair transition state with terminal ends of the fragments 2.2 Å apart. The chair transition structure optimization was then done by the first two different approaches.

<center>[[Image:XJB_guess ts.jpg|400px|center|thumb|'''Figure 4'''. ''The guessed chair transition structure of two allyl fragments'']]</center>

For the first optimization approach, the force constants were computed (the Hessian) and the chair transition state was optimized to a TS (Berny) at the '''HF/3-21G''' level of theory. The frequency calculation was carried out at the same time and gave an imaginary frequency at -818 cm<sup>-1</sup>. The electronic energy was calculated to be '''-231.61932Ha''' with C<sub>2h</sub> symmetry point group. The distance between the terminal carbons was 2.0203 Å. An animation of this transition state vibrations was generated to confirm the corresponding Cope rearrangement.

<center>[[File:XJB_chair ts animation.GIF|400px|center|thumb|'''Figure 5.''' ''Animation of the Cope rearrangement via the chair transition state'']]</center>

The chair transition state was then reoptimized using the '''B3LYP/6-31G*''' level of theory and carried out frequency calculations as well. The electronic energy calculated this time was '''-234.55698Ha''' with an imaginary frequency at -566 cm<sup>-1</sup>. We can see that with the geometries quite similar, the energies differ markedly calculated using two levels of theory. The summary and comparison of the calculations optimized at two levels of theory are summarized in '''Table 8'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46670||-234.41493
|-
| Sum of electronic and thermal Energies||-231.46134||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40807
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir1.jpg|200px]]||[[Image:XJB_chairir2.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 8.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the first approach
([[File:XJB_chairts1.1.log]], [[File:XJB_chairts1.2.log]])''</div>

The second approach is the frozen coordinate method using the Redundant Coord Editor to freeze the bond lengths of the terminal ends to 2.2 Å. First the calculation failed because of the recent update of GaussView. After editting the coordinate distance (B 5 1 2.2 F) and the frozen distance (B 5 1 F) the optimization went well this time. The structure was then optimized again using a normal guess Hessian modified including the two differentiating coordinates. This chair transition state was then reoptimized again using the '''B3LYP/6-31G*''' level of theory. The electronic energies were calculated to be the same as in the first approach and frequency calculations were summarized in '''Table 9'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46671||-234.41490
|-
| Sum of electronic and thermal Energies||-231.46135||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40806
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir3.jpg|200px]]||[[Image:XJB_chairir4.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 9.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the second approach
([[File:XJB_chairts2.1.log]], [[File:XJB_chairts2.2.log]])''</div>

As the same energy results given, the bond forming/breaking bond lengths in the chair transition structures in the above two different approaches were compared. The first approach gave the bond lengths of 2.0203 Å while the second approach gave 2.0207 Å. Considering the accuracy limit of bond length estimation in Gaussview, these two bond lengths can be seen as exactly the same. Two methods worked reasonably well because of the good assumption of transition structures. However for more complicated and larger molecule, the Hessian method may fail due to the difficulty to predict the transition geometry (different surface curvature). Thus better way to generate such transition structures is to freeze reaction coordinate.

{| align="center"
|- align="center"
|
[[Image:XJB_chairts bondlength1.jpg|350px|thumb|'''Figure 6.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by Hession'']]
|
[[Image:XJB_chairts bondlength2.jpg|350px|thumb|'''Figure 7.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by frozen coordinate'']]
|- align="center"
|}

====The Boat Transition State====

The boat transition structure was then optimized using the '''QST2''' method. The same numbering for the reactant and product molecules was important in order to specify for this reaction. The renumbering of the molecules is shown as in '''Figure 8''', but the following optimization failed and gave a twisted structure. The reactant and product geometries were then modified further by changing the central C-C-C-C dihedral angle to 0<sup>o</sup> and the inside two C-C-C (i.e. C2-C3-C4 and C3-C4-C5) angles to 100<sup>o</sup>. The job was run successfully and frequency calculations were done with the boat transition state generated. Only one imaginary frequency was detected also at -840 cm<sup>-1</sup>. The electronic energy was '''-231.60280Ha''' with C<sub>2v</sub> symmetry. The distances between the terminal carbons of the allyl fragments were examined to be both 2.140Å which were comparably longer than that of the chair transition structure.

{| align="center"
|- align="center"
|
[[Image:XJB_QST1.jpg|400px|thumb|'''Figure 8.''' ''Numbering of reactant and product'']]
|
[[Image:XJB_QST2.jpg|410px|thumb|'''Figure 9.''' ''Re-numbering of reactant and product'']]
|- align="center"
|}

<center>[[File:XJB_boat ts animation.GIF|400px|center|thumb|'''Figure 10.''' ''Animation of the Cope rearrangement via the boat transition state'']]</center>

The structure was then reoptimized at the higher '''B3LYP/6-31G*''' theory of level and the electronic energy was calculated to be '''-234.54309Ha''' with very similar geometry. The frequency calculation generated an imaginary frequency at -530 cm<sup>-1</sup>. The summary and comparison of two levels' calculations are carried out in the following table.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.45093||-234.40234
|-
| Sum of electronic and thermal Energies||-231.44530||-234.39601
|-
| Sum of electronic and thermal Enthalpie|| -231.44436||-234.39506
|-
| Sum of electronic and thermal Free Energie||-231.47977||-234.43175
|-
| Generated IR spectrum||[[Image:XJB_boatir1.jpg|200px]]||[[Image:XJB_boatir2.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 10.''' ''Frequency calculations of the boat transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by QST2
([[File:XJB_boat ts1.log]], [[File:XJB_boat ts2.log]])''</div>

====Intrinsic Reaction Coordinate====

Next a useful method called '''Intrinct Reaction Coordinate (IRC)''' is applied to the Hessian optimized chair transition structure to follow the mininmum energy path (MEP) from a transition structure down to its local minimum on a potential energy surface. IRC can be used to verify that the transition state you found actually connects the reactants and products you think it does, or to identify any reaction intermediates you haven't thought about, or to generate an animation of the reaction [ref]. The symmetrical reaction coordinate was computed only in the forward direction with force constant always calculated and the number of points along IRC 50. The last structure generated along IRC is given in '''Figure 11''' with the energy calculated as '''-231.69158Ha''' and the dihedral angle and internal angle measured as D = -67.188<sup>o</sup> A = 111.78<sup>o</sup>. This energy does not match to any of the conformations discussed in Appendix 1 thus a minimum geometry has not reached yet.

{| align="center"
|- align="center"
|
<jmol><jmolApplet><title>XJB_IRC.mol</title><color>white</color>
<size>200</size><script>zoom=5</script>
<uploadedFileContents>XJB_IRC.mol</uploadedFileContents>
</jmolApplet></jmol>
|
[[Image:XJB_irc.jpg|500px]]
|- align="center"
| align="center" | '''Figure 11.''' ''The energy minimum structure along IRC''
| align="center" | '''Figure 12.''' ''Total energy along IRC''
|}

Three improving options can be used to reach the minimum energy. The first one is to run a normal Hessian optimization of the last point on IRC; the second one involves using a larger number of points (in this case 100 was used); the last one is to redo the IRC computation specifying to compute force constant at every step. Three methods were all used and the results were summarized in '''Table 11'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Improved IRC method 1
! scope="col" | Improved IRC method 2
! scope="col" | Improved IRC method 3
|-
| [[Image:XJB_IRC1.jpg|300px|center|IRC1]]||[[Image:XJB_IRC2.jpg|300px|center|IRC3]] || [[Image:XJB_IRC3.jpg|300px|center|IRC3]]
|-
| D = -64.17<sup>o</sup> A = 112.04<sup>o</sup> || D = -66.83<sup>o</sup> A = 111.82<sup>o</sup> || D = -64.17<sup>o</sup> A = 112.04<sup>o</sup>
|-
| -231.69167Ha || -231.69150Ha || -231.69167Ha
|-
| [[File:XJB_IRC1.log]]||[[File:XJB_IRC2.log]]||[[File:XJB_IRC3.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 11.''' ''Comparison of three improvement methods of IRC computation upon the chair transition structure''</div>

It is concluded that method 1 and 3 gave the same the lowest energy which was consistent to the energy given in Appendix 1.

====Activation Energies====

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | HF/3-21G at 0K
! scope="col" | HF/3-21G at 298.15K
! scope="col" | B3LYP/6-31G* at 0K
! scope="col" | B3LYP/6-31G* at 298.15K
! scope="col" | Expt. at 0K
|-
| '''ΔE (chair)'''||45.71||44.70||34.05||33.16||33.5 ± 0.5
|-
| '''ΔE (boat)'''||55.60||54.76||41.96||41.32||44.7 ± 0.5
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 12.''' ''Summary of activation energies for the Cope rearrangement via both transition states in kcal/mol''</div>

==The Diels Alder Cycloaddition==

In this exercise, AM1 semi-empirical method is used to compute the Diels Alder cycloaddition which is a concerted pericyclic reaction. The driving force of this reaction is the formation of a new σ bond between the π orbitals of the dieneophile and the diene. The reaction is allowed with HOMO-LUMO interation and a symmetry overlap.

===Cis Butadiene===

<center>[[Image:prototype Diels Alder.jpg|500px|center|thumb|'''Figure 13'''. ''The prototype reaction between butadiene and ethylene'']]</center>

To study a prototype Diels Alder cycloaddition reaction between butadiene and ethylene, first a cis butadiene molecule was built using Gaussview and optimized using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the butadiene were plotted. As seen the HOMO of cis-butadiene is anti-symmetric ('''a''') and the LUMO is symmetric ('''s''') with respect to plane.

{| align="center"
|- align="center"
|
[[Image:XJB_HOMO.jpg|280px|thumb|'''Figure 14.''' ''Anti-symmetric HOMO of cis-butadiene'']]
|
[[Image:XJB_LUMO.jpg|300px|thumb|'''Figure 15.''' ''Symmetric HOMO of cis-butadiene'']]
|- align="center"
|}

===The Transition State of prototype reaction between ethylene and butadiene===

The Hessian method was used to estimate transition state for the reaction between ethylene and butadiene because these two reactants are relatively small and simple molecules. The transition state was drawn by first building a structure as (a) in '''Figure 16''' then removing the -CH<sub>2</sub>-CH<sub>2</sub>- group to form the envelop structure as in (b). The '''clean''' option should not be used and the dashed bond length was guessed to be larger than 2.2 Å otherwise the geometry change would cause a failure in optimization.

<center>[[Image:XJB_dielsguessts.jpg|300px|center|thumb|'''Figure 16'''. ''The guessing transition state for the reaction between butadiene and ethylene'']]</center>

<center>[[File:XJB_butadiene and ethylene ts.GIF|400px|center|thumb|'''Figure 17.''' ''Animation of the butadiene and ethylene cycloaddition transition state'']]</center>

After the Hessian optimization, the transition structure has been successfully determined and an animation was generated in '''Figure 17'''. It has a C<sub>1</sub> symmetry point group and the bond lengths of two partly formed C-C bonds were measured to be 2.12Å. The typical sp<sup>3</sup> and sp<sup>2</sup> C-C bond lengths are 1.54Å and 1.47Å respectively[ref]. The van der Waals radius of the C atom is 1.7 Å[ref]. The measured bond distance of the transition structure is larger than typical bond lengths but still within the van der Waals radius, suggesting the bond forming interation between the carbon atoms.

From the animation ('''Figure 17''') the vibration directions of two molecules are opposite and they are approaching and leaving each other at the same time. This suggests the formation of the two bonds synchronous. The vibrational frequencies were also calculated and an infrared spectrum was simulated ('''Figure 20'''). An imaginary frequency of magnitude -955 cm<sup>-1</sup> was seen and the lowest positive frequency was observed at 147 cm<sup>-1</sup> which corresponded to the reactant vibration but not the bond formation.

{| align="center"
|
[[Image:XJB_HOMOts.jpg|300px|thumb|'''Figure 18'''. HOMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_LUMOts.jpg|300px|thumb|'''Figure 19'''. LUMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_IRts.jpg|300px|thumb|'''Figure 20'''. Generated IR of transition state of reaction between cis-butadiene and ethylene]]
|}

The HOMO and LUMO molecular orbitals of this transition structure were visualized in '''Figure 18''' and '''Figure 19'''. From these two figures it can be observed that the HOMO at the transition structure is anti-symmetric ('''a''') while the LUMO is symmetric ('''s'''). Thus this anti-symmetric transition state HOMO is formed by the HOMO of cis-butadiene and the LUMO of ethylene with the knowledge that they are both anti-symmetrical. This conclusion agrees with the previous talked conditions for allowed reaction.

===The cyclohexa-1,3-diene reaction with maleic anhydride===

<center>[[Image:XJB_mechanism3.jpg|500px|center|thumb|'''Figure 21'''. ''The endo and exo products of the Diels Alder reaction between cyclohexa-1,3-diene and maleic anhydride'']]</center>

The exo and endo transition states were drawn according to '''Figure 21''' with the new C-C bond formation line dashed and bond length estimation as 2.5. Then the same Hessian method was applied to optimize the structures. The electronic energy of the endo structure was calculated to be '''-0.05150Ha''' and '''-0.05042Ha''' for the exo. This reaction is supposed to be kinetically controlled thus the lower energy endo transition state is more favoured.

The bond lengths of the partly formed C-C bond are 2.16 Å and 2.17 Å for endo and exo transition states respectively. Another measured C-C bond length is shown in '''Table 13''' as 2.89 Å for endo (C3-C8 and C2-C7) and 2.95 Å for endo (C7-C10 and C8-C9). The longer atom distance indicates the larger steric repulsion in the exo transition state because of the extra existence of hydrogen atoms on C7 and C8.

Woodward and Hoffmann also proposed a secondary orbital overlap statement as an explanation of the endo stereo-preference in Diels Alder reactions<ref name="soo">''J. Org. Chem.'', 1987, '''52(8)''', pp 1469–1474{{DOI|10.1021/jo00384a016}}</ref>. This secondary orbital overlap does not involve bond changes but π orbital interactions as the highlighted carbon atoms of the endo transition state in '''Table 13'''. Such overlap stablises the structure and lowers the energy of the endo transition state while there is no such effect existed in the exo transition state.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | endo
! scope="col" | exo
|-
| '''Animation'''||[[File:XJB_endots.GIF|300px]]||[[File:XJB_exots.GIF|300px]]
|-
| '''Sum of electronic and zero-point Energies'''||0.133493||0.134880
|-
| '''Sum of electronic and thermal Energies'''||0.143683||0.144881
|-
| '''Sum of electronic and thermal Enthalpies'''||0.144627||0.145825
|-
| '''Sum of electronic and thermal Free Energies'''||0.097349||0.099117
|-
| '''Bond lengths'''||[[Image:XJB_endomeasure.jpg|300px]]||[[Image:XJB_exomeasure.jpg|300px]]
|-
| '''HOMO'''||[[Image:XJB_endoHOMO.jpg|270px]]||[[Image:XJB_exoHOMO.jpg|255px]]
|-
| '''Secondary orbital overlap effect'''||[[Image:XJB_endosoo.jpg|300px]]||[[Image:XJB_exosoo.jpg|300px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 13.''' ''Summary of comparison of properties of endo and exo transition structures for reaction between cyclohexa-1,3-diene and maleic anhydride''</div>

File:XJB endosoo.jpg

2013-12-06T00:01:14Z

Jx1011:

Rep:Mod:XJB1130

2013-12-06T00:00:31Z

Jx1011: /* The cyclohexa-1,3-diene reaction with maleic anhydride */

==The Cope Rearrangement of 1,5-hexadiene==

The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement reaction ('''Figure 1'''). Its mechanism has been studied by a large number of experimental and computational researches, which is recently accepted that the reaction occurs via either a "chair" or a "boat" transition state structure in a concerted manner. The aim of this exercise is to locate the low-energy minimum and the transition structures on the potential energy surface by Gaussian calculation in order to study its mechanism. The reaction pathway and the barrier heights can also be calculated by this calculation.

<center>[[Image:XJB_cope rearrangement.jpg|500px|center|thumb|'''Figure 1'''. ''The Cope rearrangement'']]</center>

===Optimizing the Reactants and Products===

Different conformations of 1,5-hexadiene can be optimized using '''Gaussview'''. The optimized structure of four conformers involved in this discussion and their corresponding energies optimized at '''HF/3-21G''' level of theory are concluded in the following table ('''Table 1''').

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | anti4
! scope="col" | anti2
! scope="col" | gauche
! scope="col" | gauche3
|-
| <jmol><jmolApplet><title>XJB_anti4.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti4.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche3.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| -231.69097Ha || -231.69254Ha || -231.68772Ha ||-231.69266Ha
|}

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 1.''' ''Four conformers of 1,5-hexadiene''</div>

This exercise first involved optimizing a molecule of 1,5-hexadiene with "anti" linkage for the cetral four C atoms at the '''HF/3-21G''' level of theory. The final optimization structure had the total electronic energy of '''-231.69097Ha''' with a '''C<sub>1</sub>''' symmetry. By comparing the energy and point group to that in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1], this structure is confirmed to be ''anti4''. The compositions of energies are also listed in '''Table 2''' as i) the sum of electronic and zero-point energies (potential energy at 0K including the zero-point vibrational energy), ii) the sum of electronic and thermal energies (energy including translational, rotational and vibrational at 298.15K and 1 atm), iii) the sum of electronic and thermal enthalpies (containning an additional correction for RT), and iv) the sum of electronic and thermal free energies (including the entropic contribution to the free energy). For what we are considering, the first two terms are the most useful for further comparisons.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Formula Description'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||E = E<sub>elec</sub> + ZPE||-231.53785
|-
| Sum of electronic and thermal Energies||E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>||-231.53096
|-
| Sum of electronic and thermal Enthalpie||H = E + RT||-231.53002
|-
| Sum of electronic and thermal Free Energie||G = H - TS||-231.56891
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 2.''' ''Frequency calculations of anti4 optimized at '''HF/3-21G''' ([[File:XJB_anti4.log]])''</div>

Next another 1,5-hexadiene molecule with "gauche" linkage for the central four atoms was drawn and optimized at the '''HF/3-21G''' level of theory as before. Energy for this conformation is expected to be higher than that for the ''anti4'' conformation due to the closer position of two alkene ends thus the steric interaction. The final structure had the electronic energy of '''-231.68772Ha''' which was indeed higher than that of the ''anti4'', indicating less favour towards this gauche structure. It had a '''C<sub>2</sub>''' symmetry and was confirmed to be the ''gauche'' conformation.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53439
|-
| Sum of electronic and thermal Energies||-231.52770
|-
| Sum of electronic and thermal Enthalpie||-231.52676
|-
| Sum of electronic and thermal Free Energie||-231.56483
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 3.''' ''Frequency calculations of gauche optimized at '''HF/3-21G''' ([[File:XJB_gauche.log]])''</div>

A lowest energy conformation ''gauche3'' of the reactant molecule was drawn based on and confirmed by [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. The optimization gave a final structure with an energy of '''-231.69266Ha''' with a symmetry of '''C<sub>1</sub>'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53949
|-
| Sum of electronic and thermal Energies||-231.53265
|-
| Sum of electronic and thermal Enthalpie||-231.5317
|-
| Sum of electronic and thermal Free Energie||-231.57064
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 4.''' ''Frequency calculations of gauche3 optimized at '''HF/3-21G''' ([[File:XJB_gauche3.log]])''</div>

The next step required us to draw the C<sub>i</sub> symmetry ''anti2'' conformation of 1,5-hexadiene and optimize it at the '''HF/3-21G''' level of theory. The optimized energy was calculated to be '''-231.69254Ha''' with point group of '''C<sub>i</sub>''', which was consistent with the one given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53954
|-
| Sum of electronic and thermal Energies||-231.53257
|-
| Sum of electronic and thermal Enthalpie||-231.53162
|-
| Sum of electronic and thermal Free Energie||-231.57092
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 5.''' ''Frequency calculations of anti2 optimized at '''HF/3-21G''' ([[File:XJB_anti2.log]])''</div>

The ''anti2'' structure was then reoptimized at '''B3LYP/6-31G*''' which was a improvement over the '''HF/3-21G''' basis set. It adds polarization to all atoms and improves the modeling of core electrons thus producing more accurate description of orbitals[ref]. The reoptimized ''anti2'' structure had the total electronic energy of '''-234.61171Ha''' and the same symmetry C<sub>i</sub>. The comparison of two structures optimized using different levels of theory is summarized in '''Table 6'''. We can see that they have exactly the same dihedral angle of the central four C atoms at 180.00<sup>o</sup> and angles between three of them are very similar as 111.3<sup>o</sup> and 112.7<sup>o</sup> respectively. Thus the the geometry change using '''HF/3-21G''' or '''B3LYP/6-31G*''' is trivial.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Before re-optimization
! scope="col" | After re-optimization
|-
| <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2reoptimized</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2reoptimized.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| ''anti2 optimized at '''HF/3-21G''''' || ''anti2 optimized at '''B3LYP/6-31G*'''''
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 6.''' ''Two optimization of anti2 at '''HF/3-21G''' and '''B3LYP/6-31G*'''([[File:XJB_anti2.log]])''</div>

The frequency calculation was run as well and a list of energies is summarized in '''Table 7'''. A list of all the vibrational frequencies was also generated and no imaginary (negative) frequency was observed and these vibrations are visualized by simulating the infrared spectrum in '''Figure 2'''.

{| align="center"
|- align="center"
|
{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-234.46920
|-
| Sum of electronic and thermal Energies||-234.46186
|-
| Sum of electronic and thermal Enthalpie||-234.46091
|-
| Sum of electronic and thermal Free Energie||-234.50078
|}
|
[[Image:XJB_IR_anti2.jpg|300px]]
|- align="center"
| align="center" | '''Table 7.''' ''Frequency calculations of anti2 optimized at '''B3LYP/6-31G*'''
([[File:XJB_anti2_reoptimized.log]])''
| align="center" | '''Figure 2.''' ''The generated IR spectrum of optimized '''B3LYP/6-31G*''' structure''
|}

===Optimizing the "Chair" and "Boat" Transition Structures===

The study of the chair and boat transition states is carried out using three different approaches. The first one is called the Hessian which involves computing the force constant matrix; the second one is to freeze the reaction coordinate and then optimize the structure once the molecule is fully relaxed; the last one is to use QST2 method to specify the reactants and products then finding the transition structure.

<center>[[Image:XJB_2 ts.jpg|500px|center|thumb|'''Figure 3'''. ''The two transition states of Cope rearrangement'']]</center>

====The Chair Transition State====

In order to set up a transition state optimization, first an allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimized at the '''HF/3-21G''' level of theory. Then this half of the transition structure was duplicated with rotation to make the approximate resembling of the chair transition state with terminal ends of the fragments 2.2 Å apart. The chair transition structure optimization was then done by the first two different approaches.

<center>[[Image:XJB_guess ts.jpg|400px|center|thumb|'''Figure 4'''. ''The guessed chair transition structure of two allyl fragments'']]</center>

For the first optimization approach, the force constants were computed (the Hessian) and the chair transition state was optimized to a TS (Berny) at the '''HF/3-21G''' level of theory. The frequency calculation was carried out at the same time and gave an imaginary frequency at -818 cm<sup>-1</sup>. The electronic energy was calculated to be '''-231.61932Ha''' with C<sub>2h</sub> symmetry point group. The distance between the terminal carbons was 2.0203 Å. An animation of this transition state vibrations was generated to confirm the corresponding Cope rearrangement.

<center>[[File:XJB_chair ts animation.GIF|400px|center|thumb|'''Figure 5.''' ''Animation of the Cope rearrangement via the chair transition state'']]</center>

The chair transition state was then reoptimized using the '''B3LYP/6-31G*''' level of theory and carried out frequency calculations as well. The electronic energy calculated this time was '''-234.55698Ha''' with an imaginary frequency at -566 cm<sup>-1</sup>. We can see that with the geometries quite similar, the energies differ markedly calculated using two levels of theory. The summary and comparison of the calculations optimized at two levels of theory are summarized in '''Table 8'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46670||-234.41493
|-
| Sum of electronic and thermal Energies||-231.46134||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40807
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir1.jpg|200px]]||[[Image:XJB_chairir2.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 8.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the first approach
([[File:XJB_chairts1.1.log]], [[File:XJB_chairts1.2.log]])''</div>

The second approach is the frozen coordinate method using the Redundant Coord Editor to freeze the bond lengths of the terminal ends to 2.2 Å. First the calculation failed because of the recent update of GaussView. After editting the coordinate distance (B 5 1 2.2 F) and the frozen distance (B 5 1 F) the optimization went well this time. The structure was then optimized again using a normal guess Hessian modified including the two differentiating coordinates. This chair transition state was then reoptimized again using the '''B3LYP/6-31G*''' level of theory. The electronic energies were calculated to be the same as in the first approach and frequency calculations were summarized in '''Table 9'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46671||-234.41490
|-
| Sum of electronic and thermal Energies||-231.46135||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40806
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir3.jpg|200px]]||[[Image:XJB_chairir4.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 9.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the second approach
([[File:XJB_chairts2.1.log]], [[File:XJB_chairts2.2.log]])''</div>

As the same energy results given, the bond forming/breaking bond lengths in the chair transition structures in the above two different approaches were compared. The first approach gave the bond lengths of 2.0203 Å while the second approach gave 2.0207 Å. Considering the accuracy limit of bond length estimation in Gaussview, these two bond lengths can be seen as exactly the same. Two methods worked reasonably well because of the good assumption of transition structures. However for more complicated and larger molecule, the Hessian method may fail due to the difficulty to predict the transition geometry (different surface curvature). Thus better way to generate such transition structures is to freeze reaction coordinate.

{| align="center"
|- align="center"
|
[[Image:XJB_chairts bondlength1.jpg|350px|thumb|'''Figure 6.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by Hession'']]
|
[[Image:XJB_chairts bondlength2.jpg|350px|thumb|'''Figure 7.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by frozen coordinate'']]
|- align="center"
|}

====The Boat Transition State====

The boat transition structure was then optimized using the '''QST2''' method. The same numbering for the reactant and product molecules was important in order to specify for this reaction. The renumbering of the molecules is shown as in '''Figure 8''', but the following optimization failed and gave a twisted structure. The reactant and product geometries were then modified further by changing the central C-C-C-C dihedral angle to 0<sup>o</sup> and the inside two C-C-C (i.e. C2-C3-C4 and C3-C4-C5) angles to 100<sup>o</sup>. The job was run successfully and frequency calculations were done with the boat transition state generated. Only one imaginary frequency was detected also at -840 cm<sup>-1</sup>. The electronic energy was '''-231.60280Ha''' with C<sub>2v</sub> symmetry. The distances between the terminal carbons of the allyl fragments were examined to be both 2.140Å which were comparably longer than that of the chair transition structure.

{| align="center"
|- align="center"
|
[[Image:XJB_QST1.jpg|400px|thumb|'''Figure 8.''' ''Numbering of reactant and product'']]
|
[[Image:XJB_QST2.jpg|410px|thumb|'''Figure 9.''' ''Re-numbering of reactant and product'']]
|- align="center"
|}

<center>[[File:XJB_boat ts animation.GIF|400px|center|thumb|'''Figure 10.''' ''Animation of the Cope rearrangement via the boat transition state'']]</center>

The structure was then reoptimized at the higher '''B3LYP/6-31G*''' theory of level and the electronic energy was calculated to be '''-234.54309Ha''' with very similar geometry. The frequency calculation generated an imaginary frequency at -530 cm<sup>-1</sup>. The summary and comparison of two levels' calculations are carried out in the following table.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.45093||-234.40234
|-
| Sum of electronic and thermal Energies||-231.44530||-234.39601
|-
| Sum of electronic and thermal Enthalpie|| -231.44436||-234.39506
|-
| Sum of electronic and thermal Free Energie||-231.47977||-234.43175
|-
| Generated IR spectrum||[[Image:XJB_boatir1.jpg|200px]]||[[Image:XJB_boatir2.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 10.''' ''Frequency calculations of the boat transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by QST2
([[File:XJB_boat ts1.log]], [[File:XJB_boat ts2.log]])''</div>

====Intrinsic Reaction Coordinate====

Next a useful method called '''Intrinct Reaction Coordinate (IRC)''' is applied to the Hessian optimized chair transition structure to follow the mininmum energy path (MEP) from a transition structure down to its local minimum on a potential energy surface. IRC can be used to verify that the transition state you found actually connects the reactants and products you think it does, or to identify any reaction intermediates you haven't thought about, or to generate an animation of the reaction [ref]. The symmetrical reaction coordinate was computed only in the forward direction with force constant always calculated and the number of points along IRC 50. The last structure generated along IRC is given in '''Figure 11''' with the energy calculated as '''-231.69158Ha''' and the dihedral angle and internal angle measured as D = -67.188<sup>o</sup> A = 111.78<sup>o</sup>. This energy does not match to any of the conformations discussed in Appendix 1 thus a minimum geometry has not reached yet.

{| align="center"
|- align="center"
|
<jmol><jmolApplet><title>XJB_IRC.mol</title><color>white</color>
<size>200</size><script>zoom=5</script>
<uploadedFileContents>XJB_IRC.mol</uploadedFileContents>
</jmolApplet></jmol>
|
[[Image:XJB_irc.jpg|500px]]
|- align="center"
| align="center" | '''Figure 11.''' ''The energy minimum structure along IRC''
| align="center" | '''Figure 12.''' ''Total energy along IRC''
|}

Three improving options can be used to reach the minimum energy. The first one is to run a normal Hessian optimization of the last point on IRC; the second one involves using a larger number of points (in this case 100 was used); the last one is to redo the IRC computation specifying to compute force constant at every step. Three methods were all used and the results were summarized in '''Table 11'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Improved IRC method 1
! scope="col" | Improved IRC method 2
! scope="col" | Improved IRC method 3
|-
| [[Image:XJB_IRC1.jpg|300px|center|IRC1]]||[[Image:XJB_IRC2.jpg|300px|center|IRC3]] || [[Image:XJB_IRC3.jpg|300px|center|IRC3]]
|-
| D = -64.17<sup>o</sup> A = 112.04<sup>o</sup> || D = -66.83<sup>o</sup> A = 111.82<sup>o</sup> || D = -64.17<sup>o</sup> A = 112.04<sup>o</sup>
|-
| -231.69167Ha || -231.69150Ha || -231.69167Ha
|-
| [[File:XJB_IRC1.log]]||[[File:XJB_IRC2.log]]||[[File:XJB_IRC3.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 11.''' ''Comparison of three improvement methods of IRC computation upon the chair transition structure''</div>

It is concluded that method 1 and 3 gave the same the lowest energy which was consistent to the energy given in Appendix 1.

====Activation Energies====

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | HF/3-21G at 0K
! scope="col" | HF/3-21G at 298.15K
! scope="col" | B3LYP/6-31G* at 0K
! scope="col" | B3LYP/6-31G* at 298.15K
! scope="col" | Expt. at 0K
|-
| '''ΔE (chair)'''||45.71||44.70||34.05||33.16||33.5 ± 0.5
|-
| '''ΔE (boat)'''||55.60||54.76||41.96||41.32||44.7 ± 0.5
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 12.''' ''Summary of activation energies for the Cope rearrangement via both transition states in kcal/mol''</div>

==The Diels Alder Cycloaddition==

In this exercise, AM1 semi-empirical method is used to compute the Diels Alder cycloaddition which is a concerted pericyclic reaction. The driving force of this reaction is the formation of a new σ bond between the π orbitals of the dieneophile and the diene. The reaction is allowed with HOMO-LUMO interation and a symmetry overlap.

===Cis Butadiene===

<center>[[Image:prototype Diels Alder.jpg|500px|center|thumb|'''Figure 13'''. ''The prototype reaction between butadiene and ethylene'']]</center>

To study a prototype Diels Alder cycloaddition reaction between butadiene and ethylene, first a cis butadiene molecule was built using Gaussview and optimized using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the butadiene were plotted. As seen the HOMO of cis-butadiene is anti-symmetric ('''a''') and the LUMO is symmetric ('''s''') with respect to plane.

{| align="center"
|- align="center"
|
[[Image:XJB_HOMO.jpg|280px|thumb|'''Figure 14.''' ''Anti-symmetric HOMO of cis-butadiene'']]
|
[[Image:XJB_LUMO.jpg|300px|thumb|'''Figure 15.''' ''Symmetric HOMO of cis-butadiene'']]
|- align="center"
|}

===The Transition State of prototype reaction between ethylene and butadiene===

The Hessian method was used to estimate transition state for the reaction between ethylene and butadiene because these two reactants are relatively small and simple molecules. The transition state was drawn by first building a structure as (a) in '''Figure 16''' then removing the -CH<sub>2</sub>-CH<sub>2</sub>- group to form the envelop structure as in (b). The '''clean''' option should not be used and the dashed bond length was guessed to be larger than 2.2 Å otherwise the geometry change would cause a failure in optimization.

<center>[[Image:XJB_dielsguessts.jpg|300px|center|thumb|'''Figure 16'''. ''The guessing transition state for the reaction between butadiene and ethylene'']]</center>

<center>[[File:XJB_butadiene and ethylene ts.GIF|400px|center|thumb|'''Figure 17.''' ''Animation of the butadiene and ethylene cycloaddition transition state'']]</center>

After the Hessian optimization, the transition structure has been successfully determined and an animation was generated in '''Figure 17'''. It has a C<sub>1</sub> symmetry point group and the bond lengths of two partly formed C-C bonds were measured to be 2.12Å. The typical sp<sup>3</sup> and sp<sup>2</sup> C-C bond lengths are 1.54Å and 1.47Å respectively[ref]. The van der Waals radius of the C atom is 1.7 Å[ref]. The measured bond distance of the transition structure is larger than typical bond lengths but still within the van der Waals radius, suggesting the bond forming interation between the carbon atoms.

From the animation ('''Figure 17''') the vibration directions of two molecules are opposite and they are approaching and leaving each other at the same time. This suggests the formation of the two bonds synchronous. The vibrational frequencies were also calculated and an infrared spectrum was simulated ('''Figure 20'''). An imaginary frequency of magnitude -955 cm<sup>-1</sup> was seen and the lowest positive frequency was observed at 147 cm<sup>-1</sup> which corresponded to the reactant vibration but not the bond formation.

{| align="center"
|
[[Image:XJB_HOMOts.jpg|300px|thumb|'''Figure 18'''. HOMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_LUMOts.jpg|300px|thumb|'''Figure 19'''. LUMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_IRts.jpg|300px|thumb|'''Figure 20'''. Generated IR of transition state of reaction between cis-butadiene and ethylene]]
|}

The HOMO and LUMO molecular orbitals of this transition structure were visualized in '''Figure 18''' and '''Figure 19'''. From these two figures it can be observed that the HOMO at the transition structure is anti-symmetric ('''a''') while the LUMO is symmetric ('''s'''). Thus this anti-symmetric transition state HOMO is formed by the HOMO of cis-butadiene and the LUMO of ethylene with the knowledge that they are both anti-symmetrical. This conclusion agrees with the previous talked conditions for allowed reaction.

===The cyclohexa-1,3-diene reaction with maleic anhydride===

<center>[[Image:XJB_mechanism3.jpg|500px|center|thumb|'''Figure 21'''. ''The endo and exo products of the Diels Alder reaction between cyclohexa-1,3-diene and maleic anhydride'']]</center>

The exo and endo transition states were drawn according to '''Figure 21''' with the new C-C bond formation line dashed and bond length estimation as 2.5. Then the same Hessian method was applied to optimize the structures. The electronic energy of the endo structure was calculated to be '''-0.05150Ha''' and '''-0.05042Ha''' for the exo. This reaction is supposed to be kinetically controlled thus the lower energy endo transition state is more favoured.

The bond lengths of the partly formed C-C bond are 2.16 Å and 2.17 Å for endo and exo transition states respectively. Another measured C-C bond length is shown in '''Table 13''' as 2.89 Å for endo (C3-C8 and C2-C7) and 2.95 Å for endo (C7-C10 and C8-C9). The longer atom distance indicates the larger steric repulsion in the exo transition state because of the extra existence of hydrogen atoms on C7 and C8.

Woodward and Hoffmann also proposed a secondary orbital overlap statement as an explanation of the endo stereo-preference in Diels Alder reactions<ref name="soo">''J. Org. Chem.'', 1987, '''52(8)''', pp 1469–1474{{DOI|10.1021/jo00384a016}}</ref>. This secondary orbital overlap does not involve bond changes but π orbital interactions as the highlighted carbon atoms of the endo transition state in '''Table 13'''

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | endo
! scope="col" | exo
|-
| '''Animation'''||[[File:XJB_endots.GIF|300px]]||[[File:XJB_exots.GIF|300px]]
|-
| '''Sum of electronic and zero-point Energies'''||0.133493||0.134880
|-
| '''Sum of electronic and thermal Energies'''||0.143683||0.144881
|-
| '''Sum of electronic and thermal Enthalpies'''||0.144627||0.145825
|-
| '''Sum of electronic and thermal Free Energies'''||0.097349||0.099117
|-
| '''Bond lengths'''||[[Image:XJB_endomeasure.jpg|300px]]||[[Image:XJB_exomeasure.jpg|300px]]
|-
| '''HOMO'''||[[Image:XJB_endoHOMO.jpg|270px]]||[[Image:XJB_exoHOMO.jpg|255px]]
|-
| '''Secondary orbital overlap effect'''||[[Image:XJB_endosoo.jpg|300px]]||[[Image:XJB_exosoo.jpg|300px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 13.''' ''Summary of comparison of properties of endo and exo transition structures for reaction between cyclohexa-1,3-diene and maleic anhydride''</div>

File:XJB exomeasure.jpg

2013-12-05T18:59:51Z

Jx1011:

File:XJB endomeasure.jpg

2013-12-05T18:59:31Z

Jx1011:

Rep:Mod:XJB1130

2013-12-05T18:20:19Z

Jx1011: /* The cyclohexa-1,3-diene reaction with maleic anhydride */

==The Cope Rearrangement of 1,5-hexadiene==

The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement reaction ('''Figure 1'''). Its mechanism has been studied by a large number of experimental and computational researches, which is recently accepted that the reaction occurs via either a "chair" or a "boat" transition state structure in a concerted manner. The aim of this exercise is to locate the low-energy minimum and the transition structures on the potential energy surface by Gaussian calculation in order to study its mechanism. The reaction pathway and the barrier heights can also be calculated by this calculation.

<center>[[Image:XJB_cope rearrangement.jpg|500px|center|thumb|'''Figure 1'''. ''The Cope rearrangement'']]</center>

===Optimizing the Reactants and Products===

Different conformations of 1,5-hexadiene can be optimized using '''Gaussview'''. The optimized structure of four conformers involved in this discussion and their corresponding energies optimized at '''HF/3-21G''' level of theory are concluded in the following table ('''Table 1''').

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | anti4
! scope="col" | anti2
! scope="col" | gauche
! scope="col" | gauche3
|-
| <jmol><jmolApplet><title>XJB_anti4.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti4.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche3.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| -231.69097Ha || -231.69254Ha || -231.68772Ha ||-231.69266Ha
|}

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 1.''' ''Four conformers of 1,5-hexadiene''</div>

This exercise first involved optimizing a molecule of 1,5-hexadiene with "anti" linkage for the cetral four C atoms at the '''HF/3-21G''' level of theory. The final optimization structure had the total electronic energy of '''-231.69097Ha''' with a '''C<sub>1</sub>''' symmetry. By comparing the energy and point group to that in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1], this structure is confirmed to be ''anti4''. The compositions of energies are also listed in '''Table 2''' as i) the sum of electronic and zero-point energies (potential energy at 0K including the zero-point vibrational energy), ii) the sum of electronic and thermal energies (energy including translational, rotational and vibrational at 298.15K and 1 atm), iii) the sum of electronic and thermal enthalpies (containning an additional correction for RT), and iv) the sum of electronic and thermal free energies (including the entropic contribution to the free energy). For what we are considering, the first two terms are the most useful for further comparisons.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Formula Description'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||E = E<sub>elec</sub> + ZPE||-231.53785
|-
| Sum of electronic and thermal Energies||E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>||-231.53096
|-
| Sum of electronic and thermal Enthalpie||H = E + RT||-231.53002
|-
| Sum of electronic and thermal Free Energie||G = H - TS||-231.56891
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 2.''' ''Frequency calculations of anti4 optimized at '''HF/3-21G''' ([[File:XJB_anti4.log]])''</div>

Next another 1,5-hexadiene molecule with "gauche" linkage for the central four atoms was drawn and optimized at the '''HF/3-21G''' level of theory as before. Energy for this conformation is expected to be higher than that for the ''anti4'' conformation due to the closer position of two alkene ends thus the steric interaction. The final structure had the electronic energy of '''-231.68772Ha''' which was indeed higher than that of the ''anti4'', indicating less favour towards this gauche structure. It had a '''C<sub>2</sub>''' symmetry and was confirmed to be the ''gauche'' conformation.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53439
|-
| Sum of electronic and thermal Energies||-231.52770
|-
| Sum of electronic and thermal Enthalpie||-231.52676
|-
| Sum of electronic and thermal Free Energie||-231.56483
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 3.''' ''Frequency calculations of gauche optimized at '''HF/3-21G''' ([[File:XJB_gauche.log]])''</div>

A lowest energy conformation ''gauche3'' of the reactant molecule was drawn based on and confirmed by [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. The optimization gave a final structure with an energy of '''-231.69266Ha''' with a symmetry of '''C<sub>1</sub>'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53949
|-
| Sum of electronic and thermal Energies||-231.53265
|-
| Sum of electronic and thermal Enthalpie||-231.5317
|-
| Sum of electronic and thermal Free Energie||-231.57064
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 4.''' ''Frequency calculations of gauche3 optimized at '''HF/3-21G''' ([[File:XJB_gauche3.log]])''</div>

The next step required us to draw the C<sub>i</sub> symmetry ''anti2'' conformation of 1,5-hexadiene and optimize it at the '''HF/3-21G''' level of theory. The optimized energy was calculated to be '''-231.69254Ha''' with point group of '''C<sub>i</sub>''', which was consistent with the one given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53954
|-
| Sum of electronic and thermal Energies||-231.53257
|-
| Sum of electronic and thermal Enthalpie||-231.53162
|-
| Sum of electronic and thermal Free Energie||-231.57092
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 5.''' ''Frequency calculations of anti2 optimized at '''HF/3-21G''' ([[File:XJB_anti2.log]])''</div>

The ''anti2'' structure was then reoptimized at '''B3LYP/6-31G*''' which was a improvement over the '''HF/3-21G''' basis set. It adds polarization to all atoms and improves the modeling of core electrons thus producing more accurate description of orbitals[ref]. The reoptimized ''anti2'' structure had the total electronic energy of '''-234.61171Ha''' and the same symmetry C<sub>i</sub>. The comparison of two structures optimized using different levels of theory is summarized in '''Table 6'''. We can see that they have exactly the same dihedral angle of the central four C atoms at 180.00<sup>o</sup> and angles between three of them are very similar as 111.3<sup>o</sup> and 112.7<sup>o</sup> respectively. Thus the the geometry change using '''HF/3-21G''' or '''B3LYP/6-31G*''' is trivial.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Before re-optimization
! scope="col" | After re-optimization
|-
| <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2reoptimized</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2reoptimized.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| ''anti2 optimized at '''HF/3-21G''''' || ''anti2 optimized at '''B3LYP/6-31G*'''''
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 6.''' ''Two optimization of anti2 at '''HF/3-21G''' and '''B3LYP/6-31G*'''([[File:XJB_anti2.log]])''</div>

The frequency calculation was run as well and a list of energies is summarized in '''Table 7'''. A list of all the vibrational frequencies was also generated and no imaginary (negative) frequency was observed and these vibrations are visualized by simulating the infrared spectrum in '''Figure 2'''.

{| align="center"
|- align="center"
|
{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-234.46920
|-
| Sum of electronic and thermal Energies||-234.46186
|-
| Sum of electronic and thermal Enthalpie||-234.46091
|-
| Sum of electronic and thermal Free Energie||-234.50078
|}
|
[[Image:XJB_IR_anti2.jpg|300px]]
|- align="center"
| align="center" | '''Table 7.''' ''Frequency calculations of anti2 optimized at '''B3LYP/6-31G*'''
([[File:XJB_anti2_reoptimized.log]])''
| align="center" | '''Figure 2.''' ''The generated IR spectrum of optimized '''B3LYP/6-31G*''' structure''
|}

===Optimizing the "Chair" and "Boat" Transition Structures===

The study of the chair and boat transition states is carried out using three different approaches. The first one is called the Hessian which involves computing the force constant matrix; the second one is to freeze the reaction coordinate and then optimize the structure once the molecule is fully relaxed; the last one is to use QST2 method to specify the reactants and products then finding the transition structure.

<center>[[Image:XJB_2 ts.jpg|500px|center|thumb|'''Figure 3'''. ''The two transition states of Cope rearrangement'']]</center>

====The Chair Transition State====

In order to set up a transition state optimization, first an allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimized at the '''HF/3-21G''' level of theory. Then this half of the transition structure was duplicated with rotation to make the approximate resembling of the chair transition state with terminal ends of the fragments 2.2 Å apart. The chair transition structure optimization was then done by the first two different approaches.

<center>[[Image:XJB_guess ts.jpg|400px|center|thumb|'''Figure 4'''. ''The guessed chair transition structure of two allyl fragments'']]</center>

For the first optimization approach, the force constants were computed (the Hessian) and the chair transition state was optimized to a TS (Berny) at the '''HF/3-21G''' level of theory. The frequency calculation was carried out at the same time and gave an imaginary frequency at -818 cm<sup>-1</sup>. The electronic energy was calculated to be '''-231.61932Ha''' with C<sub>2h</sub> symmetry point group. The distance between the terminal carbons was 2.0203 Å. An animation of this transition state vibrations was generated to confirm the corresponding Cope rearrangement.

<center>[[File:XJB_chair ts animation.GIF|400px|center|thumb|'''Figure 5.''' ''Animation of the Cope rearrangement via the chair transition state'']]</center>

The chair transition state was then reoptimized using the '''B3LYP/6-31G*''' level of theory and carried out frequency calculations as well. The electronic energy calculated this time was '''-234.55698Ha''' with an imaginary frequency at -566 cm<sup>-1</sup>. We can see that with the geometries quite similar, the energies differ markedly calculated using two levels of theory. The summary and comparison of the calculations optimized at two levels of theory are summarized in '''Table 8'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46670||-234.41493
|-
| Sum of electronic and thermal Energies||-231.46134||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40807
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir1.jpg|200px]]||[[Image:XJB_chairir2.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 8.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the first approach
([[File:XJB_chairts1.1.log]], [[File:XJB_chairts1.2.log]])''</div>

The second approach is the frozen coordinate method using the Redundant Coord Editor to freeze the bond lengths of the terminal ends to 2.2 Å. First the calculation failed because of the recent update of GaussView. After editting the coordinate distance (B 5 1 2.2 F) and the frozen distance (B 5 1 F) the optimization went well this time. The structure was then optimized again using a normal guess Hessian modified including the two differentiating coordinates. This chair transition state was then reoptimized again using the '''B3LYP/6-31G*''' level of theory. The electronic energies were calculated to be the same as in the first approach and frequency calculations were summarized in '''Table 9'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46671||-234.41490
|-
| Sum of electronic and thermal Energies||-231.46135||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40806
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir3.jpg|200px]]||[[Image:XJB_chairir4.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 9.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the second approach
([[File:XJB_chairts2.1.log]], [[File:XJB_chairts2.2.log]])''</div>

As the same energy results given, the bond forming/breaking bond lengths in the chair transition structures in the above two different approaches were compared. The first approach gave the bond lengths of 2.0203 Å while the second approach gave 2.0207 Å. Considering the accuracy limit of bond length estimation in Gaussview, these two bond lengths can be seen as exactly the same. Two methods worked reasonably well because of the good assumption of transition structures. However for more complicated and larger molecule, the Hessian method may fail due to the difficulty to predict the transition geometry (different surface curvature). Thus better way to generate such transition structures is to freeze reaction coordinate.

{| align="center"
|- align="center"
|
[[Image:XJB_chairts bondlength1.jpg|350px|thumb|'''Figure 6.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by Hession'']]
|
[[Image:XJB_chairts bondlength2.jpg|350px|thumb|'''Figure 7.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by frozen coordinate'']]
|- align="center"
|}

====The Boat Transition State====

The boat transition structure was then optimized using the '''QST2''' method. The same numbering for the reactant and product molecules was important in order to specify for this reaction. The renumbering of the molecules is shown as in '''Figure 8''', but the following optimization failed and gave a twisted structure. The reactant and product geometries were then modified further by changing the central C-C-C-C dihedral angle to 0<sup>o</sup> and the inside two C-C-C (i.e. C2-C3-C4 and C3-C4-C5) angles to 100<sup>o</sup>. The job was run successfully and frequency calculations were done with the boat transition state generated. Only one imaginary frequency was detected also at -840 cm<sup>-1</sup>. The electronic energy was '''-231.60280Ha''' with C<sub>2v</sub> symmetry. The distances between the terminal carbons of the allyl fragments were examined to be both 2.140Å which were comparably longer than that of the chair transition structure.

{| align="center"
|- align="center"
|
[[Image:XJB_QST1.jpg|400px|thumb|'''Figure 8.''' ''Numbering of reactant and product'']]
|
[[Image:XJB_QST2.jpg|410px|thumb|'''Figure 9.''' ''Re-numbering of reactant and product'']]
|- align="center"
|}

<center>[[File:XJB_boat ts animation.GIF|400px|center|thumb|'''Figure 10.''' ''Animation of the Cope rearrangement via the boat transition state'']]</center>

The structure was then reoptimized at the higher '''B3LYP/6-31G*''' theory of level and the electronic energy was calculated to be '''-234.54309Ha''' with very similar geometry. The frequency calculation generated an imaginary frequency at -530 cm<sup>-1</sup>. The summary and comparison of two levels' calculations are carried out in the following table.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.45093||-234.40234
|-
| Sum of electronic and thermal Energies||-231.44530||-234.39601
|-
| Sum of electronic and thermal Enthalpie|| -231.44436||-234.39506
|-
| Sum of electronic and thermal Free Energie||-231.47977||-234.43175
|-
| Generated IR spectrum||[[Image:XJB_boatir1.jpg|200px]]||[[Image:XJB_boatir2.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 10.''' ''Frequency calculations of the boat transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by QST2
([[File:XJB_boat ts1.log]], [[File:XJB_boat ts2.log]])''</div>

====Intrinsic Reaction Coordinate====

Next a useful method called '''Intrinct Reaction Coordinate (IRC)''' is applied to the Hessian optimized chair transition structure to follow the mininmum energy path (MEP) from a transition structure down to its local minimum on a potential energy surface. IRC can be used to verify that the transition state you found actually connects the reactants and products you think it does, or to identify any reaction intermediates you haven't thought about, or to generate an animation of the reaction [ref]. The symmetrical reaction coordinate was computed only in the forward direction with force constant always calculated and the number of points along IRC 50. The last structure generated along IRC is given in '''Figure 11''' with the energy calculated as '''-231.69158Ha''' and the dihedral angle and internal angle measured as D = -67.188<sup>o</sup> A = 111.78<sup>o</sup>. This energy does not match to any of the conformations discussed in Appendix 1 thus a minimum geometry has not reached yet.

{| align="center"
|- align="center"
|
<jmol><jmolApplet><title>XJB_IRC.mol</title><color>white</color>
<size>200</size><script>zoom=5</script>
<uploadedFileContents>XJB_IRC.mol</uploadedFileContents>
</jmolApplet></jmol>
|
[[Image:XJB_irc.jpg|500px]]
|- align="center"
| align="center" | '''Figure 11.''' ''The energy minimum structure along IRC''
| align="center" | '''Figure 12.''' ''Total energy along IRC''
|}

Three improving options can be used to reach the minimum energy. The first one is to run a normal Hessian optimization of the last point on IRC; the second one involves using a larger number of points (in this case 100 was used); the last one is to redo the IRC computation specifying to compute force constant at every step. Three methods were all used and the results were summarized in '''Table 11'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Improved IRC method 1
! scope="col" | Improved IRC method 2
! scope="col" | Improved IRC method 3
|-
| [[Image:XJB_IRC1.jpg|300px|center|IRC1]]||[[Image:XJB_IRC2.jpg|300px|center|IRC3]] || [[Image:XJB_IRC3.jpg|300px|center|IRC3]]
|-
| D = -64.17<sup>o</sup> A = 112.04<sup>o</sup> || D = -66.83<sup>o</sup> A = 111.82<sup>o</sup> || D = -64.17<sup>o</sup> A = 112.04<sup>o</sup>
|-
| -231.69167Ha || -231.69150Ha || -231.69167Ha
|-
| [[File:XJB_IRC1.log]]||[[File:XJB_IRC2.log]]||[[File:XJB_IRC3.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 11.''' ''Comparison of three improvement methods of IRC computation upon the chair transition structure''</div>

It is concluded that method 1 and 3 gave the same the lowest energy which was consistent to the energy given in Appendix 1.

====Activation Energies====

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | HF/3-21G at 0K
! scope="col" | HF/3-21G at 298.15K
! scope="col" | B3LYP/6-31G* at 0K
! scope="col" | B3LYP/6-31G* at 298.15K
! scope="col" | Expt. at 0K
|-
| '''ΔE (chair)'''||45.71||44.70||34.05||33.16||33.5 ± 0.5
|-
| '''ΔE (boat)'''||55.60||54.76||41.96||41.32||44.7 ± 0.5
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 12.''' ''Summary of activation energies for the Cope rearrangement via both transition states in kcal/mol''</div>

==The Diels Alder Cycloaddition==

In this exercise, AM1 semi-empirical method is used to compute the Diels Alder cycloaddition which is a concerted pericyclic reaction. The driving force of this reaction is the formation of a new σ bond between the π orbitals of the dieneophile and the diene. The reaction is allowed with HOMO-LUMO interation and a symmetry overlap.

===Cis Butadiene===

<center>[[Image:prototype Diels Alder.jpg|500px|center|thumb|'''Figure 13'''. ''The prototype reaction between butadiene and ethylene'']]</center>

To study a prototype Diels Alder cycloaddition reaction between butadiene and ethylene, first a cis butadiene molecule was built using Gaussview and optimized using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the butadiene were plotted. As seen the HOMO of cis-butadiene is anti-symmetric ('''a''') and the LUMO is symmetric ('''s''') with respect to plane.

{| align="center"
|- align="center"
|
[[Image:XJB_HOMO.jpg|280px|thumb|'''Figure 14.''' ''Anti-symmetric HOMO of cis-butadiene'']]
|
[[Image:XJB_LUMO.jpg|300px|thumb|'''Figure 15.''' ''Symmetric HOMO of cis-butadiene'']]
|- align="center"
|}

===The Transition State of prototype reaction between ethylene and butadiene===

The Hessian method was used to estimate transition state for the reaction between ethylene and butadiene because these two reactants are relatively small and simple molecules. The transition state was drawn by first building a structure as (a) in '''Figure 16''' then removing the -CH<sub>2</sub>-CH<sub>2</sub>- group to form the envelop structure as in (b). The '''clean''' option should not be used and the dashed bond length was guessed to be larger than 2.2 Å otherwise the geometry change would cause a failure in optimization.

<center>[[Image:XJB_dielsguessts.jpg|300px|center|thumb|'''Figure 16'''. ''The guessing transition state for the reaction between butadiene and ethylene'']]</center>

<center>[[File:XJB_butadiene and ethylene ts.GIF|400px|center|thumb|'''Figure 17.''' ''Animation of the butadiene and ethylene cycloaddition transition state'']]</center>

After the Hessian optimization, the transition structure has been successfully determined and an animation was generated in '''Figure 17'''. It has a C<sub>1</sub> symmetry point group and the bond lengths of two partly formed C-C bonds were measured to be 2.12Å. The typical sp<sup>3</sup> and sp<sup>2</sup> C-C bond lengths are 1.54Å and 1.47Å respectively[ref]. The van der Waals radius of the C atom is 1.7 Å[ref]. The measured bond distance of the transition structure is larger than typical bond lengths but still within the van der Waals radius, suggesting the bond forming interation between the carbon atoms.

From the animation ('''Figure 17''') the vibration directions of two molecules are opposite and they are approaching and leaving each other at the same time. This suggests the formation of the two bonds synchronous. The vibrational frequencies were also calculated and an infrared spectrum was simulated ('''Figure 20'''). An imaginary frequency of magnitude -955 cm<sup>-1</sup> was seen and the lowest positive frequency was observed at 147 cm<sup>-1</sup> which corresponded to the reactant vibration but not the bond formation.

{| align="center"
|
[[Image:XJB_HOMOts.jpg|300px|thumb|'''Figure 18'''. HOMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_LUMOts.jpg|300px|thumb|'''Figure 19'''. LUMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_IRts.jpg|300px|thumb|'''Figure 20'''. Generated IR of transition state of reaction between cis-butadiene and ethylene]]
|}

The HOMO and LUMO molecular orbitals of this transition structure were visualized in '''Figure 18''' and '''Figure 19'''. From these two figures it can be observed that the HOMO at the transition structure is anti-symmetric ('''a''') while the LUMO is symmetric ('''s'''). Thus this anti-symmetric transition state HOMO is formed by the HOMO of cis-butadiene and the LUMO of ethylene with the knowledge that they are both anti-symmetrical. This conclusion agrees with the previous talked conditions for allowed reaction.

===The cyclohexa-1,3-diene reaction with maleic anhydride===

<center>[[Image:XJB_mechanism3.jpg|500px|center|thumb|'''Figure 21'''. ''The endo and exo products of the Diels Alder reaction between cyclohexa-1,3-diene and maleic anhydride'']]</center>

The exo and endo transition states were drawn according to '''Figure 21''' with the new C-C bond formation line dashed and bond length estimation as 2.5. Then the same Hessian method was applied to optimize the structures. The electronic energy of the endo structure was calculated to be '''-0.05150Ha''' and '''-0.05042Ha''' for the exo.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | endo
! scope="col" | exo
|-
| '''Animation'''||[[File:XJB_endots.GIF|300px]]||[[File:XJB_exots.GIF|300px]]
|-
| '''Sum of electronic and zero-point Energies'''||0.133493||0.134880
|-
| '''Sum of electronic and thermal Energies'''||0.143683||0.144881
|-
| '''Sum of electronic and thermal Enthalpies'''||0.144627||0.145825
|-
| '''Sum of electronic and thermal Free Energies'''||0.097349||0.099117
|-
| '''Bond lengths'''||[[Image:XJB_endomeasure.jpg|300px]]||[[Image:XJB_exomeasure.jpg|300px]]
|-
| '''HOMO'''||[[Image:XJB_endoHOMO.jpg|270px]]||[[Image:XJB_exoHOMO.jpg|255px]]
|-
| '''Secondary orbital overlap effect'''||YES||NO
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 12.''' ''Summary of comparison of properties of endo and exo transition structures for reaction between cyclohexa-1,3-diene and maleic anhydride''</div>

Rep:Mod:XJB1130

2013-12-05T18:16:04Z

Jx1011: /* The cyclohexa-1,3-diene reaction with maleic anhydride */

==The Cope Rearrangement of 1,5-hexadiene==

The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement reaction ('''Figure 1'''). Its mechanism has been studied by a large number of experimental and computational researches, which is recently accepted that the reaction occurs via either a "chair" or a "boat" transition state structure in a concerted manner. The aim of this exercise is to locate the low-energy minimum and the transition structures on the potential energy surface by Gaussian calculation in order to study its mechanism. The reaction pathway and the barrier heights can also be calculated by this calculation.

<center>[[Image:XJB_cope rearrangement.jpg|500px|center|thumb|'''Figure 1'''. ''The Cope rearrangement'']]</center>

===Optimizing the Reactants and Products===

Different conformations of 1,5-hexadiene can be optimized using '''Gaussview'''. The optimized structure of four conformers involved in this discussion and their corresponding energies optimized at '''HF/3-21G''' level of theory are concluded in the following table ('''Table 1''').

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | anti4
! scope="col" | anti2
! scope="col" | gauche
! scope="col" | gauche3
|-
| <jmol><jmolApplet><title>XJB_anti4.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti4.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche3.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| -231.69097Ha || -231.69254Ha || -231.68772Ha ||-231.69266Ha
|}

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 1.''' ''Four conformers of 1,5-hexadiene''</div>

This exercise first involved optimizing a molecule of 1,5-hexadiene with "anti" linkage for the cetral four C atoms at the '''HF/3-21G''' level of theory. The final optimization structure had the total electronic energy of '''-231.69097Ha''' with a '''C<sub>1</sub>''' symmetry. By comparing the energy and point group to that in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1], this structure is confirmed to be ''anti4''. The compositions of energies are also listed in '''Table 2''' as i) the sum of electronic and zero-point energies (potential energy at 0K including the zero-point vibrational energy), ii) the sum of electronic and thermal energies (energy including translational, rotational and vibrational at 298.15K and 1 atm), iii) the sum of electronic and thermal enthalpies (containning an additional correction for RT), and iv) the sum of electronic and thermal free energies (including the entropic contribution to the free energy). For what we are considering, the first two terms are the most useful for further comparisons.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Formula Description'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||E = E<sub>elec</sub> + ZPE||-231.53785
|-
| Sum of electronic and thermal Energies||E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>||-231.53096
|-
| Sum of electronic and thermal Enthalpie||H = E + RT||-231.53002
|-
| Sum of electronic and thermal Free Energie||G = H - TS||-231.56891
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 2.''' ''Frequency calculations of anti4 optimized at '''HF/3-21G''' ([[File:XJB_anti4.log]])''</div>

Next another 1,5-hexadiene molecule with "gauche" linkage for the central four atoms was drawn and optimized at the '''HF/3-21G''' level of theory as before. Energy for this conformation is expected to be higher than that for the ''anti4'' conformation due to the closer position of two alkene ends thus the steric interaction. The final structure had the electronic energy of '''-231.68772Ha''' which was indeed higher than that of the ''anti4'', indicating less favour towards this gauche structure. It had a '''C<sub>2</sub>''' symmetry and was confirmed to be the ''gauche'' conformation.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53439
|-
| Sum of electronic and thermal Energies||-231.52770
|-
| Sum of electronic and thermal Enthalpie||-231.52676
|-
| Sum of electronic and thermal Free Energie||-231.56483
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 3.''' ''Frequency calculations of gauche optimized at '''HF/3-21G''' ([[File:XJB_gauche.log]])''</div>

A lowest energy conformation ''gauche3'' of the reactant molecule was drawn based on and confirmed by [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. The optimization gave a final structure with an energy of '''-231.69266Ha''' with a symmetry of '''C<sub>1</sub>'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53949
|-
| Sum of electronic and thermal Energies||-231.53265
|-
| Sum of electronic and thermal Enthalpie||-231.5317
|-
| Sum of electronic and thermal Free Energie||-231.57064
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 4.''' ''Frequency calculations of gauche3 optimized at '''HF/3-21G''' ([[File:XJB_gauche3.log]])''</div>

The next step required us to draw the C<sub>i</sub> symmetry ''anti2'' conformation of 1,5-hexadiene and optimize it at the '''HF/3-21G''' level of theory. The optimized energy was calculated to be '''-231.69254Ha''' with point group of '''C<sub>i</sub>''', which was consistent with the one given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53954
|-
| Sum of electronic and thermal Energies||-231.53257
|-
| Sum of electronic and thermal Enthalpie||-231.53162
|-
| Sum of electronic and thermal Free Energie||-231.57092
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 5.''' ''Frequency calculations of anti2 optimized at '''HF/3-21G''' ([[File:XJB_anti2.log]])''</div>

The ''anti2'' structure was then reoptimized at '''B3LYP/6-31G*''' which was a improvement over the '''HF/3-21G''' basis set. It adds polarization to all atoms and improves the modeling of core electrons thus producing more accurate description of orbitals[ref]. The reoptimized ''anti2'' structure had the total electronic energy of '''-234.61171Ha''' and the same symmetry C<sub>i</sub>. The comparison of two structures optimized using different levels of theory is summarized in '''Table 6'''. We can see that they have exactly the same dihedral angle of the central four C atoms at 180.00<sup>o</sup> and angles between three of them are very similar as 111.3<sup>o</sup> and 112.7<sup>o</sup> respectively. Thus the the geometry change using '''HF/3-21G''' or '''B3LYP/6-31G*''' is trivial.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Before re-optimization
! scope="col" | After re-optimization
|-
| <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2reoptimized</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2reoptimized.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| ''anti2 optimized at '''HF/3-21G''''' || ''anti2 optimized at '''B3LYP/6-31G*'''''
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 6.''' ''Two optimization of anti2 at '''HF/3-21G''' and '''B3LYP/6-31G*'''([[File:XJB_anti2.log]])''</div>

The frequency calculation was run as well and a list of energies is summarized in '''Table 7'''. A list of all the vibrational frequencies was also generated and no imaginary (negative) frequency was observed and these vibrations are visualized by simulating the infrared spectrum in '''Figure 2'''.

{| align="center"
|- align="center"
|
{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-234.46920
|-
| Sum of electronic and thermal Energies||-234.46186
|-
| Sum of electronic and thermal Enthalpie||-234.46091
|-
| Sum of electronic and thermal Free Energie||-234.50078
|}
|
[[Image:XJB_IR_anti2.jpg|300px]]
|- align="center"
| align="center" | '''Table 7.''' ''Frequency calculations of anti2 optimized at '''B3LYP/6-31G*'''
([[File:XJB_anti2_reoptimized.log]])''
| align="center" | '''Figure 2.''' ''The generated IR spectrum of optimized '''B3LYP/6-31G*''' structure''
|}

===Optimizing the "Chair" and "Boat" Transition Structures===

The study of the chair and boat transition states is carried out using three different approaches. The first one is called the Hessian which involves computing the force constant matrix; the second one is to freeze the reaction coordinate and then optimize the structure once the molecule is fully relaxed; the last one is to use QST2 method to specify the reactants and products then finding the transition structure.

<center>[[Image:XJB_2 ts.jpg|500px|center|thumb|'''Figure 3'''. ''The two transition states of Cope rearrangement'']]</center>

====The Chair Transition State====

In order to set up a transition state optimization, first an allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimized at the '''HF/3-21G''' level of theory. Then this half of the transition structure was duplicated with rotation to make the approximate resembling of the chair transition state with terminal ends of the fragments 2.2 Å apart. The chair transition structure optimization was then done by the first two different approaches.

<center>[[Image:XJB_guess ts.jpg|400px|center|thumb|'''Figure 4'''. ''The guessed chair transition structure of two allyl fragments'']]</center>

For the first optimization approach, the force constants were computed (the Hessian) and the chair transition state was optimized to a TS (Berny) at the '''HF/3-21G''' level of theory. The frequency calculation was carried out at the same time and gave an imaginary frequency at -818 cm<sup>-1</sup>. The electronic energy was calculated to be '''-231.61932Ha''' with C<sub>2h</sub> symmetry point group. The distance between the terminal carbons was 2.0203 Å. An animation of this transition state vibrations was generated to confirm the corresponding Cope rearrangement.

<center>[[File:XJB_chair ts animation.GIF|400px|center|thumb|'''Figure 5.''' ''Animation of the Cope rearrangement via the chair transition state'']]</center>

The chair transition state was then reoptimized using the '''B3LYP/6-31G*''' level of theory and carried out frequency calculations as well. The electronic energy calculated this time was '''-234.55698Ha''' with an imaginary frequency at -566 cm<sup>-1</sup>. We can see that with the geometries quite similar, the energies differ markedly calculated using two levels of theory. The summary and comparison of the calculations optimized at two levels of theory are summarized in '''Table 8'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46670||-234.41493
|-
| Sum of electronic and thermal Energies||-231.46134||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40807
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir1.jpg|200px]]||[[Image:XJB_chairir2.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 8.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the first approach
([[File:XJB_chairts1.1.log]], [[File:XJB_chairts1.2.log]])''</div>

The second approach is the frozen coordinate method using the Redundant Coord Editor to freeze the bond lengths of the terminal ends to 2.2 Å. First the calculation failed because of the recent update of GaussView. After editting the coordinate distance (B 5 1 2.2 F) and the frozen distance (B 5 1 F) the optimization went well this time. The structure was then optimized again using a normal guess Hessian modified including the two differentiating coordinates. This chair transition state was then reoptimized again using the '''B3LYP/6-31G*''' level of theory. The electronic energies were calculated to be the same as in the first approach and frequency calculations were summarized in '''Table 9'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46671||-234.41490
|-
| Sum of electronic and thermal Energies||-231.46135||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40806
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir3.jpg|200px]]||[[Image:XJB_chairir4.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 9.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the second approach
([[File:XJB_chairts2.1.log]], [[File:XJB_chairts2.2.log]])''</div>

As the same energy results given, the bond forming/breaking bond lengths in the chair transition structures in the above two different approaches were compared. The first approach gave the bond lengths of 2.0203 Å while the second approach gave 2.0207 Å. Considering the accuracy limit of bond length estimation in Gaussview, these two bond lengths can be seen as exactly the same. Two methods worked reasonably well because of the good assumption of transition structures. However for more complicated and larger molecule, the Hessian method may fail due to the difficulty to predict the transition geometry (different surface curvature). Thus better way to generate such transition structures is to freeze reaction coordinate.

{| align="center"
|- align="center"
|
[[Image:XJB_chairts bondlength1.jpg|350px|thumb|'''Figure 6.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by Hession'']]
|
[[Image:XJB_chairts bondlength2.jpg|350px|thumb|'''Figure 7.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by frozen coordinate'']]
|- align="center"
|}

====The Boat Transition State====

The boat transition structure was then optimized using the '''QST2''' method. The same numbering for the reactant and product molecules was important in order to specify for this reaction. The renumbering of the molecules is shown as in '''Figure 8''', but the following optimization failed and gave a twisted structure. The reactant and product geometries were then modified further by changing the central C-C-C-C dihedral angle to 0<sup>o</sup> and the inside two C-C-C (i.e. C2-C3-C4 and C3-C4-C5) angles to 100<sup>o</sup>. The job was run successfully and frequency calculations were done with the boat transition state generated. Only one imaginary frequency was detected also at -840 cm<sup>-1</sup>. The electronic energy was '''-231.60280Ha''' with C<sub>2v</sub> symmetry. The distances between the terminal carbons of the allyl fragments were examined to be both 2.140Å which were comparably longer than that of the chair transition structure.

{| align="center"
|- align="center"
|
[[Image:XJB_QST1.jpg|400px|thumb|'''Figure 8.''' ''Numbering of reactant and product'']]
|
[[Image:XJB_QST2.jpg|410px|thumb|'''Figure 9.''' ''Re-numbering of reactant and product'']]
|- align="center"
|}

<center>[[File:XJB_boat ts animation.GIF|400px|center|thumb|'''Figure 10.''' ''Animation of the Cope rearrangement via the boat transition state'']]</center>

The structure was then reoptimized at the higher '''B3LYP/6-31G*''' theory of level and the electronic energy was calculated to be '''-234.54309Ha''' with very similar geometry. The frequency calculation generated an imaginary frequency at -530 cm<sup>-1</sup>. The summary and comparison of two levels' calculations are carried out in the following table.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.45093||-234.40234
|-
| Sum of electronic and thermal Energies||-231.44530||-234.39601
|-
| Sum of electronic and thermal Enthalpie|| -231.44436||-234.39506
|-
| Sum of electronic and thermal Free Energie||-231.47977||-234.43175
|-
| Generated IR spectrum||[[Image:XJB_boatir1.jpg|200px]]||[[Image:XJB_boatir2.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 10.''' ''Frequency calculations of the boat transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by QST2
([[File:XJB_boat ts1.log]], [[File:XJB_boat ts2.log]])''</div>

====Intrinsic Reaction Coordinate====

Next a useful method called '''Intrinct Reaction Coordinate (IRC)''' is applied to the Hessian optimized chair transition structure to follow the mininmum energy path (MEP) from a transition structure down to its local minimum on a potential energy surface. IRC can be used to verify that the transition state you found actually connects the reactants and products you think it does, or to identify any reaction intermediates you haven't thought about, or to generate an animation of the reaction [ref]. The symmetrical reaction coordinate was computed only in the forward direction with force constant always calculated and the number of points along IRC 50. The last structure generated along IRC is given in '''Figure 11''' with the energy calculated as '''-231.69158Ha''' and the dihedral angle and internal angle measured as D = -67.188<sup>o</sup> A = 111.78<sup>o</sup>. This energy does not match to any of the conformations discussed in Appendix 1 thus a minimum geometry has not reached yet.

{| align="center"
|- align="center"
|
<jmol><jmolApplet><title>XJB_IRC.mol</title><color>white</color>
<size>200</size><script>zoom=5</script>
<uploadedFileContents>XJB_IRC.mol</uploadedFileContents>
</jmolApplet></jmol>
|
[[Image:XJB_irc.jpg|500px]]
|- align="center"
| align="center" | '''Figure 11.''' ''The energy minimum structure along IRC''
| align="center" | '''Figure 12.''' ''Total energy along IRC''
|}

Three improving options can be used to reach the minimum energy. The first one is to run a normal Hessian optimization of the last point on IRC; the second one involves using a larger number of points (in this case 100 was used); the last one is to redo the IRC computation specifying to compute force constant at every step. Three methods were all used and the results were summarized in '''Table 11'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Improved IRC method 1
! scope="col" | Improved IRC method 2
! scope="col" | Improved IRC method 3
|-
| [[Image:XJB_IRC1.jpg|300px|center|IRC1]]||[[Image:XJB_IRC2.jpg|300px|center|IRC3]] || [[Image:XJB_IRC3.jpg|300px|center|IRC3]]
|-
| D = -64.17<sup>o</sup> A = 112.04<sup>o</sup> || D = -66.83<sup>o</sup> A = 111.82<sup>o</sup> || D = -64.17<sup>o</sup> A = 112.04<sup>o</sup>
|-
| -231.69167Ha || -231.69150Ha || -231.69167Ha
|-
| [[File:XJB_IRC1.log]]||[[File:XJB_IRC2.log]]||[[File:XJB_IRC3.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 11.''' ''Comparison of three improvement methods of IRC computation upon the chair transition structure''</div>

It is concluded that method 1 and 3 gave the same the lowest energy which was consistent to the energy given in Appendix 1.

====Activation Energies====

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | HF/3-21G at 0K
! scope="col" | HF/3-21G at 298.15K
! scope="col" | B3LYP/6-31G* at 0K
! scope="col" | B3LYP/6-31G* at 298.15K
! scope="col" | Expt. at 0K
|-
| '''ΔE (chair)'''||45.71||44.70||34.05||33.16||33.5 ± 0.5
|-
| '''ΔE (boat)'''||55.60||54.76||41.96||41.32||44.7 ± 0.5
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 12.''' ''Summary of activation energies for the Cope rearrangement via both transition states in kcal/mol''</div>

==The Diels Alder Cycloaddition==

In this exercise, AM1 semi-empirical method is used to compute the Diels Alder cycloaddition which is a concerted pericyclic reaction. The driving force of this reaction is the formation of a new σ bond between the π orbitals of the dieneophile and the diene. The reaction is allowed with HOMO-LUMO interation and a symmetry overlap.

===Cis Butadiene===

<center>[[Image:prototype Diels Alder.jpg|500px|center|thumb|'''Figure 13'''. ''The prototype reaction between butadiene and ethylene'']]</center>

To study a prototype Diels Alder cycloaddition reaction between butadiene and ethylene, first a cis butadiene molecule was built using Gaussview and optimized using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the butadiene were plotted. As seen the HOMO of cis-butadiene is anti-symmetric ('''a''') and the LUMO is symmetric ('''s''') with respect to plane.

{| align="center"
|- align="center"
|
[[Image:XJB_HOMO.jpg|280px|thumb|'''Figure 14.''' ''Anti-symmetric HOMO of cis-butadiene'']]
|
[[Image:XJB_LUMO.jpg|300px|thumb|'''Figure 15.''' ''Symmetric HOMO of cis-butadiene'']]
|- align="center"
|}

===The Transition State of prototype reaction between ethylene and butadiene===

The Hessian method was used to estimate transition state for the reaction between ethylene and butadiene because these two reactants are relatively small and simple molecules. The transition state was drawn by first building a structure as (a) in '''Figure 16''' then removing the -CH<sub>2</sub>-CH<sub>2</sub>- group to form the envelop structure as in (b). The '''clean''' option should not be used and the dashed bond length was guessed to be larger than 2.2 Å otherwise the geometry change would cause a failure in optimization.

<center>[[Image:XJB_dielsguessts.jpg|300px|center|thumb|'''Figure 16'''. ''The guessing transition state for the reaction between butadiene and ethylene'']]</center>

<center>[[File:XJB_butadiene and ethylene ts.GIF|400px|center|thumb|'''Figure 17.''' ''Animation of the butadiene and ethylene cycloaddition transition state'']]</center>

After the Hessian optimization, the transition structure has been successfully determined and an animation was generated in '''Figure 17'''. It has a C<sub>1</sub> symmetry point group and the bond lengths of two partly formed C-C bonds were measured to be 2.12Å. The typical sp<sup>3</sup> and sp<sup>2</sup> C-C bond lengths are 1.54Å and 1.47Å respectively[ref]. The van der Waals radius of the C atom is 1.7 Å[ref]. The measured bond distance of the transition structure is larger than typical bond lengths but still within the van der Waals radius, suggesting the bond forming interation between the carbon atoms.

From the animation ('''Figure 17''') the vibration directions of two molecules are opposite and they are approaching and leaving each other at the same time. This suggests the formation of the two bonds synchronous. The vibrational frequencies were also calculated and an infrared spectrum was simulated ('''Figure 20'''). An imaginary frequency of magnitude -955 cm<sup>-1</sup> was seen and the lowest positive frequency was observed at 147 cm<sup>-1</sup> which corresponded to the reactant vibration but not the bond formation.

{| align="center"
|
[[Image:XJB_HOMOts.jpg|300px|thumb|'''Figure 18'''. HOMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_LUMOts.jpg|300px|thumb|'''Figure 19'''. LUMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_IRts.jpg|300px|thumb|'''Figure 20'''. Generated IR of transition state of reaction between cis-butadiene and ethylene]]
|}

The HOMO and LUMO molecular orbitals of this transition structure were visualized in '''Figure 18''' and '''Figure 19'''. From these two figures it can be observed that the HOMO at the transition structure is anti-symmetric ('''a''') while the LUMO is symmetric ('''s'''). Thus this anti-symmetric transition state HOMO is formed by the HOMO of cis-butadiene and the LUMO of ethylene with the knowledge that they are both anti-symmetrical. This conclusion agrees with the previous talked conditions for allowed reaction.

===The cyclohexa-1,3-diene reaction with maleic anhydride===

<center>[[Image:XJB_mechanism3.jpg|500px|center|thumb|'''Figure 21'''. ''The endo and exo products of the Diels Alder reaction between cyclohexa-1,3-diene and maleic anhydride'']]</center>

The exo and endo transition states were drawn according to '''Figure 21''' with the new C-C bond formation line dashed and bond length estimation as 2.5. Then the same Hessian method was applied to optimize the structures. The electronic energy of the endo structure was calculated to be '''-0.05150Ha''' and '''-0.05042Ha''' for the exo.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | endo
! scope="col" | exo
|-
| '''Animation'''||[[File:XJB_endots.GIF|300px]]||[[File:XJB_exots.GIF|300px]]
|-
| '''Sum of electronic and zero-point Energies'''||0.133493||0.134880
|-
| '''Sum of electronic and thermal Energies'''||0.143683||0.144881
|-
| '''Sum of electronic and thermal Enthalpies'''||0.144627||0.145825
|-
| '''Sum of electronic and thermal Free Energies'''||0.097349||0.099117
|-
| '''Bond lengths'''||[[Image:XJB_endomeasure.jpg|270px]]||[[Image:XJB_exomeasure.jpg|300px]]
|-
| '''HOMO'''||[[Image:XJB_endoHOMO.jpg|300px]]||[[Image:XJB_exoHOMO.jpg|250px]]
|-
| '''Secondary orbital overlap effect'''||YES||NO
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 12.''' ''Summary of comparison of properties of endo and exo transition structures for reaction between cyclohexa-1,3-diene and maleic anhydride''</div>

Rep:Mod:XJB1130

2013-12-05T18:14:56Z

Jx1011: /* The cyclohexa-1,3-diene reaction with maleic anhydride */

==The Cope Rearrangement of 1,5-hexadiene==

The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement reaction ('''Figure 1'''). Its mechanism has been studied by a large number of experimental and computational researches, which is recently accepted that the reaction occurs via either a "chair" or a "boat" transition state structure in a concerted manner. The aim of this exercise is to locate the low-energy minimum and the transition structures on the potential energy surface by Gaussian calculation in order to study its mechanism. The reaction pathway and the barrier heights can also be calculated by this calculation.

<center>[[Image:XJB_cope rearrangement.jpg|500px|center|thumb|'''Figure 1'''. ''The Cope rearrangement'']]</center>

===Optimizing the Reactants and Products===

Different conformations of 1,5-hexadiene can be optimized using '''Gaussview'''. The optimized structure of four conformers involved in this discussion and their corresponding energies optimized at '''HF/3-21G''' level of theory are concluded in the following table ('''Table 1''').

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | anti4
! scope="col" | anti2
! scope="col" | gauche
! scope="col" | gauche3
|-
| <jmol><jmolApplet><title>XJB_anti4.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti4.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche3.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| -231.69097Ha || -231.69254Ha || -231.68772Ha ||-231.69266Ha
|}

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 1.''' ''Four conformers of 1,5-hexadiene''</div>

This exercise first involved optimizing a molecule of 1,5-hexadiene with "anti" linkage for the cetral four C atoms at the '''HF/3-21G''' level of theory. The final optimization structure had the total electronic energy of '''-231.69097Ha''' with a '''C<sub>1</sub>''' symmetry. By comparing the energy and point group to that in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1], this structure is confirmed to be ''anti4''. The compositions of energies are also listed in '''Table 2''' as i) the sum of electronic and zero-point energies (potential energy at 0K including the zero-point vibrational energy), ii) the sum of electronic and thermal energies (energy including translational, rotational and vibrational at 298.15K and 1 atm), iii) the sum of electronic and thermal enthalpies (containning an additional correction for RT), and iv) the sum of electronic and thermal free energies (including the entropic contribution to the free energy). For what we are considering, the first two terms are the most useful for further comparisons.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Formula Description'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||E = E<sub>elec</sub> + ZPE||-231.53785
|-
| Sum of electronic and thermal Energies||E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>||-231.53096
|-
| Sum of electronic and thermal Enthalpie||H = E + RT||-231.53002
|-
| Sum of electronic and thermal Free Energie||G = H - TS||-231.56891
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 2.''' ''Frequency calculations of anti4 optimized at '''HF/3-21G''' ([[File:XJB_anti4.log]])''</div>

Next another 1,5-hexadiene molecule with "gauche" linkage for the central four atoms was drawn and optimized at the '''HF/3-21G''' level of theory as before. Energy for this conformation is expected to be higher than that for the ''anti4'' conformation due to the closer position of two alkene ends thus the steric interaction. The final structure had the electronic energy of '''-231.68772Ha''' which was indeed higher than that of the ''anti4'', indicating less favour towards this gauche structure. It had a '''C<sub>2</sub>''' symmetry and was confirmed to be the ''gauche'' conformation.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53439
|-
| Sum of electronic and thermal Energies||-231.52770
|-
| Sum of electronic and thermal Enthalpie||-231.52676
|-
| Sum of electronic and thermal Free Energie||-231.56483
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 3.''' ''Frequency calculations of gauche optimized at '''HF/3-21G''' ([[File:XJB_gauche.log]])''</div>

A lowest energy conformation ''gauche3'' of the reactant molecule was drawn based on and confirmed by [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. The optimization gave a final structure with an energy of '''-231.69266Ha''' with a symmetry of '''C<sub>1</sub>'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53949
|-
| Sum of electronic and thermal Energies||-231.53265
|-
| Sum of electronic and thermal Enthalpie||-231.5317
|-
| Sum of electronic and thermal Free Energie||-231.57064
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 4.''' ''Frequency calculations of gauche3 optimized at '''HF/3-21G''' ([[File:XJB_gauche3.log]])''</div>

The next step required us to draw the C<sub>i</sub> symmetry ''anti2'' conformation of 1,5-hexadiene and optimize it at the '''HF/3-21G''' level of theory. The optimized energy was calculated to be '''-231.69254Ha''' with point group of '''C<sub>i</sub>''', which was consistent with the one given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53954
|-
| Sum of electronic and thermal Energies||-231.53257
|-
| Sum of electronic and thermal Enthalpie||-231.53162
|-
| Sum of electronic and thermal Free Energie||-231.57092
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 5.''' ''Frequency calculations of anti2 optimized at '''HF/3-21G''' ([[File:XJB_anti2.log]])''</div>

The ''anti2'' structure was then reoptimized at '''B3LYP/6-31G*''' which was a improvement over the '''HF/3-21G''' basis set. It adds polarization to all atoms and improves the modeling of core electrons thus producing more accurate description of orbitals[ref]. The reoptimized ''anti2'' structure had the total electronic energy of '''-234.61171Ha''' and the same symmetry C<sub>i</sub>. The comparison of two structures optimized using different levels of theory is summarized in '''Table 6'''. We can see that they have exactly the same dihedral angle of the central four C atoms at 180.00<sup>o</sup> and angles between three of them are very similar as 111.3<sup>o</sup> and 112.7<sup>o</sup> respectively. Thus the the geometry change using '''HF/3-21G''' or '''B3LYP/6-31G*''' is trivial.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Before re-optimization
! scope="col" | After re-optimization
|-
| <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2reoptimized</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2reoptimized.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| ''anti2 optimized at '''HF/3-21G''''' || ''anti2 optimized at '''B3LYP/6-31G*'''''
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 6.''' ''Two optimization of anti2 at '''HF/3-21G''' and '''B3LYP/6-31G*'''([[File:XJB_anti2.log]])''</div>

The frequency calculation was run as well and a list of energies is summarized in '''Table 7'''. A list of all the vibrational frequencies was also generated and no imaginary (negative) frequency was observed and these vibrations are visualized by simulating the infrared spectrum in '''Figure 2'''.

{| align="center"
|- align="center"
|
{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-234.46920
|-
| Sum of electronic and thermal Energies||-234.46186
|-
| Sum of electronic and thermal Enthalpie||-234.46091
|-
| Sum of electronic and thermal Free Energie||-234.50078
|}
|
[[Image:XJB_IR_anti2.jpg|300px]]
|- align="center"
| align="center" | '''Table 7.''' ''Frequency calculations of anti2 optimized at '''B3LYP/6-31G*'''
([[File:XJB_anti2_reoptimized.log]])''
| align="center" | '''Figure 2.''' ''The generated IR spectrum of optimized '''B3LYP/6-31G*''' structure''
|}

===Optimizing the "Chair" and "Boat" Transition Structures===

The study of the chair and boat transition states is carried out using three different approaches. The first one is called the Hessian which involves computing the force constant matrix; the second one is to freeze the reaction coordinate and then optimize the structure once the molecule is fully relaxed; the last one is to use QST2 method to specify the reactants and products then finding the transition structure.

<center>[[Image:XJB_2 ts.jpg|500px|center|thumb|'''Figure 3'''. ''The two transition states of Cope rearrangement'']]</center>

====The Chair Transition State====

In order to set up a transition state optimization, first an allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimized at the '''HF/3-21G''' level of theory. Then this half of the transition structure was duplicated with rotation to make the approximate resembling of the chair transition state with terminal ends of the fragments 2.2 Å apart. The chair transition structure optimization was then done by the first two different approaches.

<center>[[Image:XJB_guess ts.jpg|400px|center|thumb|'''Figure 4'''. ''The guessed chair transition structure of two allyl fragments'']]</center>

For the first optimization approach, the force constants were computed (the Hessian) and the chair transition state was optimized to a TS (Berny) at the '''HF/3-21G''' level of theory. The frequency calculation was carried out at the same time and gave an imaginary frequency at -818 cm<sup>-1</sup>. The electronic energy was calculated to be '''-231.61932Ha''' with C<sub>2h</sub> symmetry point group. The distance between the terminal carbons was 2.0203 Å. An animation of this transition state vibrations was generated to confirm the corresponding Cope rearrangement.

<center>[[File:XJB_chair ts animation.GIF|400px|center|thumb|'''Figure 5.''' ''Animation of the Cope rearrangement via the chair transition state'']]</center>

The chair transition state was then reoptimized using the '''B3LYP/6-31G*''' level of theory and carried out frequency calculations as well. The electronic energy calculated this time was '''-234.55698Ha''' with an imaginary frequency at -566 cm<sup>-1</sup>. We can see that with the geometries quite similar, the energies differ markedly calculated using two levels of theory. The summary and comparison of the calculations optimized at two levels of theory are summarized in '''Table 8'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46670||-234.41493
|-
| Sum of electronic and thermal Energies||-231.46134||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40807
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir1.jpg|200px]]||[[Image:XJB_chairir2.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 8.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the first approach
([[File:XJB_chairts1.1.log]], [[File:XJB_chairts1.2.log]])''</div>

The second approach is the frozen coordinate method using the Redundant Coord Editor to freeze the bond lengths of the terminal ends to 2.2 Å. First the calculation failed because of the recent update of GaussView. After editting the coordinate distance (B 5 1 2.2 F) and the frozen distance (B 5 1 F) the optimization went well this time. The structure was then optimized again using a normal guess Hessian modified including the two differentiating coordinates. This chair transition state was then reoptimized again using the '''B3LYP/6-31G*''' level of theory. The electronic energies were calculated to be the same as in the first approach and frequency calculations were summarized in '''Table 9'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46671||-234.41490
|-
| Sum of electronic and thermal Energies||-231.46135||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40806
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir3.jpg|200px]]||[[Image:XJB_chairir4.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 9.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the second approach
([[File:XJB_chairts2.1.log]], [[File:XJB_chairts2.2.log]])''</div>

As the same energy results given, the bond forming/breaking bond lengths in the chair transition structures in the above two different approaches were compared. The first approach gave the bond lengths of 2.0203 Å while the second approach gave 2.0207 Å. Considering the accuracy limit of bond length estimation in Gaussview, these two bond lengths can be seen as exactly the same. Two methods worked reasonably well because of the good assumption of transition structures. However for more complicated and larger molecule, the Hessian method may fail due to the difficulty to predict the transition geometry (different surface curvature). Thus better way to generate such transition structures is to freeze reaction coordinate.

{| align="center"
|- align="center"
|
[[Image:XJB_chairts bondlength1.jpg|350px|thumb|'''Figure 6.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by Hession'']]
|
[[Image:XJB_chairts bondlength2.jpg|350px|thumb|'''Figure 7.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by frozen coordinate'']]
|- align="center"
|}

====The Boat Transition State====

The boat transition structure was then optimized using the '''QST2''' method. The same numbering for the reactant and product molecules was important in order to specify for this reaction. The renumbering of the molecules is shown as in '''Figure 8''', but the following optimization failed and gave a twisted structure. The reactant and product geometries were then modified further by changing the central C-C-C-C dihedral angle to 0<sup>o</sup> and the inside two C-C-C (i.e. C2-C3-C4 and C3-C4-C5) angles to 100<sup>o</sup>. The job was run successfully and frequency calculations were done with the boat transition state generated. Only one imaginary frequency was detected also at -840 cm<sup>-1</sup>. The electronic energy was '''-231.60280Ha''' with C<sub>2v</sub> symmetry. The distances between the terminal carbons of the allyl fragments were examined to be both 2.140Å which were comparably longer than that of the chair transition structure.

{| align="center"
|- align="center"
|
[[Image:XJB_QST1.jpg|400px|thumb|'''Figure 8.''' ''Numbering of reactant and product'']]
|
[[Image:XJB_QST2.jpg|410px|thumb|'''Figure 9.''' ''Re-numbering of reactant and product'']]
|- align="center"
|}

<center>[[File:XJB_boat ts animation.GIF|400px|center|thumb|'''Figure 10.''' ''Animation of the Cope rearrangement via the boat transition state'']]</center>

The structure was then reoptimized at the higher '''B3LYP/6-31G*''' theory of level and the electronic energy was calculated to be '''-234.54309Ha''' with very similar geometry. The frequency calculation generated an imaginary frequency at -530 cm<sup>-1</sup>. The summary and comparison of two levels' calculations are carried out in the following table.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.45093||-234.40234
|-
| Sum of electronic and thermal Energies||-231.44530||-234.39601
|-
| Sum of electronic and thermal Enthalpie|| -231.44436||-234.39506
|-
| Sum of electronic and thermal Free Energie||-231.47977||-234.43175
|-
| Generated IR spectrum||[[Image:XJB_boatir1.jpg|200px]]||[[Image:XJB_boatir2.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 10.''' ''Frequency calculations of the boat transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by QST2
([[File:XJB_boat ts1.log]], [[File:XJB_boat ts2.log]])''</div>

====Intrinsic Reaction Coordinate====

Next a useful method called '''Intrinct Reaction Coordinate (IRC)''' is applied to the Hessian optimized chair transition structure to follow the mininmum energy path (MEP) from a transition structure down to its local minimum on a potential energy surface. IRC can be used to verify that the transition state you found actually connects the reactants and products you think it does, or to identify any reaction intermediates you haven't thought about, or to generate an animation of the reaction [ref]. The symmetrical reaction coordinate was computed only in the forward direction with force constant always calculated and the number of points along IRC 50. The last structure generated along IRC is given in '''Figure 11''' with the energy calculated as '''-231.69158Ha''' and the dihedral angle and internal angle measured as D = -67.188<sup>o</sup> A = 111.78<sup>o</sup>. This energy does not match to any of the conformations discussed in Appendix 1 thus a minimum geometry has not reached yet.

{| align="center"
|- align="center"
|
<jmol><jmolApplet><title>XJB_IRC.mol</title><color>white</color>
<size>200</size><script>zoom=5</script>
<uploadedFileContents>XJB_IRC.mol</uploadedFileContents>
</jmolApplet></jmol>
|
[[Image:XJB_irc.jpg|500px]]
|- align="center"
| align="center" | '''Figure 11.''' ''The energy minimum structure along IRC''
| align="center" | '''Figure 12.''' ''Total energy along IRC''
|}

Three improving options can be used to reach the minimum energy. The first one is to run a normal Hessian optimization of the last point on IRC; the second one involves using a larger number of points (in this case 100 was used); the last one is to redo the IRC computation specifying to compute force constant at every step. Three methods were all used and the results were summarized in '''Table 11'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Improved IRC method 1
! scope="col" | Improved IRC method 2
! scope="col" | Improved IRC method 3
|-
| [[Image:XJB_IRC1.jpg|300px|center|IRC1]]||[[Image:XJB_IRC2.jpg|300px|center|IRC3]] || [[Image:XJB_IRC3.jpg|300px|center|IRC3]]
|-
| D = -64.17<sup>o</sup> A = 112.04<sup>o</sup> || D = -66.83<sup>o</sup> A = 111.82<sup>o</sup> || D = -64.17<sup>o</sup> A = 112.04<sup>o</sup>
|-
| -231.69167Ha || -231.69150Ha || -231.69167Ha
|-
| [[File:XJB_IRC1.log]]||[[File:XJB_IRC2.log]]||[[File:XJB_IRC3.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 11.''' ''Comparison of three improvement methods of IRC computation upon the chair transition structure''</div>

It is concluded that method 1 and 3 gave the same the lowest energy which was consistent to the energy given in Appendix 1.

====Activation Energies====

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | HF/3-21G at 0K
! scope="col" | HF/3-21G at 298.15K
! scope="col" | B3LYP/6-31G* at 0K
! scope="col" | B3LYP/6-31G* at 298.15K
! scope="col" | Expt. at 0K
|-
| '''ΔE (chair)'''||45.71||44.70||34.05||33.16||33.5 ± 0.5
|-
| '''ΔE (boat)'''||55.60||54.76||41.96||41.32||44.7 ± 0.5
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 12.''' ''Summary of activation energies for the Cope rearrangement via both transition states in kcal/mol''</div>

==The Diels Alder Cycloaddition==

In this exercise, AM1 semi-empirical method is used to compute the Diels Alder cycloaddition which is a concerted pericyclic reaction. The driving force of this reaction is the formation of a new σ bond between the π orbitals of the dieneophile and the diene. The reaction is allowed with HOMO-LUMO interation and a symmetry overlap.

===Cis Butadiene===

<center>[[Image:prototype Diels Alder.jpg|500px|center|thumb|'''Figure 13'''. ''The prototype reaction between butadiene and ethylene'']]</center>

To study a prototype Diels Alder cycloaddition reaction between butadiene and ethylene, first a cis butadiene molecule was built using Gaussview and optimized using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the butadiene were plotted. As seen the HOMO of cis-butadiene is anti-symmetric ('''a''') and the LUMO is symmetric ('''s''') with respect to plane.

{| align="center"
|- align="center"
|
[[Image:XJB_HOMO.jpg|280px|thumb|'''Figure 14.''' ''Anti-symmetric HOMO of cis-butadiene'']]
|
[[Image:XJB_LUMO.jpg|300px|thumb|'''Figure 15.''' ''Symmetric HOMO of cis-butadiene'']]
|- align="center"
|}

===The Transition State of prototype reaction between ethylene and butadiene===

The Hessian method was used to estimate transition state for the reaction between ethylene and butadiene because these two reactants are relatively small and simple molecules. The transition state was drawn by first building a structure as (a) in '''Figure 16''' then removing the -CH<sub>2</sub>-CH<sub>2</sub>- group to form the envelop structure as in (b). The '''clean''' option should not be used and the dashed bond length was guessed to be larger than 2.2 Å otherwise the geometry change would cause a failure in optimization.

<center>[[Image:XJB_dielsguessts.jpg|300px|center|thumb|'''Figure 16'''. ''The guessing transition state for the reaction between butadiene and ethylene'']]</center>

<center>[[File:XJB_butadiene and ethylene ts.GIF|400px|center|thumb|'''Figure 17.''' ''Animation of the butadiene and ethylene cycloaddition transition state'']]</center>

After the Hessian optimization, the transition structure has been successfully determined and an animation was generated in '''Figure 17'''. It has a C<sub>1</sub> symmetry point group and the bond lengths of two partly formed C-C bonds were measured to be 2.12Å. The typical sp<sup>3</sup> and sp<sup>2</sup> C-C bond lengths are 1.54Å and 1.47Å respectively[ref]. The van der Waals radius of the C atom is 1.7 Å[ref]. The measured bond distance of the transition structure is larger than typical bond lengths but still within the van der Waals radius, suggesting the bond forming interation between the carbon atoms.

From the animation ('''Figure 17''') the vibration directions of two molecules are opposite and they are approaching and leaving each other at the same time. This suggests the formation of the two bonds synchronous. The vibrational frequencies were also calculated and an infrared spectrum was simulated ('''Figure 20'''). An imaginary frequency of magnitude -955 cm<sup>-1</sup> was seen and the lowest positive frequency was observed at 147 cm<sup>-1</sup> which corresponded to the reactant vibration but not the bond formation.

{| align="center"
|
[[Image:XJB_HOMOts.jpg|300px|thumb|'''Figure 18'''. HOMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_LUMOts.jpg|300px|thumb|'''Figure 19'''. LUMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_IRts.jpg|300px|thumb|'''Figure 20'''. Generated IR of transition state of reaction between cis-butadiene and ethylene]]
|}

The HOMO and LUMO molecular orbitals of this transition structure were visualized in '''Figure 18''' and '''Figure 19'''. From these two figures it can be observed that the HOMO at the transition structure is anti-symmetric ('''a''') while the LUMO is symmetric ('''s'''). Thus this anti-symmetric transition state HOMO is formed by the HOMO of cis-butadiene and the LUMO of ethylene with the knowledge that they are both anti-symmetrical. This conclusion agrees with the previous talked conditions for allowed reaction.

===The cyclohexa-1,3-diene reaction with maleic anhydride===

<center>[[Image:XJB_mechanism3.jpg|500px|center|thumb|'''Figure 21'''. ''The endo and exo products of the Diels Alder reaction between cyclohexa-1,3-diene and maleic anhydride'']]</center>

The exo and endo transition states were drawn according to '''Figure 21''' with the new C-C bond formation line dashed and bond length estimation as 2.5. Then the same Hessian method was applied to optimize the structures. The electronic energy of the endo structure was calculated to be '''-0.05150Ha''' and '''-0.05042Ha''' for the exo.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | endo
! scope="col" | exo
|-
| '''Animation'''||[[File:XJB_endots.GIF|300px]]||[[File:XJB_exots.GIF|300px]]
|-
| '''Sum of electronic and zero-point Energies'''||0.133493||0.134880
|-
| '''Sum of electronic and thermal Energies'''||0.143683||0.144881
|-
| '''Sum of electronic and thermal Enthalpies'''||0.144627||0.145825
|-
| '''Sum of electronic and thermal Free Energies'''||0.097349||0.099117
|-
| '''Bond lengths'''||[[Image:XJB_endomeasure.jpg|300px]]||[[Image:XJB_exomeasure.jpg|300px]]
|-
| '''HOMO'''||[[Image:XJB_endoHOMO.jpg|300px]]||[[Image:XJB_exoHOMO.jpg|250px]]
|-
| '''Secondary orbital overlap effect'''||YES||NO
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 12.''' ''Summary of comparison of properties of endo and exo transition structures for reaction between cyclohexa-1,3-diene and maleic anhydride''</div>

File:XJB exoHOMO.jpg

2013-12-05T18:14:06Z

Jx1011:

File:XJB endoHOMO.jpg

2013-12-05T18:12:34Z

Jx1011:

Rep:Mod:XJB1130

2013-12-05T18:01:14Z

Jx1011: /* The cyclohexa-1,3-diene reaction with maleic anhydride */

==The Cope Rearrangement of 1,5-hexadiene==

The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement reaction ('''Figure 1'''). Its mechanism has been studied by a large number of experimental and computational researches, which is recently accepted that the reaction occurs via either a "chair" or a "boat" transition state structure in a concerted manner. The aim of this exercise is to locate the low-energy minimum and the transition structures on the potential energy surface by Gaussian calculation in order to study its mechanism. The reaction pathway and the barrier heights can also be calculated by this calculation.

<center>[[Image:XJB_cope rearrangement.jpg|500px|center|thumb|'''Figure 1'''. ''The Cope rearrangement'']]</center>

===Optimizing the Reactants and Products===

Different conformations of 1,5-hexadiene can be optimized using '''Gaussview'''. The optimized structure of four conformers involved in this discussion and their corresponding energies optimized at '''HF/3-21G''' level of theory are concluded in the following table ('''Table 1''').

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | anti4
! scope="col" | anti2
! scope="col" | gauche
! scope="col" | gauche3
|-
| <jmol><jmolApplet><title>XJB_anti4.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti4.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche3.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| -231.69097Ha || -231.69254Ha || -231.68772Ha ||-231.69266Ha
|}

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 1.''' ''Four conformers of 1,5-hexadiene''</div>

This exercise first involved optimizing a molecule of 1,5-hexadiene with "anti" linkage for the cetral four C atoms at the '''HF/3-21G''' level of theory. The final optimization structure had the total electronic energy of '''-231.69097Ha''' with a '''C<sub>1</sub>''' symmetry. By comparing the energy and point group to that in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1], this structure is confirmed to be ''anti4''. The compositions of energies are also listed in '''Table 2''' as i) the sum of electronic and zero-point energies (potential energy at 0K including the zero-point vibrational energy), ii) the sum of electronic and thermal energies (energy including translational, rotational and vibrational at 298.15K and 1 atm), iii) the sum of electronic and thermal enthalpies (containning an additional correction for RT), and iv) the sum of electronic and thermal free energies (including the entropic contribution to the free energy). For what we are considering, the first two terms are the most useful for further comparisons.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Formula Description'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||E = E<sub>elec</sub> + ZPE||-231.53785
|-
| Sum of electronic and thermal Energies||E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>||-231.53096
|-
| Sum of electronic and thermal Enthalpie||H = E + RT||-231.53002
|-
| Sum of electronic and thermal Free Energie||G = H - TS||-231.56891
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 2.''' ''Frequency calculations of anti4 optimized at '''HF/3-21G''' ([[File:XJB_anti4.log]])''</div>

Next another 1,5-hexadiene molecule with "gauche" linkage for the central four atoms was drawn and optimized at the '''HF/3-21G''' level of theory as before. Energy for this conformation is expected to be higher than that for the ''anti4'' conformation due to the closer position of two alkene ends thus the steric interaction. The final structure had the electronic energy of '''-231.68772Ha''' which was indeed higher than that of the ''anti4'', indicating less favour towards this gauche structure. It had a '''C<sub>2</sub>''' symmetry and was confirmed to be the ''gauche'' conformation.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53439
|-
| Sum of electronic and thermal Energies||-231.52770
|-
| Sum of electronic and thermal Enthalpie||-231.52676
|-
| Sum of electronic and thermal Free Energie||-231.56483
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 3.''' ''Frequency calculations of gauche optimized at '''HF/3-21G''' ([[File:XJB_gauche.log]])''</div>

A lowest energy conformation ''gauche3'' of the reactant molecule was drawn based on and confirmed by [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. The optimization gave a final structure with an energy of '''-231.69266Ha''' with a symmetry of '''C<sub>1</sub>'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53949
|-
| Sum of electronic and thermal Energies||-231.53265
|-
| Sum of electronic and thermal Enthalpie||-231.5317
|-
| Sum of electronic and thermal Free Energie||-231.57064
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 4.''' ''Frequency calculations of gauche3 optimized at '''HF/3-21G''' ([[File:XJB_gauche3.log]])''</div>

The next step required us to draw the C<sub>i</sub> symmetry ''anti2'' conformation of 1,5-hexadiene and optimize it at the '''HF/3-21G''' level of theory. The optimized energy was calculated to be '''-231.69254Ha''' with point group of '''C<sub>i</sub>''', which was consistent with the one given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53954
|-
| Sum of electronic and thermal Energies||-231.53257
|-
| Sum of electronic and thermal Enthalpie||-231.53162
|-
| Sum of electronic and thermal Free Energie||-231.57092
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 5.''' ''Frequency calculations of anti2 optimized at '''HF/3-21G''' ([[File:XJB_anti2.log]])''</div>

The ''anti2'' structure was then reoptimized at '''B3LYP/6-31G*''' which was a improvement over the '''HF/3-21G''' basis set. It adds polarization to all atoms and improves the modeling of core electrons thus producing more accurate description of orbitals[ref]. The reoptimized ''anti2'' structure had the total electronic energy of '''-234.61171Ha''' and the same symmetry C<sub>i</sub>. The comparison of two structures optimized using different levels of theory is summarized in '''Table 6'''. We can see that they have exactly the same dihedral angle of the central four C atoms at 180.00<sup>o</sup> and angles between three of them are very similar as 111.3<sup>o</sup> and 112.7<sup>o</sup> respectively. Thus the the geometry change using '''HF/3-21G''' or '''B3LYP/6-31G*''' is trivial.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Before re-optimization
! scope="col" | After re-optimization
|-
| <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2reoptimized</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2reoptimized.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| ''anti2 optimized at '''HF/3-21G''''' || ''anti2 optimized at '''B3LYP/6-31G*'''''
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 6.''' ''Two optimization of anti2 at '''HF/3-21G''' and '''B3LYP/6-31G*'''([[File:XJB_anti2.log]])''</div>

The frequency calculation was run as well and a list of energies is summarized in '''Table 7'''. A list of all the vibrational frequencies was also generated and no imaginary (negative) frequency was observed and these vibrations are visualized by simulating the infrared spectrum in '''Figure 2'''.

{| align="center"
|- align="center"
|
{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-234.46920
|-
| Sum of electronic and thermal Energies||-234.46186
|-
| Sum of electronic and thermal Enthalpie||-234.46091
|-
| Sum of electronic and thermal Free Energie||-234.50078
|}
|
[[Image:XJB_IR_anti2.jpg|300px]]
|- align="center"
| align="center" | '''Table 7.''' ''Frequency calculations of anti2 optimized at '''B3LYP/6-31G*'''
([[File:XJB_anti2_reoptimized.log]])''
| align="center" | '''Figure 2.''' ''The generated IR spectrum of optimized '''B3LYP/6-31G*''' structure''
|}

===Optimizing the "Chair" and "Boat" Transition Structures===

The study of the chair and boat transition states is carried out using three different approaches. The first one is called the Hessian which involves computing the force constant matrix; the second one is to freeze the reaction coordinate and then optimize the structure once the molecule is fully relaxed; the last one is to use QST2 method to specify the reactants and products then finding the transition structure.

<center>[[Image:XJB_2 ts.jpg|500px|center|thumb|'''Figure 3'''. ''The two transition states of Cope rearrangement'']]</center>

====The Chair Transition State====

In order to set up a transition state optimization, first an allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimized at the '''HF/3-21G''' level of theory. Then this half of the transition structure was duplicated with rotation to make the approximate resembling of the chair transition state with terminal ends of the fragments 2.2 Å apart. The chair transition structure optimization was then done by the first two different approaches.

<center>[[Image:XJB_guess ts.jpg|400px|center|thumb|'''Figure 4'''. ''The guessed chair transition structure of two allyl fragments'']]</center>

For the first optimization approach, the force constants were computed (the Hessian) and the chair transition state was optimized to a TS (Berny) at the '''HF/3-21G''' level of theory. The frequency calculation was carried out at the same time and gave an imaginary frequency at -818 cm<sup>-1</sup>. The electronic energy was calculated to be '''-231.61932Ha''' with C<sub>2h</sub> symmetry point group. The distance between the terminal carbons was 2.0203 Å. An animation of this transition state vibrations was generated to confirm the corresponding Cope rearrangement.

<center>[[File:XJB_chair ts animation.GIF|400px|center|thumb|'''Figure 5.''' ''Animation of the Cope rearrangement via the chair transition state'']]</center>

The chair transition state was then reoptimized using the '''B3LYP/6-31G*''' level of theory and carried out frequency calculations as well. The electronic energy calculated this time was '''-234.55698Ha''' with an imaginary frequency at -566 cm<sup>-1</sup>. We can see that with the geometries quite similar, the energies differ markedly calculated using two levels of theory. The summary and comparison of the calculations optimized at two levels of theory are summarized in '''Table 8'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46670||-234.41493
|-
| Sum of electronic and thermal Energies||-231.46134||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40807
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir1.jpg|200px]]||[[Image:XJB_chairir2.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 8.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the first approach
([[File:XJB_chairts1.1.log]], [[File:XJB_chairts1.2.log]])''</div>

The second approach is the frozen coordinate method using the Redundant Coord Editor to freeze the bond lengths of the terminal ends to 2.2 Å. First the calculation failed because of the recent update of GaussView. After editting the coordinate distance (B 5 1 2.2 F) and the frozen distance (B 5 1 F) the optimization went well this time. The structure was then optimized again using a normal guess Hessian modified including the two differentiating coordinates. This chair transition state was then reoptimized again using the '''B3LYP/6-31G*''' level of theory. The electronic energies were calculated to be the same as in the first approach and frequency calculations were summarized in '''Table 9'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46671||-234.41490
|-
| Sum of electronic and thermal Energies||-231.46135||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40806
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir3.jpg|200px]]||[[Image:XJB_chairir4.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 9.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the second approach
([[File:XJB_chairts2.1.log]], [[File:XJB_chairts2.2.log]])''</div>

As the same energy results given, the bond forming/breaking bond lengths in the chair transition structures in the above two different approaches were compared. The first approach gave the bond lengths of 2.0203 Å while the second approach gave 2.0207 Å. Considering the accuracy limit of bond length estimation in Gaussview, these two bond lengths can be seen as exactly the same. Two methods worked reasonably well because of the good assumption of transition structures. However for more complicated and larger molecule, the Hessian method may fail due to the difficulty to predict the transition geometry (different surface curvature). Thus better way to generate such transition structures is to freeze reaction coordinate.

{| align="center"
|- align="center"
|
[[Image:XJB_chairts bondlength1.jpg|350px|thumb|'''Figure 6.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by Hession'']]
|
[[Image:XJB_chairts bondlength2.jpg|350px|thumb|'''Figure 7.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by frozen coordinate'']]
|- align="center"
|}

====The Boat Transition State====

The boat transition structure was then optimized using the '''QST2''' method. The same numbering for the reactant and product molecules was important in order to specify for this reaction. The renumbering of the molecules is shown as in '''Figure 8''', but the following optimization failed and gave a twisted structure. The reactant and product geometries were then modified further by changing the central C-C-C-C dihedral angle to 0<sup>o</sup> and the inside two C-C-C (i.e. C2-C3-C4 and C3-C4-C5) angles to 100<sup>o</sup>. The job was run successfully and frequency calculations were done with the boat transition state generated. Only one imaginary frequency was detected also at -840 cm<sup>-1</sup>. The electronic energy was '''-231.60280Ha''' with C<sub>2v</sub> symmetry. The distances between the terminal carbons of the allyl fragments were examined to be both 2.140Å which were comparably longer than that of the chair transition structure.

{| align="center"
|- align="center"
|
[[Image:XJB_QST1.jpg|400px|thumb|'''Figure 8.''' ''Numbering of reactant and product'']]
|
[[Image:XJB_QST2.jpg|410px|thumb|'''Figure 9.''' ''Re-numbering of reactant and product'']]
|- align="center"
|}

<center>[[File:XJB_boat ts animation.GIF|400px|center|thumb|'''Figure 10.''' ''Animation of the Cope rearrangement via the boat transition state'']]</center>

The structure was then reoptimized at the higher '''B3LYP/6-31G*''' theory of level and the electronic energy was calculated to be '''-234.54309Ha''' with very similar geometry. The frequency calculation generated an imaginary frequency at -530 cm<sup>-1</sup>. The summary and comparison of two levels' calculations are carried out in the following table.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.45093||-234.40234
|-
| Sum of electronic and thermal Energies||-231.44530||-234.39601
|-
| Sum of electronic and thermal Enthalpie|| -231.44436||-234.39506
|-
| Sum of electronic and thermal Free Energie||-231.47977||-234.43175
|-
| Generated IR spectrum||[[Image:XJB_boatir1.jpg|200px]]||[[Image:XJB_boatir2.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 10.''' ''Frequency calculations of the boat transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by QST2
([[File:XJB_boat ts1.log]], [[File:XJB_boat ts2.log]])''</div>

====Intrinsic Reaction Coordinate====

Next a useful method called '''Intrinct Reaction Coordinate (IRC)''' is applied to the Hessian optimized chair transition structure to follow the mininmum energy path (MEP) from a transition structure down to its local minimum on a potential energy surface. IRC can be used to verify that the transition state you found actually connects the reactants and products you think it does, or to identify any reaction intermediates you haven't thought about, or to generate an animation of the reaction [ref]. The symmetrical reaction coordinate was computed only in the forward direction with force constant always calculated and the number of points along IRC 50. The last structure generated along IRC is given in '''Figure 11''' with the energy calculated as '''-231.69158Ha''' and the dihedral angle and internal angle measured as D = -67.188<sup>o</sup> A = 111.78<sup>o</sup>. This energy does not match to any of the conformations discussed in Appendix 1 thus a minimum geometry has not reached yet.

{| align="center"
|- align="center"
|
<jmol><jmolApplet><title>XJB_IRC.mol</title><color>white</color>
<size>200</size><script>zoom=5</script>
<uploadedFileContents>XJB_IRC.mol</uploadedFileContents>
</jmolApplet></jmol>
|
[[Image:XJB_irc.jpg|500px]]
|- align="center"
| align="center" | '''Figure 11.''' ''The energy minimum structure along IRC''
| align="center" | '''Figure 12.''' ''Total energy along IRC''
|}

Three improving options can be used to reach the minimum energy. The first one is to run a normal Hessian optimization of the last point on IRC; the second one involves using a larger number of points (in this case 100 was used); the last one is to redo the IRC computation specifying to compute force constant at every step. Three methods were all used and the results were summarized in '''Table 11'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Improved IRC method 1
! scope="col" | Improved IRC method 2
! scope="col" | Improved IRC method 3
|-
| [[Image:XJB_IRC1.jpg|300px|center|IRC1]]||[[Image:XJB_IRC2.jpg|300px|center|IRC3]] || [[Image:XJB_IRC3.jpg|300px|center|IRC3]]
|-
| D = -64.17<sup>o</sup> A = 112.04<sup>o</sup> || D = -66.83<sup>o</sup> A = 111.82<sup>o</sup> || D = -64.17<sup>o</sup> A = 112.04<sup>o</sup>
|-
| -231.69167Ha || -231.69150Ha || -231.69167Ha
|-
| [[File:XJB_IRC1.log]]||[[File:XJB_IRC2.log]]||[[File:XJB_IRC3.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 11.''' ''Comparison of three improvement methods of IRC computation upon the chair transition structure''</div>

It is concluded that method 1 and 3 gave the same the lowest energy which was consistent to the energy given in Appendix 1.

====Activation Energies====

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | HF/3-21G at 0K
! scope="col" | HF/3-21G at 298.15K
! scope="col" | B3LYP/6-31G* at 0K
! scope="col" | B3LYP/6-31G* at 298.15K
! scope="col" | Expt. at 0K
|-
| '''ΔE (chair)'''||45.71||44.70||34.05||33.16||33.5 ± 0.5
|-
| '''ΔE (boat)'''||55.60||54.76||41.96||41.32||44.7 ± 0.5
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 12.''' ''Summary of activation energies for the Cope rearrangement via both transition states in kcal/mol''</div>

==The Diels Alder Cycloaddition==

In this exercise, AM1 semi-empirical method is used to compute the Diels Alder cycloaddition which is a concerted pericyclic reaction. The driving force of this reaction is the formation of a new σ bond between the π orbitals of the dieneophile and the diene. The reaction is allowed with HOMO-LUMO interation and a symmetry overlap.

===Cis Butadiene===

<center>[[Image:prototype Diels Alder.jpg|500px|center|thumb|'''Figure 13'''. ''The prototype reaction between butadiene and ethylene'']]</center>

To study a prototype Diels Alder cycloaddition reaction between butadiene and ethylene, first a cis butadiene molecule was built using Gaussview and optimized using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the butadiene were plotted. As seen the HOMO of cis-butadiene is anti-symmetric ('''a''') and the LUMO is symmetric ('''s''') with respect to plane.

{| align="center"
|- align="center"
|
[[Image:XJB_HOMO.jpg|280px|thumb|'''Figure 14.''' ''Anti-symmetric HOMO of cis-butadiene'']]
|
[[Image:XJB_LUMO.jpg|300px|thumb|'''Figure 15.''' ''Symmetric HOMO of cis-butadiene'']]
|- align="center"
|}

===The Transition State of prototype reaction between ethylene and butadiene===

The Hessian method was used to estimate transition state for the reaction between ethylene and butadiene because these two reactants are relatively small and simple molecules. The transition state was drawn by first building a structure as (a) in '''Figure 16''' then removing the -CH<sub>2</sub>-CH<sub>2</sub>- group to form the envelop structure as in (b). The '''clean''' option should not be used and the dashed bond length was guessed to be larger than 2.2 Å otherwise the geometry change would cause a failure in optimization.

<center>[[Image:XJB_dielsguessts.jpg|300px|center|thumb|'''Figure 16'''. ''The guessing transition state for the reaction between butadiene and ethylene'']]</center>

<center>[[File:XJB_butadiene and ethylene ts.GIF|400px|center|thumb|'''Figure 17.''' ''Animation of the butadiene and ethylene cycloaddition transition state'']]</center>

After the Hessian optimization, the transition structure has been successfully determined and an animation was generated in '''Figure 17'''. It has a C<sub>1</sub> symmetry point group and the bond lengths of two partly formed C-C bonds were measured to be 2.12Å. The typical sp<sup>3</sup> and sp<sup>2</sup> C-C bond lengths are 1.54Å and 1.47Å respectively[ref]. The van der Waals radius of the C atom is 1.7 Å[ref]. The measured bond distance of the transition structure is larger than typical bond lengths but still within the van der Waals radius, suggesting the bond forming interation between the carbon atoms.

From the animation ('''Figure 17''') the vibration directions of two molecules are opposite and they are approaching and leaving each other at the same time. This suggests the formation of the two bonds synchronous. The vibrational frequencies were also calculated and an infrared spectrum was simulated ('''Figure 20'''). An imaginary frequency of magnitude -955 cm<sup>-1</sup> was seen and the lowest positive frequency was observed at 147 cm<sup>-1</sup> which corresponded to the reactant vibration but not the bond formation.

{| align="center"
|
[[Image:XJB_HOMOts.jpg|300px|thumb|'''Figure 18'''. HOMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_LUMOts.jpg|300px|thumb|'''Figure 19'''. LUMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_IRts.jpg|300px|thumb|'''Figure 20'''. Generated IR of transition state of reaction between cis-butadiene and ethylene]]
|}

The HOMO and LUMO molecular orbitals of this transition structure were visualized in '''Figure 18''' and '''Figure 19'''. From these two figures it can be observed that the HOMO at the transition structure is anti-symmetric ('''a''') while the LUMO is symmetric ('''s'''). Thus this anti-symmetric transition state HOMO is formed by the HOMO of cis-butadiene and the LUMO of ethylene with the knowledge that they are both anti-symmetrical. This conclusion agrees with the previous talked conditions for allowed reaction.

===The cyclohexa-1,3-diene reaction with maleic anhydride===

<center>[[Image:XJB_mechanism3.jpg|500px|center|thumb|'''Figure 21'''. ''The endo and exo products of the Diels Alder reaction between cyclohexa-1,3-diene and maleic anhydride'']]</center>

The exo and endo transition states were drawn according to '''Figure 21''' with the new C-C bond formation line dashed and bond length estimation as 2.5. Then the same Hessian method was applied to optimize the structures. The electronic energy of the endo structure was calculated to be '''-0.05150Ha''' and '''-0.05042Ha''' for the exo.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | endo
! scope="col" | exo
|-
| '''Animation'''||[[File:XJB_endots.GIF|300px]]||[[File:XJB_exots.GIF|300px]]
|-
| '''Sum of electronic and zero-point Energies'''||0.133493||0.134880
|-
| '''Sum of electronic and thermal Energies'''||0.143683||0.144881
|-
| '''Sum of electronic and thermal Enthalpies'''||0.144627||0.145825
|-
| '''Sum of electronic and thermal Free Energies'''||0.097349||0.099117
|-
| '''Bond lengths'''||[[Image:XJB_endomeasure.jpg|300px]]||[[Image:XJB_exomeasure.jpg|300px]]
|-
| '''HOMO'''||[[Image:XJB_endoHOMO.jpg|300px]]||[[Image:XJB_exoHOMO.jpg|300px]]
|-
| '''Secondary orbital overlap effect'''||YES||NO
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 12.''' ''Summary of comparison of properties of endo and exo transition structures for reaction between cyclohexa-1,3-diene and maleic anhydride''</div>

File:XJB exots.GIF

2013-12-05T18:00:16Z

Jx1011:

File:XJB endots.GIF

2013-12-05T17:59:47Z

Jx1011:

Rep:Mod:XJB1130

2013-12-05T17:59:29Z

Jx1011:

==The Cope Rearrangement of 1,5-hexadiene==

The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement reaction ('''Figure 1'''). Its mechanism has been studied by a large number of experimental and computational researches, which is recently accepted that the reaction occurs via either a "chair" or a "boat" transition state structure in a concerted manner. The aim of this exercise is to locate the low-energy minimum and the transition structures on the potential energy surface by Gaussian calculation in order to study its mechanism. The reaction pathway and the barrier heights can also be calculated by this calculation.

<center>[[Image:XJB_cope rearrangement.jpg|500px|center|thumb|'''Figure 1'''. ''The Cope rearrangement'']]</center>

===Optimizing the Reactants and Products===

Different conformations of 1,5-hexadiene can be optimized using '''Gaussview'''. The optimized structure of four conformers involved in this discussion and their corresponding energies optimized at '''HF/3-21G''' level of theory are concluded in the following table ('''Table 1''').

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | anti4
! scope="col" | anti2
! scope="col" | gauche
! scope="col" | gauche3
|-
| <jmol><jmolApplet><title>XJB_anti4.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti4.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche3.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| -231.69097Ha || -231.69254Ha || -231.68772Ha ||-231.69266Ha
|}

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 1.''' ''Four conformers of 1,5-hexadiene''</div>

This exercise first involved optimizing a molecule of 1,5-hexadiene with "anti" linkage for the cetral four C atoms at the '''HF/3-21G''' level of theory. The final optimization structure had the total electronic energy of '''-231.69097Ha''' with a '''C<sub>1</sub>''' symmetry. By comparing the energy and point group to that in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1], this structure is confirmed to be ''anti4''. The compositions of energies are also listed in '''Table 2''' as i) the sum of electronic and zero-point energies (potential energy at 0K including the zero-point vibrational energy), ii) the sum of electronic and thermal energies (energy including translational, rotational and vibrational at 298.15K and 1 atm), iii) the sum of electronic and thermal enthalpies (containning an additional correction for RT), and iv) the sum of electronic and thermal free energies (including the entropic contribution to the free energy). For what we are considering, the first two terms are the most useful for further comparisons.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Formula Description'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||E = E<sub>elec</sub> + ZPE||-231.53785
|-
| Sum of electronic and thermal Energies||E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>||-231.53096
|-
| Sum of electronic and thermal Enthalpie||H = E + RT||-231.53002
|-
| Sum of electronic and thermal Free Energie||G = H - TS||-231.56891
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 2.''' ''Frequency calculations of anti4 optimized at '''HF/3-21G''' ([[File:XJB_anti4.log]])''</div>

Next another 1,5-hexadiene molecule with "gauche" linkage for the central four atoms was drawn and optimized at the '''HF/3-21G''' level of theory as before. Energy for this conformation is expected to be higher than that for the ''anti4'' conformation due to the closer position of two alkene ends thus the steric interaction. The final structure had the electronic energy of '''-231.68772Ha''' which was indeed higher than that of the ''anti4'', indicating less favour towards this gauche structure. It had a '''C<sub>2</sub>''' symmetry and was confirmed to be the ''gauche'' conformation.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53439
|-
| Sum of electronic and thermal Energies||-231.52770
|-
| Sum of electronic and thermal Enthalpie||-231.52676
|-
| Sum of electronic and thermal Free Energie||-231.56483
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 3.''' ''Frequency calculations of gauche optimized at '''HF/3-21G''' ([[File:XJB_gauche.log]])''</div>

A lowest energy conformation ''gauche3'' of the reactant molecule was drawn based on and confirmed by [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. The optimization gave a final structure with an energy of '''-231.69266Ha''' with a symmetry of '''C<sub>1</sub>'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53949
|-
| Sum of electronic and thermal Energies||-231.53265
|-
| Sum of electronic and thermal Enthalpie||-231.5317
|-
| Sum of electronic and thermal Free Energie||-231.57064
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 4.''' ''Frequency calculations of gauche3 optimized at '''HF/3-21G''' ([[File:XJB_gauche3.log]])''</div>

The next step required us to draw the C<sub>i</sub> symmetry ''anti2'' conformation of 1,5-hexadiene and optimize it at the '''HF/3-21G''' level of theory. The optimized energy was calculated to be '''-231.69254Ha''' with point group of '''C<sub>i</sub>''', which was consistent with the one given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53954
|-
| Sum of electronic and thermal Energies||-231.53257
|-
| Sum of electronic and thermal Enthalpie||-231.53162
|-
| Sum of electronic and thermal Free Energie||-231.57092
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 5.''' ''Frequency calculations of anti2 optimized at '''HF/3-21G''' ([[File:XJB_anti2.log]])''</div>

The ''anti2'' structure was then reoptimized at '''B3LYP/6-31G*''' which was a improvement over the '''HF/3-21G''' basis set. It adds polarization to all atoms and improves the modeling of core electrons thus producing more accurate description of orbitals[ref]. The reoptimized ''anti2'' structure had the total electronic energy of '''-234.61171Ha''' and the same symmetry C<sub>i</sub>. The comparison of two structures optimized using different levels of theory is summarized in '''Table 6'''. We can see that they have exactly the same dihedral angle of the central four C atoms at 180.00<sup>o</sup> and angles between three of them are very similar as 111.3<sup>o</sup> and 112.7<sup>o</sup> respectively. Thus the the geometry change using '''HF/3-21G''' or '''B3LYP/6-31G*''' is trivial.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Before re-optimization
! scope="col" | After re-optimization
|-
| <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2reoptimized</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2reoptimized.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| ''anti2 optimized at '''HF/3-21G''''' || ''anti2 optimized at '''B3LYP/6-31G*'''''
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 6.''' ''Two optimization of anti2 at '''HF/3-21G''' and '''B3LYP/6-31G*'''([[File:XJB_anti2.log]])''</div>

The frequency calculation was run as well and a list of energies is summarized in '''Table 7'''. A list of all the vibrational frequencies was also generated and no imaginary (negative) frequency was observed and these vibrations are visualized by simulating the infrared spectrum in '''Figure 2'''.

{| align="center"
|- align="center"
|
{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-234.46920
|-
| Sum of electronic and thermal Energies||-234.46186
|-
| Sum of electronic and thermal Enthalpie||-234.46091
|-
| Sum of electronic and thermal Free Energie||-234.50078
|}
|
[[Image:XJB_IR_anti2.jpg|300px]]
|- align="center"
| align="center" | '''Table 7.''' ''Frequency calculations of anti2 optimized at '''B3LYP/6-31G*'''
([[File:XJB_anti2_reoptimized.log]])''
| align="center" | '''Figure 2.''' ''The generated IR spectrum of optimized '''B3LYP/6-31G*''' structure''
|}

===Optimizing the "Chair" and "Boat" Transition Structures===

The study of the chair and boat transition states is carried out using three different approaches. The first one is called the Hessian which involves computing the force constant matrix; the second one is to freeze the reaction coordinate and then optimize the structure once the molecule is fully relaxed; the last one is to use QST2 method to specify the reactants and products then finding the transition structure.

<center>[[Image:XJB_2 ts.jpg|500px|center|thumb|'''Figure 3'''. ''The two transition states of Cope rearrangement'']]</center>

====The Chair Transition State====

In order to set up a transition state optimization, first an allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimized at the '''HF/3-21G''' level of theory. Then this half of the transition structure was duplicated with rotation to make the approximate resembling of the chair transition state with terminal ends of the fragments 2.2 Å apart. The chair transition structure optimization was then done by the first two different approaches.

<center>[[Image:XJB_guess ts.jpg|400px|center|thumb|'''Figure 4'''. ''The guessed chair transition structure of two allyl fragments'']]</center>

For the first optimization approach, the force constants were computed (the Hessian) and the chair transition state was optimized to a TS (Berny) at the '''HF/3-21G''' level of theory. The frequency calculation was carried out at the same time and gave an imaginary frequency at -818 cm<sup>-1</sup>. The electronic energy was calculated to be '''-231.61932Ha''' with C<sub>2h</sub> symmetry point group. The distance between the terminal carbons was 2.0203 Å. An animation of this transition state vibrations was generated to confirm the corresponding Cope rearrangement.

<center>[[File:XJB_chair ts animation.GIF|400px|center|thumb|'''Figure 5.''' ''Animation of the Cope rearrangement via the chair transition state'']]</center>

The chair transition state was then reoptimized using the '''B3LYP/6-31G*''' level of theory and carried out frequency calculations as well. The electronic energy calculated this time was '''-234.55698Ha''' with an imaginary frequency at -566 cm<sup>-1</sup>. We can see that with the geometries quite similar, the energies differ markedly calculated using two levels of theory. The summary and comparison of the calculations optimized at two levels of theory are summarized in '''Table 8'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46670||-234.41493
|-
| Sum of electronic and thermal Energies||-231.46134||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40807
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir1.jpg|200px]]||[[Image:XJB_chairir2.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 8.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the first approach
([[File:XJB_chairts1.1.log]], [[File:XJB_chairts1.2.log]])''</div>

The second approach is the frozen coordinate method using the Redundant Coord Editor to freeze the bond lengths of the terminal ends to 2.2 Å. First the calculation failed because of the recent update of GaussView. After editting the coordinate distance (B 5 1 2.2 F) and the frozen distance (B 5 1 F) the optimization went well this time. The structure was then optimized again using a normal guess Hessian modified including the two differentiating coordinates. This chair transition state was then reoptimized again using the '''B3LYP/6-31G*''' level of theory. The electronic energies were calculated to be the same as in the first approach and frequency calculations were summarized in '''Table 9'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46671||-234.41490
|-
| Sum of electronic and thermal Energies||-231.46135||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40806
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir3.jpg|200px]]||[[Image:XJB_chairir4.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 9.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the second approach
([[File:XJB_chairts2.1.log]], [[File:XJB_chairts2.2.log]])''</div>

As the same energy results given, the bond forming/breaking bond lengths in the chair transition structures in the above two different approaches were compared. The first approach gave the bond lengths of 2.0203 Å while the second approach gave 2.0207 Å. Considering the accuracy limit of bond length estimation in Gaussview, these two bond lengths can be seen as exactly the same. Two methods worked reasonably well because of the good assumption of transition structures. However for more complicated and larger molecule, the Hessian method may fail due to the difficulty to predict the transition geometry (different surface curvature). Thus better way to generate such transition structures is to freeze reaction coordinate.

{| align="center"
|- align="center"
|
[[Image:XJB_chairts bondlength1.jpg|350px|thumb|'''Figure 6.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by Hession'']]
|
[[Image:XJB_chairts bondlength2.jpg|350px|thumb|'''Figure 7.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by frozen coordinate'']]
|- align="center"
|}

====The Boat Transition State====

The boat transition structure was then optimized using the '''QST2''' method. The same numbering for the reactant and product molecules was important in order to specify for this reaction. The renumbering of the molecules is shown as in '''Figure 8''', but the following optimization failed and gave a twisted structure. The reactant and product geometries were then modified further by changing the central C-C-C-C dihedral angle to 0<sup>o</sup> and the inside two C-C-C (i.e. C2-C3-C4 and C3-C4-C5) angles to 100<sup>o</sup>. The job was run successfully and frequency calculations were done with the boat transition state generated. Only one imaginary frequency was detected also at -840 cm<sup>-1</sup>. The electronic energy was '''-231.60280Ha''' with C<sub>2v</sub> symmetry. The distances between the terminal carbons of the allyl fragments were examined to be both 2.140Å which were comparably longer than that of the chair transition structure.

{| align="center"
|- align="center"
|
[[Image:XJB_QST1.jpg|400px|thumb|'''Figure 8.''' ''Numbering of reactant and product'']]
|
[[Image:XJB_QST2.jpg|410px|thumb|'''Figure 9.''' ''Re-numbering of reactant and product'']]
|- align="center"
|}

<center>[[File:XJB_boat ts animation.GIF|400px|center|thumb|'''Figure 10.''' ''Animation of the Cope rearrangement via the boat transition state'']]</center>

The structure was then reoptimized at the higher '''B3LYP/6-31G*''' theory of level and the electronic energy was calculated to be '''-234.54309Ha''' with very similar geometry. The frequency calculation generated an imaginary frequency at -530 cm<sup>-1</sup>. The summary and comparison of two levels' calculations are carried out in the following table.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.45093||-234.40234
|-
| Sum of electronic and thermal Energies||-231.44530||-234.39601
|-
| Sum of electronic and thermal Enthalpie|| -231.44436||-234.39506
|-
| Sum of electronic and thermal Free Energie||-231.47977||-234.43175
|-
| Generated IR spectrum||[[Image:XJB_boatir1.jpg|200px]]||[[Image:XJB_boatir2.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 10.''' ''Frequency calculations of the boat transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by QST2
([[File:XJB_boat ts1.log]], [[File:XJB_boat ts2.log]])''</div>

====Intrinsic Reaction Coordinate====

Next a useful method called '''Intrinct Reaction Coordinate (IRC)''' is applied to the Hessian optimized chair transition structure to follow the mininmum energy path (MEP) from a transition structure down to its local minimum on a potential energy surface. IRC can be used to verify that the transition state you found actually connects the reactants and products you think it does, or to identify any reaction intermediates you haven't thought about, or to generate an animation of the reaction [ref]. The symmetrical reaction coordinate was computed only in the forward direction with force constant always calculated and the number of points along IRC 50. The last structure generated along IRC is given in '''Figure 11''' with the energy calculated as '''-231.69158Ha''' and the dihedral angle and internal angle measured as D = -67.188<sup>o</sup> A = 111.78<sup>o</sup>. This energy does not match to any of the conformations discussed in Appendix 1 thus a minimum geometry has not reached yet.

{| align="center"
|- align="center"
|
<jmol><jmolApplet><title>XJB_IRC.mol</title><color>white</color>
<size>200</size><script>zoom=5</script>
<uploadedFileContents>XJB_IRC.mol</uploadedFileContents>
</jmolApplet></jmol>
|
[[Image:XJB_irc.jpg|500px]]
|- align="center"
| align="center" | '''Figure 11.''' ''The energy minimum structure along IRC''
| align="center" | '''Figure 12.''' ''Total energy along IRC''
|}

Three improving options can be used to reach the minimum energy. The first one is to run a normal Hessian optimization of the last point on IRC; the second one involves using a larger number of points (in this case 100 was used); the last one is to redo the IRC computation specifying to compute force constant at every step. Three methods were all used and the results were summarized in '''Table 11'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Improved IRC method 1
! scope="col" | Improved IRC method 2
! scope="col" | Improved IRC method 3
|-
| [[Image:XJB_IRC1.jpg|300px|center|IRC1]]||[[Image:XJB_IRC2.jpg|300px|center|IRC3]] || [[Image:XJB_IRC3.jpg|300px|center|IRC3]]
|-
| D = -64.17<sup>o</sup> A = 112.04<sup>o</sup> || D = -66.83<sup>o</sup> A = 111.82<sup>o</sup> || D = -64.17<sup>o</sup> A = 112.04<sup>o</sup>
|-
| -231.69167Ha || -231.69150Ha || -231.69167Ha
|-
| [[File:XJB_IRC1.log]]||[[File:XJB_IRC2.log]]||[[File:XJB_IRC3.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 11.''' ''Comparison of three improvement methods of IRC computation upon the chair transition structure''</div>

It is concluded that method 1 and 3 gave the same the lowest energy which was consistent to the energy given in Appendix 1.

====Activation Energies====

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | HF/3-21G at 0K
! scope="col" | HF/3-21G at 298.15K
! scope="col" | B3LYP/6-31G* at 0K
! scope="col" | B3LYP/6-31G* at 298.15K
! scope="col" | Expt. at 0K
|-
| '''ΔE (chair)'''||45.71||44.70||34.05||33.16||33.5 ± 0.5
|-
| '''ΔE (boat)'''||55.60||54.76||41.96||41.32||44.7 ± 0.5
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 12.''' ''Summary of activation energies for the Cope rearrangement via both transition states in kcal/mol''</div>

==The Diels Alder Cycloaddition==

In this exercise, AM1 semi-empirical method is used to compute the Diels Alder cycloaddition which is a concerted pericyclic reaction. The driving force of this reaction is the formation of a new σ bond between the π orbitals of the dieneophile and the diene. The reaction is allowed with HOMO-LUMO interation and a symmetry overlap.

===Cis Butadiene===

<center>[[Image:prototype Diels Alder.jpg|500px|center|thumb|'''Figure 13'''. ''The prototype reaction between butadiene and ethylene'']]</center>

To study a prototype Diels Alder cycloaddition reaction between butadiene and ethylene, first a cis butadiene molecule was built using Gaussview and optimized using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the butadiene were plotted. As seen the HOMO of cis-butadiene is anti-symmetric ('''a''') and the LUMO is symmetric ('''s''') with respect to plane.

{| align="center"
|- align="center"
|
[[Image:XJB_HOMO.jpg|280px|thumb|'''Figure 14.''' ''Anti-symmetric HOMO of cis-butadiene'']]
|
[[Image:XJB_LUMO.jpg|300px|thumb|'''Figure 15.''' ''Symmetric HOMO of cis-butadiene'']]
|- align="center"
|}

===The Transition State of prototype reaction between ethylene and butadiene===

The Hessian method was used to estimate transition state for the reaction between ethylene and butadiene because these two reactants are relatively small and simple molecules. The transition state was drawn by first building a structure as (a) in '''Figure 16''' then removing the -CH<sub>2</sub>-CH<sub>2</sub>- group to form the envelop structure as in (b). The '''clean''' option should not be used and the dashed bond length was guessed to be larger than 2.2 Å otherwise the geometry change would cause a failure in optimization.

<center>[[Image:XJB_dielsguessts.jpg|300px|center|thumb|'''Figure 16'''. ''The guessing transition state for the reaction between butadiene and ethylene'']]</center>

<center>[[File:XJB_butadiene and ethylene ts.GIF|400px|center|thumb|'''Figure 17.''' ''Animation of the butadiene and ethylene cycloaddition transition state'']]</center>

After the Hessian optimization, the transition structure has been successfully determined and an animation was generated in '''Figure 17'''. It has a C<sub>1</sub> symmetry point group and the bond lengths of two partly formed C-C bonds were measured to be 2.12Å. The typical sp<sup>3</sup> and sp<sup>2</sup> C-C bond lengths are 1.54Å and 1.47Å respectively[ref]. The van der Waals radius of the C atom is 1.7 Å[ref]. The measured bond distance of the transition structure is larger than typical bond lengths but still within the van der Waals radius, suggesting the bond forming interation between the carbon atoms.

From the animation ('''Figure 17''') the vibration directions of two molecules are opposite and they are approaching and leaving each other at the same time. This suggests the formation of the two bonds synchronous. The vibrational frequencies were also calculated and an infrared spectrum was simulated ('''Figure 20'''). An imaginary frequency of magnitude -955 cm<sup>-1</sup> was seen and the lowest positive frequency was observed at 147 cm<sup>-1</sup> which corresponded to the reactant vibration but not the bond formation.

{| align="center"
|
[[Image:XJB_HOMOts.jpg|300px|thumb|'''Figure 18'''. HOMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_LUMOts.jpg|300px|thumb|'''Figure 19'''. LUMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_IRts.jpg|300px|thumb|'''Figure 20'''. Generated IR of transition state of reaction between cis-butadiene and ethylene]]
|}

The HOMO and LUMO molecular orbitals of this transition structure were visualized in '''Figure 18''' and '''Figure 19'''. From these two figures it can be observed that the HOMO at the transition structure is anti-symmetric ('''a''') while the LUMO is symmetric ('''s'''). Thus this anti-symmetric transition state HOMO is formed by the HOMO of cis-butadiene and the LUMO of ethylene with the knowledge that they are both anti-symmetrical. This conclusion agrees with the previous talked conditions for allowed reaction.

===The cyclohexa-1,3-diene reaction with maleic anhydride===

<center>[[Image:XJB_mechanism3.jpg|500px|center|thumb|'''Figure 21'''. ''The endo and exo products of the Diels Alder reaction between cyclohexa-1,3-diene and maleic anhydride'']]</center>

The exo and endo transition states were drawn according to '''Figure 21''' with the new C-C bond formation line dashed and bond length estimation as 2.5. Then the same Hessian method was applied to optimize the structures. The electronic energy of the endo structure was calculated to be '''-0.05150Ha''' and '''-0.05042Ha''' for the exo.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | endo
! scope="col" | exo
|-
| '''Animation'''||[[File:XJB_endots.GIF|300px]]||[[File:XJB_exots.GIF|300px]]
|-
| '''Sum of electronic and zero-point Energies'''||0.133493||0.134880
|-
| '''Sum of electronic and thermal Energies'''||0.143683||0.144881
|-
| '''Sum of electronic and thermal Enthalpies'''||0.144627||0.145825
|-
| '''Sum of electronic and thermal Free Energies'''||0.097349||0.099117
|-
| ''''''Bond lengths||[[Image:XJB_endomeasure.jpg|300px]]||[[Image:XJB_exomeasure.jpg|300px]]
|-
| '''HOMO'''||[[Image:XJB_endoHOMO.jpg|300px]]||[[Image:XJB_exoHOMO.jpg|300px]]
|-
| '''Secondary orbital overlap effect'''||YES||NO
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 12.''' ''Summary of comparison of properties of endo and exo transition structures for reaction between cyclohexa-1,3-diene and maleic anhydride''</div>

Rep:Mod:XJB1130

2013-12-05T17:41:05Z

Jx1011: /* The cyclohexa-1,3-diene reaction with maleic anhydride */

==The Cope Rearrangement of 1,5-hexadiene==

The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement reaction ('''Figure 1'''). Its mechanism has been studied by a large number of experimental and computational researches, which is recently accepted that the reaction occurs via either a "chair" or a "boat" transition state structure in a concerted manner. The aim of this exercise is to locate the low-energy minimum and the transition structures on the potential energy surface by Gaussian calculation in order to study its mechanism. The reaction pathway and the barrier heights can also be calculated by this calculation.

<center>[[Image:XJB_cope rearrangement.jpg|500px|center|thumb|'''Figure 1'''. ''The Cope rearrangement'']]</center>

===Optimizing the Reactants and Products===

Different conformations of 1,5-hexadiene can be optimized using '''Gaussview'''. The optimized structure of four conformers involved in this discussion and their corresponding energies optimized at '''HF/3-21G''' level of theory are concluded in the following table ('''Table 1''').

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | anti4
! scope="col" | anti2
! scope="col" | gauche
! scope="col" | gauche3
|-
| <jmol><jmolApplet><title>XJB_anti4.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti4.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche3.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| -231.69097Ha || -231.69254Ha || -231.68772Ha ||-231.69266Ha
|}

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 1.''' ''Four conformers of 1,5-hexadiene''</div>

This exercise first involved optimizing a molecule of 1,5-hexadiene with "anti" linkage for the cetral four C atoms at the '''HF/3-21G''' level of theory. The final optimization structure had the total electronic energy of '''-231.69097Ha''' with a '''C<sub>1</sub>''' symmetry. By comparing the energy and point group to that in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1], this structure is confirmed to be ''anti4''. The compositions of energies are also listed in '''Table 2''' as i) the sum of electronic and zero-point energies (potential energy at 0K including the zero-point vibrational energy), ii) the sum of electronic and thermal energies (energy including translational, rotational and vibrational at 298.15K and 1 atm), iii) the sum of electronic and thermal enthalpies (containning an additional correction for RT), and iv) the sum of electronic and thermal free energies (including the entropic contribution to the free energy). For what we are considering, the first two terms are the most useful for further comparisons.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Formula Description'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||E = E<sub>elec</sub> + ZPE||-231.53785
|-
| Sum of electronic and thermal Energies||E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>||-231.53096
|-
| Sum of electronic and thermal Enthalpie||H = E + RT||-231.53002
|-
| Sum of electronic and thermal Free Energie||G = H - TS||-231.56891
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 2.''' ''Frequency calculations of anti4 optimized at '''HF/3-21G''' ([[File:XJB_anti4.log]])''</div>

Next another 1,5-hexadiene molecule with "gauche" linkage for the central four atoms was drawn and optimized at the '''HF/3-21G''' level of theory as before. Energy for this conformation is expected to be higher than that for the ''anti4'' conformation due to the closer position of two alkene ends thus the steric interaction. The final structure had the electronic energy of '''-231.68772Ha''' which was indeed higher than that of the ''anti4'', indicating less favour towards this gauche structure. It had a '''C<sub>2</sub>''' symmetry and was confirmed to be the ''gauche'' conformation.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53439
|-
| Sum of electronic and thermal Energies||-231.52770
|-
| Sum of electronic and thermal Enthalpie||-231.52676
|-
| Sum of electronic and thermal Free Energie||-231.56483
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 3.''' ''Frequency calculations of gauche optimized at '''HF/3-21G''' ([[File:XJB_gauche.log]])''</div>

A lowest energy conformation ''gauche3'' of the reactant molecule was drawn based on and confirmed by [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. The optimization gave a final structure with an energy of '''-231.69266Ha''' with a symmetry of '''C<sub>1</sub>'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53949
|-
| Sum of electronic and thermal Energies||-231.53265
|-
| Sum of electronic and thermal Enthalpie||-231.5317
|-
| Sum of electronic and thermal Free Energie||-231.57064
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 4.''' ''Frequency calculations of gauche3 optimized at '''HF/3-21G''' ([[File:XJB_gauche3.log]])''</div>

The next step required us to draw the C<sub>i</sub> symmetry ''anti2'' conformation of 1,5-hexadiene and optimize it at the '''HF/3-21G''' level of theory. The optimized energy was calculated to be '''-231.69254Ha''' with point group of '''C<sub>i</sub>''', which was consistent with the one given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53954
|-
| Sum of electronic and thermal Energies||-231.53257
|-
| Sum of electronic and thermal Enthalpie||-231.53162
|-
| Sum of electronic and thermal Free Energie||-231.57092
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 5.''' ''Frequency calculations of anti2 optimized at '''HF/3-21G''' ([[File:XJB_anti2.log]])''</div>

The ''anti2'' structure was then reoptimized at '''B3LYP/6-31G*''' which was a improvement over the '''HF/3-21G''' basis set. It adds polarization to all atoms and improves the modeling of core electrons thus producing more accurate description of orbitals[ref]. The reoptimized ''anti2'' structure had the total electronic energy of '''-234.61171Ha''' and the same symmetry C<sub>i</sub>. The comparison of two structures optimized using different levels of theory is summarized in '''Table 6'''. We can see that they have exactly the same dihedral angle of the central four C atoms at 180.00<sup>o</sup> and angles between three of them are very similar as 111.3<sup>o</sup> and 112.7<sup>o</sup> respectively. Thus the the geometry change using '''HF/3-21G''' or '''B3LYP/6-31G*''' is trivial.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Before re-optimization
! scope="col" | After re-optimization
|-
| <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2reoptimized</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2reoptimized.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| ''anti2 optimized at '''HF/3-21G''''' || ''anti2 optimized at '''B3LYP/6-31G*'''''
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 6.''' ''Two optimization of anti2 at '''HF/3-21G''' and '''B3LYP/6-31G*'''([[File:XJB_anti2.log]])''</div>

The frequency calculation was run as well and a list of energies is summarized in '''Table 7'''. A list of all the vibrational frequencies was also generated and no imaginary (negative) frequency was observed and these vibrations are visualized by simulating the infrared spectrum in '''Figure 2'''.

{| align="center"
|- align="center"
|
{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-234.46920
|-
| Sum of electronic and thermal Energies||-234.46186
|-
| Sum of electronic and thermal Enthalpie||-234.46091
|-
| Sum of electronic and thermal Free Energie||-234.50078
|}
|
[[Image:XJB_IR_anti2.jpg|300px]]
|- align="center"
| align="center" | '''Table 7.''' ''Frequency calculations of anti2 optimized at '''B3LYP/6-31G*'''
([[File:XJB_anti2_reoptimized.log]])''
| align="center" | '''Figure 2.''' ''The generated IR spectrum of optimized '''B3LYP/6-31G*''' structure''
|}

===Optimizing the "Chair" and "Boat" Transition Structures===

The study of the chair and boat transition states is carried out using three different approaches. The first one is called the Hessian which involves computing the force constant matrix; the second one is to freeze the reaction coordinate and then optimize the structure once the molecule is fully relaxed; the last one is to use QST2 method to specify the reactants and products then finding the transition structure.

<center>[[Image:XJB_2 ts.jpg|500px|center|thumb|'''Figure 3'''. ''The two transition states of Cope rearrangement'']]</center>

====The Chair Transition State====

In order to set up a transition state optimization, first an allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimized at the '''HF/3-21G''' level of theory. Then this half of the transition structure was duplicated with rotation to make the approximate resembling of the chair transition state with terminal ends of the fragments 2.2 Å apart. The chair transition structure optimization was then done by the first two different approaches.

<center>[[Image:XJB_guess ts.jpg|400px|center|thumb|'''Figure 4'''. ''The guessed chair transition structure of two allyl fragments'']]</center>

For the first optimization approach, the force constants were computed (the Hessian) and the chair transition state was optimized to a TS (Berny) at the '''HF/3-21G''' level of theory. The frequency calculation was carried out at the same time and gave an imaginary frequency at -818 cm<sup>-1</sup>. The electronic energy was calculated to be '''-231.61932Ha''' with C<sub>2h</sub> symmetry point group. The distance between the terminal carbons was 2.0203 Å. An animation of this transition state vibrations was generated to confirm the corresponding Cope rearrangement.

<center>[[File:XJB_chair ts animation.GIF|400px|center|thumb|'''Figure 5.''' ''Animation of the Cope rearrangement via the chair transition state'']]</center>

The chair transition state was then reoptimized using the '''B3LYP/6-31G*''' level of theory and carried out frequency calculations as well. The electronic energy calculated this time was '''-234.55698Ha''' with an imaginary frequency at -566 cm<sup>-1</sup>. We can see that with the geometries quite similar, the energies differ markedly calculated using two levels of theory. The summary and comparison of the calculations optimized at two levels of theory are summarized in '''Table 8'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46670||-234.41493
|-
| Sum of electronic and thermal Energies||-231.46134||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40807
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir1.jpg|200px]]||[[Image:XJB_chairir2.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 8.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the first approach
([[File:XJB_chairts1.1.log]], [[File:XJB_chairts1.2.log]])''</div>

The second approach is the frozen coordinate method using the Redundant Coord Editor to freeze the bond lengths of the terminal ends to 2.2 Å. First the calculation failed because of the recent update of GaussView. After editting the coordinate distance (B 5 1 2.2 F) and the frozen distance (B 5 1 F) the optimization went well this time. The structure was then optimized again using a normal guess Hessian modified including the two differentiating coordinates. This chair transition state was then reoptimized again using the '''B3LYP/6-31G*''' level of theory. The electronic energies were calculated to be the same as in the first approach and frequency calculations were summarized in '''Table 9'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46671||-234.41490
|-
| Sum of electronic and thermal Energies||-231.46135||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40806
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir3.jpg|200px]]||[[Image:XJB_chairir4.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 9.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the second approach
([[File:XJB_chairts2.1.log]], [[File:XJB_chairts2.2.log]])''</div>

As the same energy results given, the bond forming/breaking bond lengths in the chair transition structures in the above two different approaches were compared. The first approach gave the bond lengths of 2.0203 Å while the second approach gave 2.0207 Å. Considering the accuracy limit of bond length estimation in Gaussview, these two bond lengths can be seen as exactly the same. Two methods worked reasonably well because of the good assumption of transition structures. However for more complicated and larger molecule, the Hessian method may fail due to the difficulty to predict the transition geometry (different surface curvature). Thus better way to generate such transition structures is to freeze reaction coordinate.

{| align="center"
|- align="center"
|
[[Image:XJB_chairts bondlength1.jpg|350px|thumb|'''Figure 6.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by Hession'']]
|
[[Image:XJB_chairts bondlength2.jpg|350px|thumb|'''Figure 7.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by frozen coordinate'']]
|- align="center"
|}

====The Boat Transition State====

The boat transition structure was then optimized using the '''QST2''' method. The same numbering for the reactant and product molecules was important in order to specify for this reaction. The renumbering of the molecules is shown as in '''Figure 8''', but the following optimization failed and gave a twisted structure. The reactant and product geometries were then modified further by changing the central C-C-C-C dihedral angle to 0<sup>o</sup> and the inside two C-C-C (i.e. C2-C3-C4 and C3-C4-C5) angles to 100<sup>o</sup>. The job was run successfully and frequency calculations were done with the boat transition state generated. Only one imaginary frequency was detected also at -840 cm<sup>-1</sup>. The electronic energy was '''-231.60280Ha''' with C<sub>2v</sub> symmetry. The distances between the terminal carbons of the allyl fragments were examined to be both 2.140Å which were comparably longer than that of the chair transition structure.

{| align="center"
|- align="center"
|
[[Image:XJB_QST1.jpg|400px|thumb|'''Figure 8.''' ''Numbering of reactant and product'']]
|
[[Image:XJB_QST2.jpg|410px|thumb|'''Figure 9.''' ''Re-numbering of reactant and product'']]
|- align="center"
|}

<center>[[File:XJB_boat ts animation.GIF|400px|center|thumb|'''Figure 10.''' ''Animation of the Cope rearrangement via the boat transition state'']]</center>

The structure was then reoptimized at the higher '''B3LYP/6-31G*''' theory of level and the electronic energy was calculated to be '''-234.54309Ha''' with very similar geometry. The frequency calculation generated an imaginary frequency at -530 cm<sup>-1</sup>. The summary and comparison of two levels' calculations are carried out in the following table.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.45093||-234.40234
|-
| Sum of electronic and thermal Energies||-231.44530||-234.39601
|-
| Sum of electronic and thermal Enthalpie|| -231.44436||-234.39506
|-
| Sum of electronic and thermal Free Energie||-231.47977||-234.43175
|-
| Generated IR spectrum||[[Image:XJB_boatir1.jpg|200px]]||[[Image:XJB_boatir2.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 10.''' ''Frequency calculations of the boat transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by QST2
([[File:XJB_boat ts1.log]], [[File:XJB_boat ts2.log]])''</div>

====Intrinsic Reaction Coordinate====

Next a useful method called '''Intrinct Reaction Coordinate (IRC)''' is applied to the Hessian optimized chair transition structure to follow the mininmum energy path (MEP) from a transition structure down to its local minimum on a potential energy surface. IRC can be used to verify that the transition state you found actually connects the reactants and products you think it does, or to identify any reaction intermediates you haven't thought about, or to generate an animation of the reaction [ref]. The symmetrical reaction coordinate was computed only in the forward direction with force constant always calculated and the number of points along IRC 50. The last structure generated along IRC is given in '''Figure 11''' with the energy calculated as '''-231.69158Ha''' and the dihedral angle and internal angle measured as D = -67.188<sup>o</sup> A = 111.78<sup>o</sup>. This energy does not match to any of the conformations discussed in Appendix 1 thus a minimum geometry has not reached yet.

{| align="center"
|- align="center"
|
<jmol><jmolApplet><title>XJB_IRC.mol</title><color>white</color>
<size>200</size><script>zoom=5</script>
<uploadedFileContents>XJB_IRC.mol</uploadedFileContents>
</jmolApplet></jmol>
|
[[Image:XJB_irc.jpg|500px]]
|- align="center"
| align="center" | '''Figure 11.''' ''The energy minimum structure along IRC''
| align="center" | '''Figure 12.''' ''Total energy along IRC''
|}

Three improving options can be used to reach the minimum energy. The first one is to run a normal Hessian optimization of the last point on IRC; the second one involves using a larger number of points (in this case 100 was used); the last one is to redo the IRC computation specifying to compute force constant at every step. Three methods were all used and the results were summarized in '''Table 11'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Improved IRC method 1
! scope="col" | Improved IRC method 2
! scope="col" | Improved IRC method 3
|-
| [[Image:XJB_IRC1.jpg|300px|center|IRC1]]||[[Image:XJB_IRC2.jpg|300px|center|IRC3]] || [[Image:XJB_IRC3.jpg|300px|center|IRC3]]
|-
| D = -64.17<sup>o</sup> A = 112.04<sup>o</sup> || D = -66.83<sup>o</sup> A = 111.82<sup>o</sup> || D = -64.17<sup>o</sup> A = 112.04<sup>o</sup>
|-
| -231.69167Ha || -231.69150Ha || -231.69167Ha
|-
| [[File:XJB_IRC1.log]]||[[File:XJB_IRC2.log]]||[[File:XJB_IRC3.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 11.''' ''Comparison of three improvement methods of IRC computation upon the chair transition structure''</div>

It is concluded that method 1 and 3 gave the same the lowest energy which was consistent to the energy given in Appendix 1.

====Activation Energies====

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | HF/3-21G at 0K
! scope="col" | HF/3-21G at 298.15K
! scope="col" | B3LYP/6-31G* at 0K
! scope="col" | B3LYP/6-31G* at 298.15K
! scope="col" | Expt. at 0K
|-
| '''ΔE (chair)'''||45.71||44.70||34.05||33.16||33.5 ± 0.5
|-
| '''ΔE (boat)'''||55.60||54.76||41.96||41.32||44.7 ± 0.5
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 12.''' ''Summary of activation energies for the Cope rearrangement via both transition states in kcal/mol''</div>

==The Diels Alder Cycloaddition==

In this exercise, AM1 semi-empirical method is used to compute the Diels Alder cycloaddition which is a concerted pericyclic reaction. The driving force of this reaction is the formation of a new σ bond between the π orbitals of the dieneophile and the diene. The reaction is allowed with HOMO-LUMO interation and a symmetry overlap.

===Cis Butadiene===

<center>[[Image:prototype Diels Alder.jpg|500px|center|thumb|'''Figure 13'''. ''The prototype reaction between butadiene and ethylene'']]</center>

To study a prototype Diels Alder cycloaddition reaction between butadiene and ethylene, first a cis butadiene molecule was built using Gaussview and optimized using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the butadiene were plotted. As seen the HOMO of cis-butadiene is anti-symmetric ('''a''') and the LUMO is symmetric ('''s''') with respect to plane.

{| align="center"
|- align="center"
|
[[Image:XJB_HOMO.jpg|280px|thumb|'''Figure 14.''' ''Anti-symmetric HOMO of cis-butadiene'']]
|
[[Image:XJB_LUMO.jpg|300px|thumb|'''Figure 15.''' ''Symmetric HOMO of cis-butadiene'']]
|- align="center"
|}

===The Transition State of prototype reaction between ethylene and butadiene===

The Hessian method was used to estimate transition state for the reaction between ethylene and butadiene because these two reactants are relatively small and simple molecules. The transition state was drawn by first building a structure as (a) in '''Figure 16''' then removing the -CH<sub>2</sub>-CH<sub>2</sub>- group to form the envelop structure as in (b). The '''clean''' option should not be used and the dashed bond length was guessed to be larger than 2.2 Å otherwise the geometry change would cause a failure in optimization.

<center>[[Image:XJB_dielsguessts.jpg|300px|center|thumb|'''Figure 16'''. ''The guessing transition state for the reaction between butadiene and ethylene'']]</center>

<center>[[File:XJB_butadiene and ethylene ts.GIF|400px|center|thumb|'''Figure 17.''' ''Animation of the butadiene and ethylene cycloaddition transition state'']]</center>

After the Hessian optimization, the transition structure has been successfully determined and an animation was generated in '''Figure 17'''. It has a C<sub>1</sub> symmetry point group and the bond lengths of two partly formed C-C bonds were measured to be 2.12Å. The typical sp<sup>3</sup> and sp<sup>2</sup> C-C bond lengths are 1.54Å and 1.47Å respectively[ref]. The van der Waals radius of the C atom is 1.7 Å[ref]. The measured bond distance of the transition structure is larger than typical bond lengths but still within the van der Waals radius, suggesting the bond forming interation between the carbon atoms.

From the animation ('''Figure 17''') the vibration directions of two molecules are opposite and they are approaching and leaving each other at the same time. This suggests the formation of the two bonds synchronous. The vibrational frequencies were also calculated and an infrared spectrum was simulated ('''Figure 20'''). An imaginary frequency of magnitude -955 cm<sup>-1</sup> was seen and the lowest positive frequency was observed at 147 cm<sup>-1</sup> which corresponded to the reactant vibration but not the bond formation.

{| align="center"
|
[[Image:XJB_HOMOts.jpg|300px|thumb|'''Figure 18'''. HOMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_LUMOts.jpg|300px|thumb|'''Figure 19'''. LUMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_IRts.jpg|300px|thumb|'''Figure 20'''. Generated IR of transition state of reaction between cis-butadiene and ethylene]]
|}

The HOMO and LUMO molecular orbitals of this transition structure were visualized in '''Figure 18''' and '''Figure 19'''. From these two figures it can be observed that the HOMO at the transition structure is anti-symmetric ('''a''') while the LUMO is symmetric ('''s'''). Thus this anti-symmetric transition state HOMO is formed by the HOMO of cis-butadiene and the LUMO of ethylene with the knowledge that they are both anti-symmetrical. This conclusion agrees with the previous talked conditions for allowed reaction.

===The cyclohexa-1,3-diene reaction with maleic anhydride===

<center>[[Image:XJB_mechanism3.jpg|500px|center|thumb|'''Figure 21'''. ''The endo and exo products of the Diels Alder reaction between cyclohexa-1,3-diene and maleic anhydride'']]</center>

The exo and endo transition states were drawn according to '''Figure 21''' with the new C-C bond formation line dashed and bond length estimation as 2.5. Then the same Hessian method was applied to optimize the structures. The electronic energy of the endo structure was calculated to be '''-0.05150Ha''' and '''-0.05042Ha''' for the exo .

File:XJB mechanism3.jpg

2013-12-05T17:03:38Z

Jx1011: uploaded a new version of "File:XJB mechanism3.jpg"

File:XJB mechanism3.jpg

2013-12-05T17:01:33Z

Jx1011: uploaded a new version of "File:XJB mechanism3.jpg"

File:XJB mechanism3.jpg

2013-12-05T17:01:00Z

Jx1011:

Rep:Mod:XJB1130

2013-12-05T17:00:42Z

Jx1011:

==The Cope Rearrangement of 1,5-hexadiene==

The Cope rearrangement of 1,5-hexadiene is a [3,3]-sigmatropic rearrangement reaction ('''Figure 1'''). Its mechanism has been studied by a large number of experimental and computational researches, which is recently accepted that the reaction occurs via either a "chair" or a "boat" transition state structure in a concerted manner. The aim of this exercise is to locate the low-energy minimum and the transition structures on the potential energy surface by Gaussian calculation in order to study its mechanism. The reaction pathway and the barrier heights can also be calculated by this calculation.

<center>[[Image:XJB_cope rearrangement.jpg|500px|center|thumb|'''Figure 1'''. ''The Cope rearrangement'']]</center>

===Optimizing the Reactants and Products===

Different conformations of 1,5-hexadiene can be optimized using '''Gaussview'''. The optimized structure of four conformers involved in this discussion and their corresponding energies optimized at '''HF/3-21G''' level of theory are concluded in the following table ('''Table 1''').

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | anti4
! scope="col" | anti2
! scope="col" | gauche
! scope="col" | gauche3
|-
| <jmol><jmolApplet><title>XJB_anti4.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti4.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_gauche3.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;</script>
<uploadedFileContents>XJB_gauche3.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| -231.69097Ha || -231.69254Ha || -231.68772Ha ||-231.69266Ha
|}

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 1.''' ''Four conformers of 1,5-hexadiene''</div>

This exercise first involved optimizing a molecule of 1,5-hexadiene with "anti" linkage for the cetral four C atoms at the '''HF/3-21G''' level of theory. The final optimization structure had the total electronic energy of '''-231.69097Ha''' with a '''C<sub>1</sub>''' symmetry. By comparing the energy and point group to that in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1], this structure is confirmed to be ''anti4''. The compositions of energies are also listed in '''Table 2''' as i) the sum of electronic and zero-point energies (potential energy at 0K including the zero-point vibrational energy), ii) the sum of electronic and thermal energies (energy including translational, rotational and vibrational at 298.15K and 1 atm), iii) the sum of electronic and thermal enthalpies (containning an additional correction for RT), and iv) the sum of electronic and thermal free energies (including the entropic contribution to the free energy). For what we are considering, the first two terms are the most useful for further comparisons.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Formula Description'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||E = E<sub>elec</sub> + ZPE||-231.53785
|-
| Sum of electronic and thermal Energies||E = E + E<sub>vib</sub> + E<sub>rot</sub> + E<sub>trans</sub>||-231.53096
|-
| Sum of electronic and thermal Enthalpie||H = E + RT||-231.53002
|-
| Sum of electronic and thermal Free Energie||G = H - TS||-231.56891
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 2.''' ''Frequency calculations of anti4 optimized at '''HF/3-21G''' ([[File:XJB_anti4.log]])''</div>

Next another 1,5-hexadiene molecule with "gauche" linkage for the central four atoms was drawn and optimized at the '''HF/3-21G''' level of theory as before. Energy for this conformation is expected to be higher than that for the ''anti4'' conformation due to the closer position of two alkene ends thus the steric interaction. The final structure had the electronic energy of '''-231.68772Ha''' which was indeed higher than that of the ''anti4'', indicating less favour towards this gauche structure. It had a '''C<sub>2</sub>''' symmetry and was confirmed to be the ''gauche'' conformation.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53439
|-
| Sum of electronic and thermal Energies||-231.52770
|-
| Sum of electronic and thermal Enthalpie||-231.52676
|-
| Sum of electronic and thermal Free Energie||-231.56483
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 3.''' ''Frequency calculations of gauche optimized at '''HF/3-21G''' ([[File:XJB_gauche.log]])''</div>

A lowest energy conformation ''gauche3'' of the reactant molecule was drawn based on and confirmed by [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1]. The optimization gave a final structure with an energy of '''-231.69266Ha''' with a symmetry of '''C<sub>1</sub>'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53949
|-
| Sum of electronic and thermal Energies||-231.53265
|-
| Sum of electronic and thermal Enthalpie||-231.5317
|-
| Sum of electronic and thermal Free Energie||-231.57064
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 4.''' ''Frequency calculations of gauche3 optimized at '''HF/3-21G''' ([[File:XJB_gauche3.log]])''</div>

The next step required us to draw the C<sub>i</sub> symmetry ''anti2'' conformation of 1,5-hexadiene and optimize it at the '''HF/3-21G''' level of theory. The optimized energy was calculated to be '''-231.69254Ha''' with point group of '''C<sub>i</sub>''', which was consistent with the one given in [https://wiki.ch.ic.ac.uk/wiki/index.php?title=Mod:phys3#Appendix_1 Appendix 1].

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
|-
| Sum of electronic and zero-point Energies||-231.53954
|-
| Sum of electronic and thermal Energies||-231.53257
|-
| Sum of electronic and thermal Enthalpie||-231.53162
|-
| Sum of electronic and thermal Free Energie||-231.57092
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 5.''' ''Frequency calculations of anti2 optimized at '''HF/3-21G''' ([[File:XJB_anti2.log]])''</div>

The ''anti2'' structure was then reoptimized at '''B3LYP/6-31G*''' which was a improvement over the '''HF/3-21G''' basis set. It adds polarization to all atoms and improves the modeling of core electrons thus producing more accurate description of orbitals[ref]. The reoptimized ''anti2'' structure had the total electronic energy of '''-234.61171Ha''' and the same symmetry C<sub>i</sub>. The comparison of two structures optimized using different levels of theory is summarized in '''Table 6'''. We can see that they have exactly the same dihedral angle of the central four C atoms at 180.00<sup>o</sup> and angles between three of them are very similar as 111.3<sup>o</sup> and 112.7<sup>o</sup> respectively. Thus the the geometry change using '''HF/3-21G''' or '''B3LYP/6-31G*''' is trivial.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Before re-optimization
! scope="col" | After re-optimization
|-
| <jmol><jmolApplet><title>XJB_anti2.mol</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2.mol</uploadedFileContents>
</jmolApplet></jmol> || <jmol><jmolApplet><title>XJB_anti2reoptimized</title><color>white</color>
<size>200</size><script>moveto 4 0 2 0 90 120;spin 2;measure 4 6 9 12;measure 6 9 12</script>
<uploadedFileContents>XJB_anti2reoptimized.mol</uploadedFileContents>
</jmolApplet></jmol>
|-
| ''anti2 optimized at '''HF/3-21G''''' || ''anti2 optimized at '''B3LYP/6-31G*'''''
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 6.''' ''Two optimization of anti2 at '''HF/3-21G''' and '''B3LYP/6-31G*'''([[File:XJB_anti2.log]])''</div>

The frequency calculation was run as well and a list of energies is summarized in '''Table 7'''. A list of all the vibrational frequencies was also generated and no imaginary (negative) frequency was observed and these vibrations are visualized by simulating the infrared spectrum in '''Figure 2'''.

{| align="center"
|- align="center"
|
{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-234.46920
|-
| Sum of electronic and thermal Energies||-234.46186
|-
| Sum of electronic and thermal Enthalpie||-234.46091
|-
| Sum of electronic and thermal Free Energie||-234.50078
|}
|
[[Image:XJB_IR_anti2.jpg|300px]]
|- align="center"
| align="center" | '''Table 7.''' ''Frequency calculations of anti2 optimized at '''B3LYP/6-31G*'''
([[File:XJB_anti2_reoptimized.log]])''
| align="center" | '''Figure 2.''' ''The generated IR spectrum of optimized '''B3LYP/6-31G*''' structure''
|}

===Optimizing the "Chair" and "Boat" Transition Structures===

The study of the chair and boat transition states is carried out using three different approaches. The first one is called the Hessian which involves computing the force constant matrix; the second one is to freeze the reaction coordinate and then optimize the structure once the molecule is fully relaxed; the last one is to use QST2 method to specify the reactants and products then finding the transition structure.

<center>[[Image:XJB_2 ts.jpg|500px|center|thumb|'''Figure 3'''. ''The two transition states of Cope rearrangement'']]</center>

====The Chair Transition State====

In order to set up a transition state optimization, first an allyl fragment (CH<sub>2</sub>CHCH<sub>2</sub>) was drawn and optimized at the '''HF/3-21G''' level of theory. Then this half of the transition structure was duplicated with rotation to make the approximate resembling of the chair transition state with terminal ends of the fragments 2.2 Å apart. The chair transition structure optimization was then done by the first two different approaches.

<center>[[Image:XJB_guess ts.jpg|400px|center|thumb|'''Figure 4'''. ''The guessed chair transition structure of two allyl fragments'']]</center>

For the first optimization approach, the force constants were computed (the Hessian) and the chair transition state was optimized to a TS (Berny) at the '''HF/3-21G''' level of theory. The frequency calculation was carried out at the same time and gave an imaginary frequency at -818 cm<sup>-1</sup>. The electronic energy was calculated to be '''-231.61932Ha''' with C<sub>2h</sub> symmetry point group. The distance between the terminal carbons was 2.0203 Å. An animation of this transition state vibrations was generated to confirm the corresponding Cope rearrangement.

<center>[[File:XJB_chair ts animation.GIF|400px|center|thumb|'''Figure 5.''' ''Animation of the Cope rearrangement via the chair transition state'']]</center>

The chair transition state was then reoptimized using the '''B3LYP/6-31G*''' level of theory and carried out frequency calculations as well. The electronic energy calculated this time was '''-234.55698Ha''' with an imaginary frequency at -566 cm<sup>-1</sup>. We can see that with the geometries quite similar, the energies differ markedly calculated using two levels of theory. The summary and comparison of the calculations optimized at two levels of theory are summarized in '''Table 8'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46670||-234.41493
|-
| Sum of electronic and thermal Energies||-231.46134||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40807
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir1.jpg|200px]]||[[Image:XJB_chairir2.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 8.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the first approach
([[File:XJB_chairts1.1.log]], [[File:XJB_chairts1.2.log]])''</div>

The second approach is the frozen coordinate method using the Redundant Coord Editor to freeze the bond lengths of the terminal ends to 2.2 Å. First the calculation failed because of the recent update of GaussView. After editting the coordinate distance (B 5 1 2.2 F) and the frozen distance (B 5 1 F) the optimization went well this time. The structure was then optimized again using a normal guess Hessian modified including the two differentiating coordinates. This chair transition state was then reoptimized again using the '''B3LYP/6-31G*''' level of theory. The electronic energies were calculated to be the same as in the first approach and frequency calculations were summarized in '''Table 9'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.46671||-234.41490
|-
| Sum of electronic and thermal Energies||-231.46135||-234.40901
|-
| Sum of electronic and thermal Enthalpie|| -231.46040||-234.40806
|-
| Sum of electronic and thermal Free Energie||-231.49521||-234.44381
|-
| Generated IR spectrum||[[Image:XJB_chairir3.jpg|200px]]||[[Image:XJB_chairir4.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 9.''' ''Frequency calculations of the chair transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by the second approach
([[File:XJB_chairts2.1.log]], [[File:XJB_chairts2.2.log]])''</div>

As the same energy results given, the bond forming/breaking bond lengths in the chair transition structures in the above two different approaches were compared. The first approach gave the bond lengths of 2.0203 Å while the second approach gave 2.0207 Å. Considering the accuracy limit of bond length estimation in Gaussview, these two bond lengths can be seen as exactly the same. Two methods worked reasonably well because of the good assumption of transition structures. However for more complicated and larger molecule, the Hessian method may fail due to the difficulty to predict the transition geometry (different surface curvature). Thus better way to generate such transition structures is to freeze reaction coordinate.

{| align="center"
|- align="center"
|
[[Image:XJB_chairts bondlength1.jpg|350px|thumb|'''Figure 6.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by Hession'']]
|
[[Image:XJB_chairts bondlength2.jpg|350px|thumb|'''Figure 7.''' ''Bond lengths of terminal ends at '''HF/3-21G''' by frozen coordinate'']]
|- align="center"
|}

====The Boat Transition State====

The boat transition structure was then optimized using the '''QST2''' method. The same numbering for the reactant and product molecules was important in order to specify for this reaction. The renumbering of the molecules is shown as in '''Figure 8''', but the following optimization failed and gave a twisted structure. The reactant and product geometries were then modified further by changing the central C-C-C-C dihedral angle to 0<sup>o</sup> and the inside two C-C-C (i.e. C2-C3-C4 and C3-C4-C5) angles to 100<sup>o</sup>. The job was run successfully and frequency calculations were done with the boat transition state generated. Only one imaginary frequency was detected also at -840 cm<sup>-1</sup>. The electronic energy was '''-231.60280Ha''' with C<sub>2v</sub> symmetry. The distances between the terminal carbons of the allyl fragments were examined to be both 2.140Å which were comparably longer than that of the chair transition structure.

{| align="center"
|- align="center"
|
[[Image:XJB_QST1.jpg|400px|thumb|'''Figure 8.''' ''Numbering of reactant and product'']]
|
[[Image:XJB_QST2.jpg|410px|thumb|'''Figure 9.''' ''Re-numbering of reactant and product'']]
|- align="center"
|}

<center>[[File:XJB_boat ts animation.GIF|400px|center|thumb|'''Figure 10.''' ''Animation of the Cope rearrangement via the boat transition state'']]</center>

The structure was then reoptimized at the higher '''B3LYP/6-31G*''' theory of level and the electronic energy was calculated to be '''-234.54309Ha''' with very similar geometry. The frequency calculation generated an imaginary frequency at -530 cm<sup>-1</sup>. The summary and comparison of two levels' calculations are carried out in the following table.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |'''Types of Energies'''
! scope="col" |'''Energy/Hartrees (HF/3-21G)'''
! scope="col" |'''Energy/Hartrees (B3LYP/6-31G*)'''
|-
| Sum of electronic and zero-point Energies||-231.45093||-234.40234
|-
| Sum of electronic and thermal Energies||-231.44530||-234.39601
|-
| Sum of electronic and thermal Enthalpie|| -231.44436||-234.39506
|-
| Sum of electronic and thermal Free Energie||-231.47977||-234.43175
|-
| Generated IR spectrum||[[Image:XJB_boatir1.jpg|200px]]||[[Image:XJB_boatir2.jpg|200px]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 10.''' ''Frequency calculations of the boat transition structure optimized at '''HF/3-21G''' and '''(B3LYP/6-31G*)''' by QST2
([[File:XJB_boat ts1.log]], [[File:XJB_boat ts2.log]])''</div>

====Intrinsic Reaction Coordinate====

Next a useful method called '''Intrinct Reaction Coordinate (IRC)''' is applied to the Hessian optimized chair transition structure to follow the mininmum energy path (MEP) from a transition structure down to its local minimum on a potential energy surface. IRC can be used to verify that the transition state you found actually connects the reactants and products you think it does, or to identify any reaction intermediates you haven't thought about, or to generate an animation of the reaction [ref]. The symmetrical reaction coordinate was computed only in the forward direction with force constant always calculated and the number of points along IRC 50. The last structure generated along IRC is given in '''Figure 11''' with the energy calculated as '''-231.69158Ha''' and the dihedral angle and internal angle measured as D = -67.188<sup>o</sup> A = 111.78<sup>o</sup>. This energy does not match to any of the conformations discussed in Appendix 1 thus a minimum geometry has not reached yet.

{| align="center"
|- align="center"
|
<jmol><jmolApplet><title>XJB_IRC.mol</title><color>white</color>
<size>200</size><script>zoom=5</script>
<uploadedFileContents>XJB_IRC.mol</uploadedFileContents>
</jmolApplet></jmol>
|
[[Image:XJB_irc.jpg|500px]]
|- align="center"
| align="center" | '''Figure 11.''' ''The energy minimum structure along IRC''
| align="center" | '''Figure 12.''' ''Total energy along IRC''
|}

Three improving options can be used to reach the minimum energy. The first one is to run a normal Hessian optimization of the last point on IRC; the second one involves using a larger number of points (in this case 100 was used); the last one is to redo the IRC computation specifying to compute force constant at every step. Three methods were all used and the results were summarized in '''Table 11'''.

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" | Improved IRC method 1
! scope="col" | Improved IRC method 2
! scope="col" | Improved IRC method 3
|-
| [[Image:XJB_IRC1.jpg|300px|center|IRC1]]||[[Image:XJB_IRC2.jpg|300px|center|IRC3]] || [[Image:XJB_IRC3.jpg|300px|center|IRC3]]
|-
| D = -64.17<sup>o</sup> A = 112.04<sup>o</sup> || D = -66.83<sup>o</sup> A = 111.82<sup>o</sup> || D = -64.17<sup>o</sup> A = 112.04<sup>o</sup>
|-
| -231.69167Ha || -231.69150Ha || -231.69167Ha
|-
| [[File:XJB_IRC1.log]]||[[File:XJB_IRC2.log]]||[[File:XJB_IRC3.log]]
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 11.''' ''Comparison of three improvement methods of IRC computation upon the chair transition structure''</div>

It is concluded that method 1 and 3 gave the same the lowest energy which was consistent to the energy given in Appendix 1.

====Activation Energies====

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+
! scope="col" |
! scope="col" | HF/3-21G at 0K
! scope="col" | HF/3-21G at 298.15K
! scope="col" | B3LYP/6-31G* at 0K
! scope="col" | B3LYP/6-31G* at 298.15K
! scope="col" | Expt. at 0K
|-
| '''ΔE (chair)'''||45.71||44.70||34.05||33.16||33.5 ± 0.5
|-
| '''ΔE (boat)'''||55.60||54.76||41.96||41.32||44.7 ± 0.5
|}
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Table 12.''' ''Summary of activation energies for the Cope rearrangement via both transition states in kcal/mol''</div>

==The Diels Alder Cycloaddition==

In this exercise, AM1 semi-empirical method is used to compute the Diels Alder cycloaddition which is a concerted pericyclic reaction. The driving force of this reaction is the formation of a new σ bond between the π orbitals of the dieneophile and the diene. The reaction is allowed with HOMO-LUMO interation and a symmetry overlap.

===Cis Butadiene===

<center>[[Image:prototype Diels Alder.jpg|500px|center|thumb|'''Figure 13'''. ''The prototype reaction between butadiene and ethylene'']]</center>

To study a prototype Diels Alder cycloaddition reaction between butadiene and ethylene, first a cis butadiene molecule was built using Gaussview and optimized using the AM1 semi-empirical molecular orbital method. The HOMO and LUMO of the butadiene were plotted. As seen the HOMO of cis-butadiene is anti-symmetric ('''a''') and the LUMO is symmetric ('''s''') with respect to plane.

{| align="center"
|- align="center"
|
[[Image:XJB_HOMO.jpg|280px|thumb|'''Figure 14.''' ''Anti-symmetric HOMO of cis-butadiene'']]
|
[[Image:XJB_LUMO.jpg|300px|thumb|'''Figure 15.''' ''Symmetric HOMO of cis-butadiene'']]
|- align="center"
|}

===The Transition State of prototype reaction between ethylene and butadiene===

The Hessian method was used to estimate transition state for the reaction between ethylene and butadiene because these two reactants are relatively small and simple molecules. The transition state was drawn by first building a structure as (a) in '''Figure 16''' then removing the -CH<sub>2</sub>-CH<sub>2</sub>- group to form the envelop structure as in (b). The '''clean''' option should not be used and the dashed bond length was guessed to be larger than 2.2 Å otherwise the geometry change would cause a failure in optimization.

<center>[[Image:XJB_dielsguessts.jpg|300px|center|thumb|'''Figure 16'''. ''The guessing transition state for the reaction between butadiene and ethylene'']]</center>

<center>[[File:XJB_butadiene and ethylene ts.GIF|400px|center|thumb|'''Figure 17.''' ''Animation of the butadiene and ethylene cycloaddition transition state'']]</center>

After the Hessian optimization, the transition structure has been successfully determined and an animation was generated in '''Figure 17'''. It has a C<sub>1</sub> symmetry point group and the bond lengths of two partly formed C-C bonds were measured to be 2.12Å. The typical sp<sup>3</sup> and sp<sup>2</sup> C-C bond lengths are 1.54Å and 1.47Å respectively[ref]. The van der Waals radius of the C atom is 1.7 Å[ref]. The measured bond distance of the transition structure is larger than typical bond lengths but still within the van der Waals radius, suggesting the bond forming interation between the carbon atoms.

From the animation ('''Figure 17''') the vibration directions of two molecules are opposite and they are approaching and leaving each other at the same time. This suggests the formation of the two bonds synchronous. The vibrational frequencies were also calculated and an infrared spectrum was simulated ('''Figure 20'''). An imaginary frequency of magnitude -955 cm<sup>-1</sup> was seen and the lowest positive frequency was observed at 147 cm<sup>-1</sup> which corresponded to the reactant vibration but not the bond formation.

{| align="center"
|
[[Image:XJB_HOMOts.jpg|300px|thumb|'''Figure 18'''. HOMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_LUMOts.jpg|300px|thumb|'''Figure 19'''. LUMO of transition state of reaction between cis-butadiene and ethylene]]
|
[[Image:XJB_IRts.jpg|300px|thumb|'''Figure 20'''. Generated IR of transition state of reaction between cis-butadiene and ethylene]]
|}

The HOMO and LUMO molecular orbitals of this transition structure were visualized in '''Figure 18''' and '''Figure 19'''. From these two figures it can be observed that the HOMO at the transition structure is anti-symmetric ('''a''') while the LUMO is symmetric ('''s'''). Thus this anti-symmetric transition state HOMO is formed by the HOMO of cis-butadiene and the LUMO of ethylene with the knowledge that they are both anti-symmetrical. This conclusion agrees with the previous talked conditions for allowed reaction.

===The cyclohexa-1,3-diene reaction with maleic anhydride===

<center>[[Image:XJB_mechanism3.jpg|500px|center|thumb|'''Figure 21'''. ''The endo and exo products of the Diels Alder reaction between cyclohexa-1,3-diene and maleic anhydride'']]</center>