ChemWiki - User contributions [en]

Rep:Mod2:hulahoops312

2011-01-17T19:50:16Z

Dh08: /* '''Module 2:Inorganic Computational Chemistry Lab''' */

='''Module 2:Inorganic Computational Chemistry Lab'''=

=='''Objectives'''==

The aim of this experiment is to use basis sets within Guassian to calculate the total energies, molecular orbitals, geometrical parameters, IR frequencies and distribution of electronic charge for a range of molecules. In addition to this a mini project is carried out to investigate the favoured stereoisomers within the Schlenk equilibrium for a Grignard reaction.

= '''Boron Molecules''' =

=='''Borane, BH3'''==

Borane, BH3 was drawn out on Gaussview 5.0 and was optimised with the calculation method: B3LYP, calculation type: OPT and the basis set: 3-21G. From this the optimised borane structure was produced:

<table border="0" cell padding="0" align="center">
<tr>
<td align="left">Optimised borane structure</td>
<tr>
<td>
<jmol>
<jmolApplet>
<title>Product 7</title><color>white</color>
<size>250</size>
<script>zoom 150</script>
<uploadedFileContents>BH3_optjmoldh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
</td>
</table>

The molecular orbitals were then calculated from the optimised BH3 molecule on Gaussian and were viewed through Gauss View 5.0, which are shown below.
{| align=center
|+ '''Molecular Orbitals of BH3'''<ref name="BH3"> Digital repository for BH3 MO analysis data. [http://hdl.handle.net/10042/to-5273]</ref>
| [[Image:MO_1_(HOMO-3)adh08.jpg|thumb|200x150px|'''MO 1 (HOMO-3)''']]
| [[Image:MO_2_(HOMO-2)adh08.jpg|thumb|200x150px|'''MO 2 (HOMO-2)''']]
| [[Image:MO_3_(HOMO-1)adh08.jpg|thumb|200x150px|'''MO 3 (HOMO-1)''']]
| [[Image:MO_4_(HOMO)adh08.jpg|thumb|200x150px|'''MO 4 (HOMO)''']]
|}
{| align=center
|+
| [[Image:MO_5_(LUMO)adh08.jpg|thumb|200x150px|'''MO 5 (LUMO)''']]
| [[Image:MO_6_(LUMO+1)adh08.jpg|thumb|200x150px|'''MO 6 (LUMO+1)''']]
| [[Image:MO_7_(LUMO+2)adh08.jpg|thumb|200x150px|'''MO 7 (LUMO+2)''']]
| [[Image:MO_8_(LUMO+3)adh08.jpg|thumb|200x150px|'''MO 8 (LUMO+3)''']]
|}

[[Image:BH3_MObdh08.jpg|center]]

From studying the images above it can be seen that there is quite good correlation between the calculated quantitative molecular orbitals and the qualitative LCAO molecular orbitals. This is seen in all the images above apart from MO 8 where they are quite different in appearance and therefore shows that the qualitative approach gives chemists a good idea of the MOs but it still has its limitations.

NBO analysis produced charge distribution for BH3 where the boron atom has a positive charge of +0.198 Debye and the hydrogen atoms have a negative charge of -0.066 Debye each. This therefore means that the hydrogen atoms possess more electron density to that of the central boron due to it being more electropositive.

[[Image:NMO_numbers_BH3dh08.jpg|center|thumb|200x200px|'''Natural Bond Orbital charges for boron and hydrogen''' ]]

'''Frequency'''

This was followed by the vibrational frequencies being calculated from the optimised BH3 molecule and the summary information of this calculation is shown below in table 1. From this it can be seen that the energy of the optimised borane molecule was -26.46 atomic units (-69.49x103 KJ mol-1).

{| class="wikitable" border="1" align=center
|+ '''Table 1: Summary information on the optimised BH3
| Filename || DOUGLASHUNT_BH3_freq
|-
| File Type ||.log
|-
| Calculation Type || FREQ
|-
| Calculation Method ||RB3LYP
|-
| Basis Set ||3-21G
|-
| Charge ||0
|-
| Spin ||Singlet
|-
| E(RB3LYP) || -26.46226338 a.u.
|-
| RMS Gradient Norm ||0.00020662 a.u.
|-
| Imaginary Freq || 0
|-
| Dipole Moment ||0.0000 Debye
|-
| Point Group || D3H
|}

[[Image:IR_Spectra2dh08.jpg|thumb|250px| '''IR spectrum for BH3''' ]]

There are 6 calcualted frequencies for BH3 and we can tell there are 6 frequencies from the 3n-6 rule for non linear molecules.
3 peaks are shown on the IR spectrum since 2 pairs of vibrational states are degenerate and vibration 4 has an intensity of 0, therefore it does not appear. The vibrations are assigned on the spectrum image on the right.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 2: Motions, frequency and intensity of each vibration for trigonal planar BH3
| ''' Vibration No.''' || '''Form of the vibration''' || '''Frequency (cm-1)''' || '''Intensity''' || '''Symmetry of D3h point group'''
|-
| '''1''' || [[Image:Vibration_1_bh3dh08.jpg|thumb| 200px| ]]|| 1145 || 93 || a" 2
|-
| '''2''' || [[Image: Vibration_2_bh3dh08.jpg|thumb|200px| ]]|| 1204 || 12 || e'
|-
| '''3''' ||[[Image: Vibration_3_bh3dh08.jpg|thumb|200px| ]] || 1204 || 12 || e'
|-
| '''4''' ||[[Image: Vibration_4_bh3dh08.jpg|thumb|200px| ]] || 2598 || 0 || a' 1
|-
| '''5''' ||[[Image: Vibration_5_bh3dh08.jpg|thumb|200px| ]]|| 2737 || 104 || e'
|-
| '''6''' ||[[Image: Vibration_6_bh3dh08.jpg|thumb|200px| ]] || 2737 || 104 || e'
|}

=='''Thallium Tribromide, TlBr3'''==

[[Image:Tlbr3chemdrawdh08.jpg|thumb|200x150px|<jmol>
<jmolAppletButton>
<uploadedFileContents>Tlbr3dh08.mol</uploadedFileContents>
<text>Thallium Tribromide, TlBr3</text>
<color>white</color>
</jmolAppletButton>
</jmol>]]

Thallium Tribromide was optimised on Guassian and the summary file (shown below in table 3) shows that the calculation method was RB3LYP, the calculation type was FOPT, the basis set was LANL2DZ and the convergence was achieved at -91.22 atomic units (-239.5 KJ mol-1).

The same method and basis set was used for the frequency calculation so it is comparable to the optimisations calculations.

Frequency analysis is essential within this calculation due to it being the second derivative of the potential energy surface. This means that if all the frequencies are all positive the system is at a minimum, if one of them is negative the optimised system is at a transition state, and if more than one are negative then the calculation has failed to find a critical point, which means the optimisation has failed. This therefore gives us a method of seeing whether the calculation has worked.

{| class="wikitable" border="1" align=center
|+ '''Table 3: Summary information on the optimised TlBr3'''<ref name="TlBr3"> Digital repository for TlBr3 summary information. [http://hdl.handle.net/10042/to-5272]</ref>
| Filename || douglashunt_tlbr3_optimisation
|-
| File Type ||.log
|-
| Calculation Type || FOPT
|-
| Calculation Method ||RB3LYP
|-
| Basis Set ||LANL2DZ
|-
| Charge ||0
|-
| Spin ||Singlet
|-
| E(RB3LYP) ||-91.21812851 a.u.
|-
| RMS Gradient Norm ||0.00000090 a.u.
|-
| Imaginary Freq ||
|-
| Dipole Moment ||0.0000 Debye
|-
| Point Group || D3H
|-
| Job cpu time || days 0 hours 0 minutes 38.0 seconds
|}

By looking at table 4 it can be seen that none of the frequencies are below -5 cm-1 and so shows that the optimisation has been reached and is at a minimum and not a maximum. Therefore the geometry of TlBr3 has been successfully optimised.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 4: Frequencies of TlBr3'''
| '''Low frequencies (cm-1)''' ||-3.42 || -0.0026 ||-0.0004 ||0.0015|| 3.9367 || 3.9367
|-
| '''"Real" normal frequencies (cm-1)''' || 46 || 46 || 52|| 165 ||211 ||211
|}

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5: Optimised parameters of TlBr3'''
! Tl-Br bond distance !! Br-Tl-Br bond angle
|-
| 2.65 Å || 120.0 o
|}

The literature value<ref name="TlBr3 bond distance"> J. Blixt, J. Glaser, J. Mink, I. Persson, P. Persson, M. Sandstroem, J. Am. Chem. Soc., 1995, 117 (18), pp. 5089–5104. [http://pubs.acs.org/doi/abs/10.1021/ja00123a011]</ref>''
for the bond length is 2.41 Å.

Even though the optimised Th-Br bond distance is 0.24 Å longer than the literature length it is of the right magnitude and still very close, which shows that the result is not unreasonable.
Gaussview draws bonds based on distance criteria and bonds can “disappear” on Gaussview due to the distance exceeding a set value. However this does mean the bonds are still there but just that Gaussview has not drawn them. The only reason they are even shown is due to them being a structural convenience to us chemists.

In this case a chemical bond is an attraction between atoms which allows the formation of molecules, which contain two or more atoms. A chemical bond is the attraction caused by the electromagnetic force between opposing charges or between nuclei and electrons and the strength of bonds varies significantly. We chemists represent this interaction as a line (or stick) even though it does not physically resemble this however this helps us understand the molecule in a qualitative way.

= '''An organometallic complex: Isomers of Mo(CO)4L2''' =

=='''Optimisation'''==

The cis isomer of <jmol>
<jmolAppletButton>
<uploadedFileContents>First_roughoptcisdh08.mol2</uploadedFileContents>
<text>cis-Mo(CO)4(PCl3)2</text>
<color>white</color>
</jmolAppletButton>
</jmol> and the trans isomer of <jmol>
<jmolAppletButton>
<uploadedFileContents>1stroughmodel_transdh08.mol</uploadedFileContents>
<text>trans-Mo(CO)4(PCl3)2</text>
<color>white</color>
</jmolAppletButton>
</jmol> were drawn out on Gaussview and were optimised with the B3LYP method, and the level basis set LANL2MB to get the rough geometry. In addition to this opt=loose was typed into the additional key words section<ref name="cis first opt"> Digital repository for the first rough optimisation of cis-Mo(CO)4(PCl3)2. [http://hdl.handle.net/10042/to-5313]</ref> <ref name="trans first opt"> Digital repository for the first optimisation of trans-Mo(CO)4(PCl3)2. [http://hdl.handle.net/10042/to-5312]</ref>.

After this the cis conformer had one chlorine on one group positioned so the bond was parallel with the top axial Mo-C bond while the other chlorine on the other group was positioned so the bond was parallel with the bottom axial bond. Which is shown in the link and image below.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Log_first_opt_cis_right_moveddh08.mol</uploadedFileContents>
<text>cis-Mo(CO)4(PCl3)2</text>
<color>white</color>
</jmolAppletButton>
</jmol>

[[Image:2nd_conformers_cisdh08.jpg |center| 600x300px]]

The trans conformer had both the PCl3 groups eclipsed to each other with both PCl3 groups having one chlorine parallel to one M-C bond. Which is shown in the link and image below.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Trans_refined_roughdh08.mol2</uploadedFileContents>
<text>trans-Mo(CO)4(PCl3)2</text>
<color>white</color>
</jmolAppletButton>
</jmol>

[[Image: 2nd_conformers_transdh08.jpg|center| 600x300px]]

This was followed by the optimisation calculation being carried out again but using the LANL2DZ pseudo-potential and basis sets with "int=ultrafine scf=conver=9" entered within the additional key words section<ref name="cis second opt"> Digital repository for the second optimisation of cis-Mo(CO)4(PCl3)2. [http://hdl.handle.net/10042/to-5315]</ref> <ref name="trans second opt"> Digital repository for the second optimisation of trans-Mo(CO)4(PCl3)2. [http://hdl.handle.net/10042/to-5321]</ref>.

<table border="0" cell padding="0" align="center">
<tr>
<td align="center">Fully optimised cis-Mo(CO)4(PCl3)2</td>
<td align="center">Fully optimised trans-Mo(CO)4(PCl3)2</td>

<tr>
<td>
<jmol>
<jmolApplet>
<title>Fully optimised cis-Mo(CO)4(PCl3)2</title><color>white</color>
<size>200</size>
<script>zoom 150</script>
<uploadedFileContents>Fullyopt_cisdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
</td>
<td>
<jmol>
<jmolApplet>
<title>Fully optimised trans-Mo(CO)4(PCl3)2</title><color>white</color>
<size>200</size>
<script>zoom 150</script>
<uploadedFileContents>Last_opt_transdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
</td>
</table>

From studying table 6 below it can be deduced that the cis isomer is more stable by 2.73 KJ mol-1, which is not a very large energy difference when compared to the literature value of 72.98 KJ mol-1. The reason why my calculated difference is a lot smaller will most likely be due to the literature using the PPh3 ligands and the steric hindrance will have more of an effect upon the energy since PPh3 are much bulkier ligands.

The reason why the cis isomer is more stable is most likely due to the the angle between the Mo and the other ligands being distorted to accommodate the PCl3 ligands and therefore lowering the energy of the complex.

A way to make the trans isomer more stable to that of the cis would be by making the ligand L=PR3 bulkier by using PPh3, PtBu3 or other bulky ligands. This would lead to large amounts of steric hindrance between the PR3 ligands on the cis isomer, leading to it destabilising and so making the trans isomer more stable when compared to the cis isomer.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 6: Total energies of the cis and trans isomers of Mo(CO)4(PCl3)2
| '''Molecule''' || '''Total energy, atomic units''' || '''ΔE, atomic units (cis-trans)''' || '''ΔE, KJ mol-1 (cis-trans)'''
|-
| cis-Mo(CO)4(PCl3)2 || -623.57707196 || -1.04086 x 10-3 || -2.73
|-
| trans-Mo(CO)4(PCl3)2 || -623.57603110
|-
| '''Literature''' <ref name="lit energy values"> D. W. Bennett, T. A. Siddiquee, D. T. Haworth, S. E. Kabir and F. Camellia. J. Chem. Cryst., 34 (6), 2004 pp. 353-359. [http://www.springerlink.com/content/u5383x017700g761/fulltext.pdf]</ref>

|-
| cis-Mo(CO)4(PPh3)2 || -1923.05810468 || -27.79717 x 10-3 || -72.98
|-
| trans-Mo(CO)4(PPh3)2 || -1923.03030751
|}

The bond lengths and angles below are not exactly the same as the literature values due to them being for the complexes with PPh3 ligands.

The reason why the angles for P-Mo-P are not exactly 95 o (cis) and 180 o (trans) for both complexes is due to the electron density on the PCl3 ligands repelling other ligands electrons and hence distorting the angle. This effect is further emphasised within the literature values by the PPh3 ligands being larger and therefore distorting the angle more.

Another point to be made is that the equatorial C-Mo bond length is shorter to that of the axial within the cis isomer. The reason for this is due to the CO acting as an electron withdrawing group to the PCl3 and so gaining electron density. This therefore makes the bond shorter.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 7: Geometric parameters of the cis and trans isomers of Mo(CO)4(PCl3)2'''
| '''cis bonds''' || '''length (Å)''' || '''cis literature values''' <ref name="cis geo lit"> F. A. Cotton, D. J. Darensbourg, S. Klein, B. W. S. Kolthammer, Inorg. Chem., 21, 1982, pp. 294-299. [http://pubs.acs.org/doi/abs/10.1021/ic00131a055]</ref> || '''trans bonds'''|| '''length (Å)''' || '''trans literature values''' <ref name="trans geo lit"> G. Hogarth, T. Norman, Inorganica Chimica Acta., 254, (1), 1997, pp. 167-171. [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TG5-3S9DKVH-V&_user=217827&_coverDate=01%2F01%2F1997&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_searchStrId=1522251498&_rerunOrigin=google&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=d1c6528f637a0440d1fb93dbbf21a9ec&searchtype=a]</ref>
|-
| axial Mo-C || 2.06 || 2.022 and 2.059 || axial Mo-C || 2.45 || -
|-
| equatorial Mo-C || 2.01 || 1.972 || equatorial Mo-C || 2.45 || 2.01
|-
|P-Mo || 2.51 || 2.58 || P-Mo || 2.45 || 2.50
|-
| P-Cl || 2.24 || - || P-Cl || 2.24 || -
|-
| triple bond CO || 1.17 || - || triple bond CO || 1.17 || -
|-
| '''cis bonds''' || '''angle''' || || '''trans bonds''' || '''angle''' ||
|-
| P-Mo-P || 94.2 o || 104.62 o || P-Mo-P || 177.4 o || -
|-
| Cl-P-Mo-C (dihedral angle) || 179.8 o || - ||Cl-P-Mo-C (dihedral angle) || 174.7 o || -
|}

=='''Frequency'''==

The vibrational frequencies were then calculated with "int=ultrafine scf=conver=9" as additional keywords.

Below in table 8 and 9 are the lowest frequencies for the vibrations of the two isomers. Within the animations below it can be seen that the chlorine atoms are rotating around their phosphorus and the molybdenum centre. This would happen due to the amount of energy required to rotate a bond being very small (in comparison to stretching or bending) and therefore would be taking place at room temperature.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 8: Motions, frequency and intensity for the low energy vibrations of cis-Mo(CO)4(PCl3)2<ref name="cis freq"> Digital repository for the frequency calculation of cis-Mo(CO)4(PCl3)2. [http://hdl.handle.net/10042/to-5325]</ref>
| '''Animation''' || '''Annotation''' || '''Frequency (cm-1)''' || '''Intensity'''
|-
| [[Image:10_73_cis_animationdh08.gif]] ||[[Image: 10.73_rotatedh08.jpg|thumb|200px| ]] || 11|| 0
|-
| [[Image:17_61_cis_animationdh08.gif]] || [[Image:17.61_rotatedh08.jpg|thumb|200px| ]] || 18 || 0
|}

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9: Motions, frequency and intensity for the low energy vibrations of trans-Mo(CO)4(PCl3)2<ref name="trans freq"> Digital repository for the frequency calculation of trans-Mo(CO)4(PCl3)2. [http://hdl.handle.net/10042/to-5322]</ref>
| '''Animation''' || '''Annotation''' || '''Frequency (cm-1)''' || '''Intensity'''
|-
| [[Image:5_06_trans_animationdh08.gif|5_06_trans_animationdh08.gif]] ||[[Image: 5.06dh08.jpg|thumb|200px| ]] || 5 || 0
|-
| [[Image:6_05_trans_animationdh08.gif ]] || [[Image:6.05_rotatedh08.jpg|thumb|200px| ]] || 6 || 0
|}

Within the cis isomer there are four distinctive peaks and within the trans isomer there are only three. This is also mentioned within literature<ref name="C=O lit2"> A. D Allen, P.F. Barrett, Ca. J. Chem., 1968, 46, pp. 1650-1651. [http://article.pubs.nrc-cnrc.gc.ca/RPAS/rpv?hm=HInit&calyLang=eng&journal=cjc&volume=46&afpf=v68-276.pdf]</ref> and was seen within my second year experiment. The reason why the trans isomer has one less peak is due to it having D4h symmetry and not C2v, which leads to the trans isomer having two chemically equivalent stretches (seen in the animations below) at 1950.28 and 1950.91 cm-1 that are therefore symmetrically degenerate. This gives this one peak a very large intensity compared to the other two peaks.

The other two peaks within the trans isomer are very small in intensity and this is due to the isomer having high symmetry. Therefore the C=O bonds cancel each other out, which is shown in the animations below and so the intensity of these stretches is close to zero.

However within the cis isomer (C2v point group) the peaks at 1945.31 and 1948.66 are chemically inequivalent C=O bonds that are stretching and therefore they give two distinctive peaks.

The literature and experimental frequencies are also higher to that of the calculated frequencies. One reason for this will be due to basis set over compensating within the calculations and so over estimating the frequencies. Additionally the calculation may not take into account back bonding within the metal complexes between the CO ligands and the metal centre which would weaken the CO bond and so lower the CO frequencies.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10: Motions, frequency and intensity for the carbonyl vibrations of the cis and trans isomers of Mo(CO)4(PCl3)2
| '''cis-Mo(CO)4(PCl3)2'''
|-
| '''Animation''' || '''Annotation''' || '''Frequency (cm-1)''' || '''Intensity''' || '''Literature values (for the PPh3 ligand)(cm-1)''' <ref name="C=O lit"> J. Shamir, A. Givan, M. Ardon, G. Ashkenazi, J. Raman Spect., 24, 1993, pp. 101-105.[http://onlinelibrary.wiley.com/doi/10.1002/jrs.1250240208/abstract]</ref> || '''2nd year experimental values (for the PPh3 ligand)(cm-1)'''
|-
| [[Image:1945_31_CO_andh08.gif]] ||[[Image: 1945_31_COdh08.jpg|thumb|200px| ]] || 1945 || 763 || 1867 || 1840
|-
| [[Image:1948_66_CO_andh08.gif]] || [[Image:1948_66_COdh08.jpg |thumb|200px| ]] || 1948 || 1498 || 1896 || 1870
|-
| [[Image:1958_36_CO_andh08.gif]] ||[[Image:1958_36_COdh08.jpg|thumb|200px| ]] || 1958 || 633 || 1924 || 1896
|-
| [[Image:2023_31_CO_andh08.gif]] || [[Image:2023_31_COdh08.jpg|thumb|200px| ]] || 2023 || 598 || 2026 || 2024
|-
| '''trans-Mo(CO)4(PCl3)2'''
|-
| '''Animation''' || '''Annotation''' || '''Frequency (cm-1)''' || '''Intensity''' || '''Literature values (for the PPh3 ligand) (cm-1)''' <ref name="C=O lit"> J. Shamir, A. Givan, M. Ardon, G. Ashkenazi, J. Raman Spect., 24, 1993, pp. 101-105.[http://onlinelibrary.wiley.com/doi/10.1002/jrs.1250240208/abstract]</ref> || '''2nd year experimental values (for the PPh3 ligand)(cm-1)'''
|-
| [[Image:1950_28_CO_andh08.gif]] ||[[Image: 1950_28_COdh08.jpg|thumb|200px| ]] || 1950 || 1475 || 1889 || 1890
|-
| [[Image:1951_91_CO_andh08.gif]] || [[Image:1950_91_COdh08.jpg |thumb|200px| ]] || 1951 || 1467 || 1889 || 1890
|-
| [[Image:1977_19_CO_andh08.gif]] ||[[Image:1977_19_COdh08.jpg|thumb|200px| ]] || 1977 || 1 || 1924|| 1943
|-
| [[Image:2030_98_CO_andh08.gif]] || [[Image:2030_98_COdh08.jpg|thumb|200px| ]] || 2031 || 4 || 2026 || 2019
|}

{| align=center
|+
| [[Image:IR_spectra_Cisdh08.jpg|thumb|250px| '''IR spectrum for cis-Mo(CO)4(PCl3)2''' ]]
| [[Image:Trans_freq_spectrumdh08.jpg|thumb|250px| '''IR spectrum for trans-Mo(CO)4(PCl3)2''' ]]
|}

= '''Mini Project''' =

Grignard reagents are alkyl/aryl-magnesium halides which act as nucleophiles by attacking electrophilic carbon atoms that are within polar bonds to create a carbon-carbon bond. They were discovered by Barbier and Grignard in 1900 and were such a significant tool in the formation of carbon-carbon bonds Grignard won a shore of the Nobel prize in 1912.

They are generally solvated by ether, which can lead to halide bridging within solution. A concise description of the situation is given by the Schlenk equilibrium. This is where the solutions of Grignard reagents can contain a variety of chemical species interlinked by equilibria, which is shown below.

[[Image:Schlenk_IMAGEdh08.jpg|center|600px]]

The relative amount of each species present depends upon:

- The size and electronic nature of the R group.

- The size and electronic nature of the halide group, X.

- The nature of the solvent.

- Concentration and temperature.

Since the X group changes the ratio of the two isomers within the equilibrium four structures will be drawn out with two being with chlorine bridging atoms (cis and trans) and the other two having bromine as the bridging atoms. From this the thermodynamically favoured isomer can be determined, the IR spectra and bond lengths/angles can be compared and the trans chlorine isomer's molecular orbitals can be analysed.

Therefore the main aim of this mini project is to see whether the bridging halides favour one isomer within the Schlenk equilibrium over another.

=='''Optimisation'''==
The
<jmol>
<jmolAppletButton>
<uploadedFileContents>Trans_Cl_drawndh08.mol</uploadedFileContents>
<text>trans chloride isomer</text>
<color>white</color>
</jmolAppletButton>
</jmol>, <jmol>
<jmolAppletButton>
<uploadedFileContents>Trans_Br_drawndh08.mol</uploadedFileContents>
<text>trans bromide isomer</text>
<color>white</color>
</jmolAppletButton>
</jmol>, <jmol>
<jmolAppletButton>
<uploadedFileContents>Cis_Cl_drawndh08.mol</uploadedFileContents>
<text>cis chloride isomer</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Cis_Br_drawndh08.mol</uploadedFileContents>
<text>cis bromide isomer</text>
<color>white</color>
</jmolAppletButton>
</jmol>
were all drawn on Gaussview 3.0 and were optimised with the B3LYP method, and the basis set 6-311G (d, p).

However the optimisation of the cis bromide isomer and cis chloride isomer failed at the first few steps of the optimisation. This was due to the hydrogen atoms on the diethyl ether being too close (error in the internal coordinate system) and so the Br/Cl-Mg bonds were increased to 3 Å and the diethyl ethers were tilted diagonally so they did not clash. The jmol files of both of these new conformers are below.

<jmol>
<jmolAppletButton>
<uploadedFileContents>Cis_br_opt_twodh08.mol</uploadedFileContents>
<text>cis bromide isomer</text>
<color>white</color>
</jmolAppletButton>
</jmol> <jmol>
<jmolAppletButton>
<uploadedFileContents>Cis_Cl_opt_twodh08.mol</uploadedFileContents>
<text>cis chloride isomer</text>
<color>white</color>
</jmolAppletButton>
</jmol>

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11: Total energies of the cis and trans chloride isomers
| '''Molecule''' || '''Total energy, atomic units''' || '''ΔE, atomic units (cis-trans)''' || '''ΔE, KJ mol-1 (cis-trans)'''
|-
| <jmol>
<jmolAppletButton>
<uploadedFileContents>Cis_Cl_opt_two_afterdh08.mol</uploadedFileContents>
<text>cis </text>
<color>white</color>
</jmolAppletButton>
</jmol> <ref name="cis chloride opt"> Digital repository for the cis chloride isomer optimisation data. [http://hdl.handle.net/10042/to-5422]</ref> || -1868.21241703 || 4.4579 x 10-3 || 11.71
|-
| <jmol>
<jmolAppletButton>
<uploadedFileContents>Trans_Cl_optdh08.mol</uploadedFileContents>
<text>trans </text>
<color>white</color>
</jmolAppletButton>
</jmol> <ref name="trans chloride opt"> Digital repository for the trans chloride isomer optimisation data. [http://hdl.handle.net/10042/to-5421]</ref> || -1868.21687493
|}

From the table 11 it can be deduced that trans chloride isomer is more stable in terms of it's total energy compared to the cis conformer. This will most likely be due to the steric hindrance between the diethyl ether substituents in the cis conformer which would destabilise it. Therefore the "nature" of the solvent favours the trans conformation.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12: Total energies of the cis and trans bromide isomers
| '''Molecule''' || '''Total energy, atomic units''' || '''ΔE, atomic units (cis-trans)''' || '''ΔE, KJ mol-1 (cis-trans)'''
|-
| <jmol>
<jmolAppletButton>
<uploadedFileContents>Cis_br_opt_two_afterdh08.mol</uploadedFileContents>
<text>cis </text>
<color>white</color>
</jmolAppletButton>
</jmol> <ref name="cis bromide opt"> Digital repository for the cis bromide isomer optimisation data. [http://hdl.handle.net/10042/to-5434]</ref> || -6096.05182941 || 6.27241 x 10-3 || 16.47
|-
| <jmol>
<jmolAppletButton>
<uploadedFileContents>Trans_br_optdh08.mol</uploadedFileContents>
<text>trans </text>
<color>white</color>
</jmolAppletButton>
</jmol> <ref name="trans bromide opt"> Digital repository for the trans bromide isomer optimisation data. [http://hdl.handle.net/10042/to-5423]</ref> || -6096.05810182
|}

By studying table 12 it can be seen that the trans bromide isomer is also more stable by comparing total energies. This again is due to the steric clash between the large diethyl ether substituents and so the "nature" of the solvent overrides any affect of the bridging halide groups. This therefore means that the diethyl ether solvent favours the trans conformation.

The bromide Grignard reagent in both conformations within the Schlenk equilibrium are considerably more stable to the chloride Grignard reagent. This is quantitatively shown by the bromo isomer being 4227.841227 atomic units (11.1 x 106 KJ mol-1) lower in energy.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 13: Geometric parameters of the cis and trans chloride isomers'''
| '''bonds''' || '''cis length (Å)''' || '''trans length (Å)''' ||'''literature values''' <ref name="chloro geo lit"> P. Sobota, T. P. Ski, T. Lis, Inst. of Chem., 14, 1983, pp. 45-47. [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TH8-42VWKYW-HK&_user=217827&_coverDate=12%2F31%2F1984&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_searchStrId=1522972144&_rerunOrigin=google&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=47ee8b16115d4b99b57651524b05027f&searchtype=a]</ref>
|-
| Mg-Cl || 2.46 || 2.48 || 2.501
|-
|Mg-Me || 2.10 || 2.10 || -
|-
| Mg-O || 2.14 || 2.14 || 2.111
|-
| || '''cis angle''' || '''trans angle''' ||
|-
| Cl-Mg-Cl || 88.7 o || 89.3 o || -
|-
| Mg-Cl-Mg || 89.9 o || 90.4 o || -
|-
| Cl-Mg-O || 100.1 o || 101.7 o|| -
|-
| Cl-Mg-Me || 124.2 o || 124.3 o|| -
|}

From looking at table 13 it can be stated that the positioning of the L groups and R groups does not change the bond lengths/angles around the bridging halides. The only slight difference is the trans Mg-Cl length being 0.02 Å longer, which is such a small difference it could be down the inaccuracies of the simulation. However from these geometries it can deduced that the optimisation went well by the bond lengths being the right order of magnitude when compared to the literature values.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 14: Geometric parameters of the cis and trans bromide isomers'''
| '''bonds''' || '''cis length (Å)''' || '''trans length (Å)''' ||'''literature values''' <ref name="bromo geo lit"> S. Mantey, C. Liebenow, E. Hecht, Z. Anorg. Allg. Chem., 627, 2001, pp. 128-130. [http://onlinelibrary.wiley.com/doi/10.1002/1521-3749%28200102%29627:2%3C128::AID-ZAAC128%3E3.0.CO;2-C/pdf]</ref>
|-
| Mg-Br || 2.64 || 2.35|| 2.6008
|-
|Mg-Me || 2.11 || 2.13 || -
|-
| Mg-O || 2.14 || 2.02 || -
|-
| || '''cis angle''' || '''trans angle''' ||
|-
| Br-Mg-Br || 91.9 o || 97.1 o || -
|-
| Mg-Br-Mg || 86.9 o || 80.7 o || -
|-
| Br-Mg-O || 102.4 o || 112.3 o|| -
|-
| Br-Mg-Me || 123.1 o || 114.4 o|| -
|}

From examining the parameters in table 14 it can be seen that they differ between the cis and trans bromide isomers. This mainly shown by the cis Mg-Br bond being 0.29 Å longer. This is most likely due to steric hindrance between the diethyl ethers with bridging bromides within such a small amount of space and so the structure distorts. It does this by lengthening the bridging bonds to lower the amount of repulsion within the conformation. This may possibly be the same reason to why the Mg-O bond is longer too in the cis isomer.
However there is a possibility that the accuracy of the calculation was very low and so one of the sets of data is wrong but no errors showed up in any of the log files.

=='''Frequency'''==

Again the frequency calculation was carried out with the B3LYP method, and the basis set 6-311G (d, p).

Within both forms of literature<ref name="lit grignard"> R. I. Yousef, B. Walfort, T. Ruffer, Journ. Organomet. Chem., 690, 2005, pp. 1178-1191. [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGW-4F4NYJ2-4&_user=217827&_coverDate=03%2F01%2F2005&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_searchStrId=1522675733&_rerunOrigin=google&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=e0db19c9be9dc4ef6913a5dac0543e9c&searchtype=a]</ref> <ref name="lit grignard2"> J. Tammiku-Taul, P. Burk, A. Tuulmets, J. Phys. Chem., 108, 2004, pp. 133-139. [http://pubs.acs.org/doi/abs/10.1021/jp035653r]</ref> the academics did not carry out any IR spectroscopy and mainly used 1H NMR and X-ray crystallography to analyse their Grignard compounds. However the frequencies were calculated within this mini project and then examined.

The vast majority of the vibrations within the four isomers were from the diethyl ether substituents and the methyl groups. However the ones which we should be interested in are ones which involve any halogen or magnesium bond vibrating with a significant intensity. In table 15 and 16 below only one of these vibrations were found in which the magnesium atoms were moving without a low intensity. Within both trans isomers the same stretch had higher intensities to that of the cis. This could be due to the two methyl groups on either side of the trans isomers constructively adding to the vibration of the magnesium atoms.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 15: Motions, frequency and intensity for the vibrations of the cis<ref name="cis freq chloro"> Digital repository for the frequency calculation of the cis chloro isomer. [http://hdl.handle.net/10042/to-5430]</ref> and trans<ref name="trans freq chloro"> Digital repository for the frequency calculation of the trans chloro isomer.[http://hdl.handle.net/10042/to-5431]</ref> chloride isomers
| '''Isomer''' ||'''Animation''' || '''Frequency (cm-1)''' || '''Intensity'''
|-
| trans || [[Image:No_39_541_130dh08.gif]] || 541 || 130
|-
| cis || [[Image:No_39_539_96dh08.gif]] || 539 || 96
|}

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 16: Motions, frequency and intensity for the vibrations of the cis<ref name="cis freq bromo"> Digital repository for the frequency calculation of the cis bromo isomer. [http://hdl.handle.net/10042/to-5439]</ref> and trans<ref name="trans freq bromo"> Digital repository for the frequency calculation of the trans bromo isomer. [http://hdl.handle.net/10042/to-5440]</ref> bromide isomers
| '''Isomer''' ||'''Animation''' || '''Frequency (cm-1)''' || '''Intensity'''
|-
| trans || [[Image:No_39_535_125.gif]] || 535 || 125
|-
| cis || [[Image:No_39_533_80dh08.gif]] || 533 || 80
|}

The low frequencies were also checked to see whether all four molecules optimised successfully. Ideally we would want the lowest frequency to be no less than -5 cm-1, however since there are a large number of hydrogens on the molecules this will be hard to achieve. The reason for this is that the hydrogens require a low level of energy to rotate and so you obtain such low frequencies.

Even though 3 of the 4 low frequencies are less than -5 cm-1 they are significantly smaller negative numbers that what were originally calculated in the first attempt of the optimisation of both cis isomers.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 17: Low frequencies for all four isomers
| '''Isomer''' ||'''Low frequency (cm-1) :''' || '''1''' || '''2''' || '''3''' || '''4''' || '''5''' || '''6'''
|-
| trans chloro || ||-1.1083 || -0.0012 || -0.0011 || -0.0011 || 6.5409 || 8.6270
|-
| cis chloro || ||-12.9612 || -0.0028 || -0.0027 || -0.0023 || 2.3104 || 7.5693
|-
| trans bromo || || -13.5727 || -5.5025 || -0.0043 || -0.0032 || 0.0050 || 5.0600
|-
| cis bromo || || -6.6564 || -0.0081 || -0.0081 || -0.0068 || 4.4325 || 5.8360
|}

=='''Molecular Orbitals and Natural Bond Orbitals'''==

The molecular orbitals were calculated by setting the job type as energy with the B3LYP method, and the basis set 6-311G (d, p). In addition to this the "Full NBO" option was chosen under the NBO tab and "pop=full" was added as additional key words.

From studying the table below it can be seen that after molecular orbital number 38 the interactions become much more anti boning to that of bonding by there being more nodes within the molecule. This would destabilise the entire molecule since these orbitals are occupied and so explains why this molecule only exists temporarily in equilibrium while in solution. The orbitals on diethyl ether substituents within the cis chloride isomer could interact across the halide bridging and further destabilise the molecule by having more anti-boning interactions. Another feature of the molecular orbitals is that the R and L groups do not interact with the magnesium or halide bridging until the higher valence electrons.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 18: Molecular Orbitals of the trans chloride isomer'''<ref name="trans chloride"> Digital repository for the trans chloride isomer MO analysis data. [http://hdl.handle.net/10042/to-5419]</ref>'''
| '''Molecular orbital number''' || '''Energy''' || '''Images''' || '''Description'''
|-
| 1 to 4 || -101.50454 to -46.76677 || [[Image:Mo_4_nothingdh08.jpg|thumb|200x200px]] || No molecular orbitals are being shown
|-
| 5 to 16 || -19.17335 to -10.08695 || [[Image:Mo_7_CARBONdh08.jpg |thumb|200x200px]] || Oxygen (molecular orbital numbers 5-6) and carbon S orbitals
|-
| 17 to 18 || -9.41979 to -9.41758 || [[Image:Mo_18_CLdh08.jpg |thumb|200x200px]] || S orbitals of chlorine
|-
| 19 to 24 || -7.17860 to -7.17458 || [[Image:Mo_19_CL_P_orbitaldh08.jpg |thumb|200x200px]] || P orbitals of chlorine
|-
| 25 to 26 || -3.07208 to -3.07102 || [[Image:Mo_25_mg_S_orbitaldh08.jpg |thumb|200x200px]] || S orbitals of magnesium
|-
| 27 to 32 || -1.81923 to -1.81376 || [[Image:Mo_28_mg_P_orbitaldh08.jpg |thumb|200x200px]] || P orbitals of magnesium
|-
| 33 to 34 || -1.06952 to -1.06947 || [[Image:Mo_34dh08.jpg |thumb|200x200px]] || S Orbitals have merged on the diethyl ethers
|-
| 35 to 36 || -0.80315 to -0.79860 || [[Image:Mo_35_AB_orbitals_DEdh08.jpg |thumb|200x200px]] || Anti bonding on opposite ends the diethyl ether substituents by the S orbitals merging
|-
| 37 to 38 || -0.78004 to -0.76977 || [[Image:Mo_38_AB_orbitals_Cldh08.jpg |thumb|200x200px]] || Bonding interaction (37) and anti bonding (38) interaction between chlorine atoms
|-
| 39 to 62 || -0.75981 to -0.38839 || [[Image: Mo_60_AB_orbitals_DE_and_Medh08.jpg|thumb|200x200px]] || Anti bonding orbitals on the diethyl ether substituents between the hydrogen S orbitals
|-
| 63 to 78 || -0.38038 to -0.29564 || [[Image: Mo_74_AB_AND_B_orbitals_DE,_Me_and_Cl_P_orbitalsdh08.jpg|thumb|200x200px]] || A mixture of bonding and anti bonding interactions between diethyl ether, the methyl groups and the chlorine atoms. This is the first evidence that the R and L groups are now bonding with the bridging halides.
|-
| 79 (HOMO-1) || -0.21924 || [[Image: Mo_79_mg,_cl,_medh08.jpg|thumb|200x200px]] || Anti bonding between methyl groups and magnesium, which is the first bonding interaction between the R/L groups and magnesium. Chlorine atoms are non-bonding
|-
| 80 (HOMO) || -0.21276 || [[Image:Mo_80_mg,_cl,_me_more_ABdh08.jpg |thumb|200x200px]] || Anti bonding between methyl groups and magnesium atoms with chlorine this time having more of an anti bonding role. If any electrophile were to attack it would most likely do so at the Mg-Me bond since it has the highest amount of electron density within the HOMO.
|-
|81 (LUMO) || +0.00617 || [[Image:Mo_81_LUMO_AB_DE_Me_Cl_with_Mgdh08.jpg |thumb|200x200px]] || Anti bonding between the magnesium atoms with the diethyl ethers, methyl groups and chlorine atoms. Here the electrons are far more delocalised than any of the previous molecular orbitals.
|}

{| align=center
|+ '''Natural Bond Orbitals for the trans chlorine isomer'''
| [[Image:NBO1dh08.jpg|thumb|200x150px|'''NBO from view 1''']]
| [[Image:NBO2dh08.jpg|thumb|200x150px|'''NBO from view 2''']]
|}

By examining the images above it can be distinguished that the chlorine bridging atoms and the oxygen atoms have a large amount of electron density. This will be of course due to them being more electronegative to the hydrogen, carbon and magnesium atoms. In fact they get the vast majority of their electron density from the electropositive magnesium and therefore there will be a considerable amount of ionic character within the bonds.

=='''Conclusion'''==

I conclude that the type of halide bridging does not favour one isomer within the Schlenk equilibrium with the L group (diethyl ether) and R group (methyl group) I picked. This is due to the solvent being too bulky and therefore favours the trans conformer due to steric factors and so the size and electronic structure of the halide bridging it not a decisive factor. From my optimisation total energies I can also conclude that the bromide Grignard reagent is more stable to the chloride one. The energy difference is quite a large value and again is most likely to do with the bromine atoms being bigger and so the L group are further apart so there is no clash in electron orbitals.

From this I also learnt that the vibrational analysis did not tell me a lot about the electronics of the molecule and so understood why in the literature they did not run IR upon their Grignard reagents but carried out x-ray crystallography to determine the structure of the reagents.

='''References'''=
<references/>

Rep:Mod:hampluscheese

2011-01-17T19:49:36Z

Dh08: /* '''Module 3:Physical Computational Chemistry Lab''' */

='''Module 3:Physical Computational Chemistry Lab'''=

=='''Objectives'''==

The aim of this experiment is to is to use various levels of theory (HF/3-21G, DFT/B3LYP/6-31G (d) and AM1/semi-empirical) on Gaussian to calculate various energies, molecular orbitals, geometrical parameters, reaction coordinate pathways and low and imaginary frequencies for a range of molecules and transition states. This will include using different optimisation methods throughout and will lead to a range of findings in various areas of chemical interest.

=='''The Cope Rearrangement'''==

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangement] is a [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement |3,3|-sigmatropic rearrangement] that takes place through a single [http://en.wikipedia.org/wiki/Pericyclic pericyclic], concerted [http://en.wikipedia.org/wiki/Transition_state transition state]. Within this rearrangement a σ bond is formed in concert with another being broken within a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] π system.

The Cope rearrangement we will be studying will be that of 1,5-hexadiene, which is shown below.

[[Image:Cope_rearrangement_schemedh08.jpg|300x300px|center]]

The objective is to locate the low-energy minima and transition structures on the C6H10 potential energy surface and to determine the preferred reaction mechanism by using [http://en.wikipedia.org/wiki/GAUSSIAN Gaussian], with the [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)] methods.

==='''Optimisation of 1,5-hexadiene Conformers'''===

Rough <jmol>
<jmolAppletButton>
<uploadedFileContents>Anti_2_drawndh08.mol</uploadedFileContents>
<text>anti</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche3_drawndh08.mol</uploadedFileContents>
<text>gauche</text>
<color>white</color>
</jmolAppletButton>
</jmol> conformers of 1,5-hexadiene were drawn on Gaussview and their structures were tidied by the clean function. They were then further optimised by the HF/3-21G method and were identified which conformer they were from their point group and appendix 1 on the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3 physical lab page].

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 1: Optimisation and comparison of anti and gauche 1,5-hexadiene conformers using HF/3-21G methods
| ''' Conformer''' || '''Optimised Structure''' || '''Energy (Atomic Units)''' || '''[http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1 Energies (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (Anti2 - Gauche3)''' || '''Point group'''
|-
| '''Anti2''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Anti_2_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692535 || -231.69254 ||-6.08 x 105 || 1.2606 x 10-4 a. u|| Ci
|-
| '''Gauche3''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Gauche3_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692661 || -231.69266 || -6.08 x 105 || 3.31 x 10-1 KJ mol-1 || C1
|}

From studying table 1 it can be seen that the gauche conformer is 0.33 KJ mol-1 more stable than the anti conformation. One would assume that the anti2 conformer would be more stable due to [http://en.wikipedia.org/wiki/Steric steric] factors/repulsion within the gauche3 conformation. However within the gauche3 conformer there is a favourable donation of [http://en.wikipedia.org/wiki/Electron_density electron density] from the C=C π orbital to the C-H σ* orbital from the adjacent proton, which stabilises the conformer<ref name="ref 1">B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446.[http://www.informaworld.com/smpp/content~db=all?content=10.1080/00268970110081412]</ref>.

 

The anti2 conformer was then optimised using the DFT/B3LYP/6-31G* (6-31G (d)) method.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 2: Comparison of the anti2 conformer optimisations carried out at different calculation methods
| ''' Method''' || '''Structure and bond distances (Å)''' || '''Dihedral angle (C1 to C4)''' || '''Energy (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (HF/3-21G - DFT/B3LYP/6-31G*)'''
|-
| HF/3-21G || [[Image:Anti2_opt1dh08.jpg|300x300px]] || 114.6 o || -231.692535 || -6.08 x 105 || 2.919171 a. u
|-
| DFT/B3LYP/6-31G* || [[Image:Anti2_opt2DH08.jpg|300x300px]] || 118.8 o || -234.611716 || -6.16 x 105 || 7.66 x 103 KJ mol-1
|}

From examining table 2 at first it is possible to see that the bond lengths are very similar to each other and that of the literature bond lengths<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref> (C1=C2: 1.340 Å, C2-C3: 1.508 Å and C3-C4: 1.538 Å). It is also possible to see that the C=C bond length is more accurate for the DFT/B3LYP/6-31G* method by it being 0.01 Å closer to the literature value. Another main difference in terms of geometry is that the DFT/B3LYP/6-31G* method gives a larger [http://en.wikipedia.org/wiki/Dihedral_angle dihedral angle] (and most likely more accurate) of 118.8o compared to that of 114.6o.

In terms of energy the DFT/B3LYP/6-31G* method produces a conformer of a higher stability by it possessing a lower energy with a difference of 7.66 x 103 KJ mol-1.

==='''Frequency analysis of 1,5-hexadiene Conformers'''===

[[Image:Vibrations_anti2dh08.jpg|right|300x300px]]

For the frequency analysis the DFT/B3LYP/6-31G* method was used.

Frequency analysis is essential within this calculation due to it being the second derivative of the potential energy surface. This means that if all the frequencies are all positive the system is at a minimum, if one of them is negative the optimised system is at a transition state, and if more than one are negative then the calculation has failed to find a critical point, which means the optimisation has failed. This therefore gives us a method of seeing whether the calculation has worked.

From looking to the image on the right it can deduced that the optimisation has taken place successfully by there being no negative vibrations and therefore the conformation is at a minimum.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 3: Energies under the thermochemistry section at 298.15 K and 0 K within the frequency logfiles'''
| '''Energies''' || '''298.15 K (atomic units)''' ||| '''0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (KJ mol-1)''' || '''Discussion'''
|-
| Sum of electronic and zero-point energies || -234.469177 || -234.466673 || -2.504 x 10-3 || -6.57 || This is the potential energy at 0 K, which includes the zero-point vibrational energy (E = Eelec + ZPE).

|-
| Sum of electronic and thermal energies || -234.461835 || -234.461085 || -7.5 x 10-4|| -1.97 || This is the energy at 298.15 K (0 K for 0 K calculation) and 1 atm of pressure, which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans).

|-
| Sum of electronic and thermal enthalpies || -234.460891 || -234.460141 || -7.5 x 10-4 || -1.97 || This contains an additional correction for RT (H = E + RT), which is especially important when looking at dissociation reactions.
|-
| Sum of electronic and thermal free energies || -234.500739 || -234.495931 || 4.808 x 10-3 || -12.62 || This includes the entropic contribution to the free energy (G = H - TS).
|}

{|align=right
|+
| [[Image:Anti2_freq_IRdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 298.15 K''']]
| [[Image:Anti2_freq_0Kdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 0 K''']]
|}

From looking at table 3 above it can be deduced that the anti2 conformer is more stable at room temperature to that of 0 K with respect to all the types of energies. The spectra also seem to not change much between the two temperatures but the main change is the change in the sum of electronic and thermal free energies, which is heavily dependent of temperature by it including the entropic contribution to the free energy (G = H - '''T'''S).

 
 
 
 

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

Within the Cope Rearrangement there are two possible transition states, the chair (C2h) and the boat (C2v), which are shown within [https://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_2 appendix two ].

==='''Comparison of optimisation methods for the chair transition state'''===

An <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2hdh08.mol</uploadedFileContents>
<text>allyl fragment</text>
<color>white</color>
</jmolAppletButton>
</jmol>, CH2CHCH2 was drawn on Gaussview and was optimized through the HF/3-21G method. After this the original allyl fragment was copied and pasted into a new window and placed exactly above the first fragment with a bond distance of 2.20 Å between the terminal end carbons. This therefore produced the <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2h_fulldh08.mol</uploadedFileContents>
<text>chair transition structure.</text>
<color>white</color>
</jmolAppletButton>
</jmol>

Gaussian was then set up for an optimisation for the transition state with the additional key words "Opt=NoEigen", the optimisation set to TS (Berny), the force constants set to "once" and using the HF/3-21G method.

After this the "chair" transition state was optimised an alternative way by freezing the coordinates of the two sets of terminal carbons (where the bonds form/break during the rearrangement) within the Redundant Coord Editor on Gauusview.
Following this a second calculation was run where the freeze coordinates option was changed to "Derivative" and the force constants were set to "never".

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 4: Comparison of chair transition state optimisation methods
| || '''Optimised through a TS (Berny)''' || '''Optimised through a frozen coordinate method'''
|-
| '''Optimised TS Structure''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>C2h_optfreq_TS_Bdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| '''Terminal C-C bond length (Å)''' || 2.02 || 2.02
|-
|'''Energy (atomic units)''' || -231.61932247 || -231.61932190
|-
| '''ΔE, TS (Berny) - frozen coordinate''' || 5.7 x 10-7 a.u || 1.50 x 10-3 KJ mol-1
|-
| '''Imaginary frequency (cm-1)''' || -818 || -818
|-
| '''Animation''' ||[[Image:818dh08TSB.gif|350x350px]] || [[Image:818cm-1Fdh08.gif]]
|}

The data in table 4 shows that both methods give the same results and they both give the same transition state. They both produced the same energy with a negligible energy difference between them and the bond lengths between the terminal carbons are both 2.02 Å. In addition to this the imaginary frequencies were both -818cm-1 and within the animations above it is possible to see that when one C-C bond is forming the other is breaking. This means it is a asynchronous bond formation, which is expected for the Cope Rearrangement since it is a pericyclic reaction.

The only difference between the two optimisations is the way in which the transition states were calculated. This is due to the TS (Berny) method calculating the force constant matrix within the first optimisation step and is then being recalculated as the optimisation carries on. When the frozen coordinate method freezes the reaction coordinate in order for the rest of the structure to be optimised around it, which does not require the whole hessian (force constant matrix) to be calculated but only that the reaction coordinate is differentiated (second step). The main advantage of this, is that you do not need to calculate the entire force constant matrix and just need to differentiate the reaction coordinate in order to obtain a good enough guess for the initial force constant matrix. The frozen coordinate method will therefore take less time to compute the transition state, which can be very expensive within certain systems. However to carry out this method one needs to have an idea of how far apart the terminal carbons are.

==='''Boat transition state'''===

===='''QST2 optimisation'''====

The optimisation for the boat transition state had a different method and this was the QST2 method. This calculation interpolates between two structures (product and reactant) and tries to find a transition state between them.

The reactants and products were numbered in the same way. Therefore both 1,5-hexadiene molecules had their numbering manually changed for the product molecule so it corresponded to the numbering of the reactant if it rearranged. This is illustrated in the reaction scheme below.

[[Image:Boat_rerrangmentdh08.jpg|300x300px|center]]

The chk file for the optimized anti2 conformer was opened and copied into a new window. A second window was opened within the new window and the same anti2 conformer was copied into the new window. The numbering was changed within the second molecule so the two looked like they do in the image below.

[[Image:TS_(QST2)_numbering_1dh08.jpg|400x400px|center]]

The job type was set as "Opt+Freq" and "TS (QST2)" was chosen to optimise the transition state. However this failed and produced
a structure a bit like a <jmol>
<jmolAppletButton>
<uploadedFileContents>Trialone_BPdh08.mol</uploadedFileContents>
<text>chair transition structure </text>
<color>white</color>
</jmolAppletButton>
</jmol> but more dissociated. This is because it interpolated between the two structures and translated the top allyl fragment and therefore did not consider the possibility of a rotation around the central bonds.

To get around this the C-C-C-C dihedral angle (C2-C3-C4-C5) was set to 0 o and the C-C-C angle (C2-C3-C4 and C3-C4-C5 for the product molecule) was set to 100 o for both molecules. This gave rise to the two structures below.

[[Image:TS_(QST2)_numbering_2dh08.jpg|400x400px|center]]

After this the QST2 calculation was run again and the results in table 5 were produced. The imaginary frequency at -839 cm-1 shows that while one bond is forming the other one is breaking, which is again an asynchronous bond formation. It is also possible to see that the boat transition state has a longer terminal C-C bond length by 0.12 Å to that of the chair.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5: Data for the boat transition state from the QST2 optimisation method
| Optimised TS Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.14
|-
|Energy (atomic units) || -231.603
|-
| Imaginary frequency/ cm-1 || -840
|-
| Animation ||[[Image:]]
|}

===='''Intrinsic Reaction Coordinate'''====

It is not possible to predict which product the chair and boat transition state will lead to. To solve this problem within Gaussian one can use the IRC method, which allows one to follow the minimum energy path from a transition structure down to its local minimum on a potential energy surface. This creates a succession of points by taking small geometry steps in the direction to where the gradient of the energy surface is at its steepest until the minimum in energy is reached.

The chk file for the optimized chair transition structure was opened and the job type selected was "IRC", the reaction coordinate was computed in the forward direction (since molecule is symmetric) with 50 points along the IRC. This was done with two type of force constants being calculated, once and always.

Within the "once" method the output showed no bond formation process by the 26th step and therefore the calculation was incomplete, since a complete 1,5-hexadiene structure was supposed to be formed. From looking at the IRC pathway it can be seen that the minimum on the potential energy surface was not reached at that point, since the points on the pathway did not form an asymptote and the gradient did not go close enough to zero.

For the "always" method the computation of force constants were carried out at every step. This therefore lets the IRC calculation distinguish whether it is moving along the minimum energy pathway while going down the potential energy surface. While this is happening it can make changes to the pathway if necessary. Within the "always" case the bond forming arrives at 1,5-hexadiene and this is backed up by the IRC pathway reaching a minimum within its energy by the gradient going close to zero and the pathway reaches an asymptote.

The structure produced through the always method had an energy of -231.69199434 atomic units, which is very close to the energy of the gauche2 conformer in the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 appendix 1]. This suggests that the reaction takes place through the chair transition state, which later produces the gauche2 conformer of 1,5-hexadiene.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 6: Comparison of IRC calculations for the chair structure
| || '''Calculate force constants once''' || '''Calculate force constants always'''
|-
| '''Number of points along IRC''' || 26 || 47
|-
| '''Structure at the last point of IRC''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Once_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Always_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|'''Energy (atomic units)''' || -231.68906923 || -231.69199434
|-
| '''ΔE, once - always''' || 2.92511 x 10-3 a.u || 7.68 KJ mol-1
|-
| '''IRC pathway''' || [[Image:One_pathwaydh08.jpg|400x400px]] || [[Image:Always_pathwaydh08.jpg|400x400px]]
|-
| '''IRC gradient''' || [[Image:One_gradientdh08.jpg|400x400px]]|| [[Image:Always_gradientdh08.jpg|400x400px]]
|}

 
 
 

===='''Comparison of activation energies for the transition states'''====

The HF/3-21G optimised chair and boat transition structures were both reoptimised using the B3LYP/6-31G* (B3LYP/6-31G (d)) method. This gives a more accurate minimisation however this method takes longer to run. The values for the zero-point energies and thermal energies were found within the thermochemistry section of the log file.

The values for the HF/3-21G reactant (anti2) optimisation were calculated and the B3LYP/6-31G* values were taken from the "Frequency analysis of 1,5-hexadiene Conformers" earlier in this module.

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 7: Energy data for chair, boat and the anti2 conformation from the HF/3-21G and B3LYP/6-31G* methods of optimisation '''
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466677
| align="center" | -231.461320
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450936
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402342
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539524
| align="center" | -231.532562
| align="center" | -234.611716
| align="center" | -234.469177
| align="center" | -234.461835
|}
Digital repositories

Chair:
HF/3-21G: http://hdl.handle.net/10042/to-5569
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5568

Boat:
HF/3-21G: http://hdl.handle.net/10042/to-5570
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5571

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 8: Summary of activation energies for the chair and boat conformations (ΔE = chair TS or boat TS minus anti2)'''
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| align="center" | 1 a.u = 627.509 kcal mol-1
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE, chair (a.u)'''
| align="center" | 0.072847
| align="center" | 0.071242
| align="center" | 0.054248
| align="center" | 0.052827
| align="center" | -
|-
| '''ΔE, chair (kcal mol-1)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE, boat (a. u)'''
| align="center" | 0.088588
| align="center" | 0.087259
| align="center" | 0.066835
| align="center" | 0.065827
| align="center" | -
|-
| '''ΔE, boat (kcal mol-1)'''
| align="center" | 55.59
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.31
| align="center" | 44.7 ± 2.0
|}

By examining table 8 it is clear to see that the B3LYP/6-31G* method gives values which are much closer to the experimental ones. This is due to the B3LYP/6-31G* giving a higher level of optimisation than the HF/3-21G method. Table 8 also shows that the chair transition state has a lower activation energy to that of the boat transition state by 8.16 kcal mol-1. This suggests that the Cope rearrangement mechanism would most likely proceed through the chair transition state to that of the boat. Therefore the reaction would take place from the anti2 conformer to the chair transition state and then onto the gauche2 conformer within the Cope rearrangement.

It is also possible to see that the calculated energies are very close to the literature values with the chair conformation being closer to its literature value to that of the boat.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9: Geometric parameters for the optimised chair and boat structures '''
| || ''' HF/3-21G chair TS''' || ''' B3LYP/6-31G* chair TS''' || '''HF/3-21G boat TS''' || '''B3LYP/6-31G* boat TS'''
|-
| Optimised Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Chair_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Boat_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.02 (C5-C10 or C3-C14)|| 1.97 (C5-C10 or C3-C14) || 2.14 (C3-C4 or C1-C6) || 2.21 (C3-C4 or C1-C6)
|-
| Fragment angle || 120 o (C3-C1-C5) || 120 o (C3-C1-C5) || 122 o (C4-C5-C6) || 122 o (C4-C5-C6)
|-
| Dihedral angle || 55 o (C1-C3-C14-C11)|| 54 o (C1-C3-C14-C11)|| 0 o (C2-C1-C6-C5) || 0 o (C2-C1-C6-C5)
|-
| Numbering of carbons || [[Image:Numbering_chairdh08.jpg|200x200px]]|| [[Image:Numbering_chairdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]]
|}

From studying table 9 it can be seen that the C-C bond breaking/forming length has changed between the two different types of optimisation methods. However the angles between the bonds remains unchanged.

=='''The Diels Alder Cycloaddition'''==

The Diels-Alder reaction is a cycloaddition between a conjugated diene and a dienophile (alkene system), which forms a cyclohexene system. It is often referred to as a [4s+2s] cycloaddition and proceeds through a single cyclic, concerted transition state where 2 σ bonds are formed in between the terminal carbons of the 2 п systems. This is shown in the reaction scheme below where ethene is reacting with cis-butadiene.

[[Image:Reaction_scheme2dh08.jpg|300x300px|center]]

The π orbitals of the dienophile form new σ bonds with the π orbitals of the diene. The number of π electrons involved determine whether the reaction proceeds in a concerted stereospecific fashion (allowed) or not (forbidden). The HOMO/LUMO of one fragment interacts with the HOMO/LUMO of the other fragment to form two new bonding and anti-bonding MOs. The nodal properties allow one to predict how the reaction will occur.

Semi-Empirical/AM1 method and the B3LYP/6-31G* method of optimisation will be used on Gaussian to determine particular properties of the reactants to predict the way in which the reaction will proceed.

==='''cis-butadiene and ethylene: reagents'''===

<jmol>
<jmolAppletButton>
<uploadedFileContents>Cis_butadienedh08.mol</uploadedFileContents>
<text>Cis-butadiene</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Ethenedh08.mol</uploadedFileContents>
<text>ethylene</text>
<color>white</color>
</jmolAppletButton>
</jmol> were optimised separately using the AM1 semi-empirical method on Gaussview.The cis-butadiene had an energy of 0.04879725 a. u (1.28 x 102 KJ mol-1) and ethylene had an energy of 0.02619027 a. u (68.76 KJ mol-1).

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10: Frontier molecular orbitals of cis-butadiene and ethylene '''
| '''Reagent''' || '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| cis-butadiene || LUMO || [[Image:LUMO1_cis_butadieneDH08.jpg|100x100px]] || [[Image:LUMO2_cis_butadienedh08.jpg|100x100px]]|| Symmetric || +0.017
|-
| cis-butadiene || HOMO || [[Image:HOMO1_cis_butadienedh08.jpg|100x100px]] || [[Image:HOMO2_cis_butadienedh08.jpg|100x100px]]|| Anti-symmetric || -0.343
|-
| ethylene || LUMO || [[Image:LUMO1_ethenedh08.jpg|100x100px]] || [[Image:LUMO2_ethenedh08.jpg|100x100px]]|| Anti-symmetric || +0.052
|-
| ethylene || HOMO || [[Image:HOMO1_ethenedh08.jpg|100x100px]] || [[Image:HOMO2_ethenedh08.jpg|100x100px]]|| Symmetric || -0.387
|}

 
The symmetries of the LUMO and HOMO orbitals for cis-butadiene are opposite of those for the ethylene. This therefore means that the only interactions that can take place are those between ethylene's LUMO and cis-butadiene's HOMO and in addition to this the interaction between cis-butadiene's LUMO and ethylene's HOMO. This is because orbital symmetry has to be conserved (the conservation of orbital symmetry states that the symmetry of an new orbital formed by the overlap of the 2 original orbitals must be the same as the symmetry of the initial orbitals).

==='''cis-butadiene and ethylene: transition state'''===

A <jmol>
<jmolAppletButton>
<uploadedFileContents>Bicyclic_startingdh08.mol</uploadedFileContents>
<text>bicyclo system</text>
<color>white</color>
</jmolAppletButton>
</jmol> was drawn and one of the -CH2-CH2- fragments were removed. Three C=C bonds were added and the previous one was taken away so a <jmol>
<jmolAppletButton>
<uploadedFileContents>Frozen_coorddh08.mol</uploadedFileContents>
<text>guess transition state</text>
<color>white</color>
</jmolAppletButton>
</jmol> was produced.

After this an optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while bonds between the termini of the fragments were frozen at 2.1Å. Following this the resulting file had two separate optimisation and frequency calculations run on it using the TS (Berny) with one using the semi-empirical/AM1 method and the other using the DFT/B3LYP/6-31G* level of theory. Both of these optimisations had the bonds unfrozen and the "derivative" option was chosen instead.

The DFT/B3LYP/6-31G* level of theory was used since it gave more accurate results in the calculations before and the energies it produces are consistent with the rest of the report (semi-empirical/AM1 method gives very different values).

The QST2 optimisation was not used since it required numbering the atoms, which would of taken a lot of time.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11: Comparison of two methods of optimsations for the transition state '''
| || '''semi-empirical/AM1 method''' || '''DFT/B3LYP/6-31G* method'''
|-
| Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>SE_AM1_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>DFT_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Energy (a. u) || 0.11165608 || -234.54389573
|-
| Bond lengths || [[Image:DistancesSEdh08.jpg|150x150px]]|| [[Image:DistancesDFTdh08.jpg|150x150px]]
|-
|Imaginary frequency (cm-1) || -957 || -525
|-
|Animation || [[Image:-957dh08.gif]] || [[Image:-525dh08.gif]]
|-
| Lowest real frequency (cm-1) || 148 || 136
|-
|Animation || [[Image: 148dh08.gif]] || [[Image: 136dh08.gif]]
|}

The C-C bond forming length between the terminal carbons is 0.15 Å larger for the DFT/B3LYP/6-31G* level of theory and this is because this method gives carries out the calculation to a greater degree of accuracy. This larger bond length leads a lower potential energy gradient to the transition state, which gives a lower total energy for the transition state. Therefore the given values below will be from the DFT/B3LYP/6-31G* and not the semi-empirical/AM1 method.

A typical sp3 C-C bond length is 1.54 Å and a typical sp2 C=C bond length is 1.34 Å<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref>. Within the optimised transition molecule above the C-C bond lengths are 1.40 Å and the C=C bond length is 1.38 Å. This suggests that the π-bonds are breaking in order for the C-C bonds to form and additionally to this a π-bond is forming in between the two sp3 carbons.

The Van der Waals radius of a C-atom is 1.72 Å<ref name="ref 3">M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer, D. G. Truhlar, 1995, 346, pp. 63-69.[http://pubs.acs.org/doi/abs/10.1021/jp8111556]</ref>. The partly formed C-C length is 2.27 Å, which is shorter than two times the the van der Waals radius of a carbon atom. Therefore the carbon atoms are within a close enough distance to each other to share electron density.

From looking at the animations for the imaginary frequency above it can be seen that the formation of the two bonds between the termini of the reactants is synchronous. This is expected of the Diels-Alder reaction since the mechanism is concerted. However the lowest real frequency is asynchronous with respect to the two bonds forming between the termini.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12: Frontier molecular orbitals of the transition state '''
| '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| LUMO || [[Image:LUMO1TSdh08.jpg|100x100px]] || [[Image:LUMO2TSdh08.jpg|100x100px]]|| Symmetric || +0.023
|-
| HOMO || [[Image:HOMO1TSdh08.jpg|100x100px]] || [[Image:HOMO2TSdh08.jpg|100x100px]]|| Anti-symmetric || -0.323
|}

From examining the images from table 12 above it can be deduced that the LUMO of the transition state originates from the cis-butadiene's LUMO and the ethylene's HOMO, which represents the σ*-antibonding orbitals. However the HOMO of the transition state originates from the cis-butadiene's HOMO and the ethylene's LUMO, which represents the σ-bonding orbitals within the two single C-C bonds that are formed within the reaction.

==='''cis-butadiene and ethylene: activation energy'''===

The cis-butadiene and ethylene were re-optimised at the DFT/B3LYP/6-31G* level of theory so the activation energy of the reaction could be calculated.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 13: Calculation for the activation energy for the reaction'''
| '''molecule''' || '''DFT/B3LYP/6-31G* energy (a. u)''' || '''Calculations''' || '''Energy '''
|-
| cis-butadiene || -155.98595591 || transition state || -234.54389573 a. u
|-
| ethylene || -78.58745828 || ΔE, activation energy || 0.02951846 a. u
|-
| cis-butadiene and ethylene combined || -234.57341419 || ΔE, activation energy || 77.50 KJ mol-1
|}

The calculated activation energy is 77.50 KJ mol-1, which is of the right magnitude when compared to the literature value (115 ± 8 KJ mol-1 at 0 K<ref name="ref 5">V. Guner, K. S. Khuong, A. G. Leach, P. S. Lee, M. D. Bartberger, K. N. Houk, J. Phys. Chem. A, 2003, 107 (51), pp. 11445–11459.[http://pubs.acs.org/doi/abs/10.1021/jp035501w]</ref>) but there is still an error of 37.50 KJ mol-1. This shows that Gaussview is not perfect but still gives an idea of the activation energy required for the reaction to take place.

==='''Cyclohexa-1,3-diene reaction with maleic anhydride'''===

The reaction scheme below shows the stereoselective Diels-Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene. The reaction is kinetically controlled with the major product being the endo product, which will be explained later on in the exercise.

[[Image:Reaction_schemedh081.jpg|center|500x500px]]

The <jmol>
<jmolAppletButton>
<uploadedFileContents>FrozenTS22dh08.mol</uploadedFileContents>
<text>exo</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_frozenTS22dh08.mol</uploadedFileContents>
<text>endo</text>
<color>white</color>
</jmolAppletButton>
</jmol> transition states were drawn on Gaussview and then the optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while the distance of the forming bonds were frozen at 2.1Å between the respective carbons. After this the structures changed slightly (<jmol>
<jmolAppletButton>
<uploadedFileContents>TS2_afteropt1_exo1dh08.mol</uploadedFileContents>
<text>exo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_TS2_afteropt1dh08.mol</uploadedFileContents>
<text>endo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol>) and the resulting files had an optimisation and frequency calculation run on it using the TA (Berny) with using the semi-empirical/AM1 level of theory while the unfrozen option as changed to "derivative". After this the endo and exo optimised transition state structures were re-optimised as they were before but using the DFT/B3LYP/6-31G* level of theory instead with the additional keyword "Opt=NoEigen".

{| class="wikitable" border="1" align="center"
|+ '''Table 14: Comparison of the exo and endo transition states '''
! rowspan="2"|-!! colspan="2"|''exo'' !! colspan="2"|''endo''
|-
| Semi-empirical/AM1 || DFT/B3LYP/6-31G* || Semi-empirical/AM1 || DFT/B3LYP/6-31G*
|-
| align="center"|Structure ||align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>After_opt2_exodh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>Endo_TS2_afteropt2dh081.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| align="center"| Energy (a. u) ||align="center"| -0.05041975 ||align="center"| -612.67931095 ||align="center"| -0.05150470 ||align="center"| -612.68339684
|-
| C-C bond forming length (Å) ||align="center"| 2.17 ||align="center"| 2.29 ||align="center"| 2.16 ||align="center"| 2.27
|-align="center"
| Imaginary frequency (cm-1) ||align="center"| -812 ||align="center"| -448 ||align="center"| -806 ||align="center"| -447
|-align="center"
| Animation ||colspan="2"|[[Image:-812_SEdh08.gif]] || colspan="2"|[[Image:-806_cm-1dh08.gif]]
|}

Again the C-C bond forming length between the termini carbons is larger (by 0.11 and 0.12 Å) for the DFT/B3LYP/6-31G* level of theory and this is due the same reasons talked about above. However the C-C bond forming length between the termini carbons are very similar in both the endo and exo transition states (0.02 Å difference).

From looking at the animations for the imaginary frequency above it can also be seen that the formation of the two bonds between the carbons in this case as well is synchronous. Again this is expected of the Diels-Alder reaction since it is concerted.

{| align=center
|+ '''Strain diagrams for the endo and exo transition states'''
| [[Image:Exostraindh08.jpg|thumb|200x200px|'''Exo TS''']]
| [[Image:Strainendodh08.jpg|thumb|200x200px|'''Endo TS''']]
|}

From studying the energies in table 14 it can be seen that the endo transition state is more stable than the exo by 4.08589 x 10-3 a. u (10.73 KJ mol-1). However by looking at the strain of both transition states one would expect the exo TS to be more stable. This is because the distance between the cyclohexa-1,3-diene ring and the -(C=O)-O-(C=O)- maleic anhydride fragment is 2.99 Å in the endo TS and 3.03 Å in the exo TS. Thus the endo form would have more strain due to steric hindrance.

However within the exo TS the vicinal hydrogens on the CH2-CH2 (carbon numbers 18 and 21) fragment on the cyclohexa-1,3-diene ring are found below the plane and next to the maleic anhydride. This leads to steric hindrance is this is where the strain originates from within the exo TS. Therefore this disfavours the exo form, which makes the endo more stable.

Another factor making the endo TS more favourable is secondary orbital overlap. These orbitals contribute to the bonds forming and they interact with each other through space. Within the endo TS below there are 2 more orbitals interacting (secondary orbitals) than within the exo TS. This therefore makes the endo TS more stable and therefore more favoured.

[[Image:Secondary_orbital_shizzledh08.jpg|500x500px|center]]

{| class="wikitable" border="1" align="center"
|+ '''Table 15: Frontier molecular orbitals of the the transition state for the cyclohexa-1,3-diene reaction '''
! rowspan="2"|Molecular Orbital !! colspan="4"|''exo'' !! colspan="4"|''endo''
|- align="center"
| '''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''||'''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''
|- align="center"
| LUMO || [[Image:LUMO1exoh081.jpg|150x150px]]|| [[Image:LUMO2exodh081.jpg|150x150px]]|| Anti-symmetric || -0.078 || [[Image:LUMO1ENDOdh08.jpg|150x150px]]||[[Image:LUMO2ENDOdh08.jpg|150x150px]] || Anti-symmetric || -0.067
|- align="center"
| HOMO || [[Image:HOMO1exodh081.jpg|150x150px]]||[[Image:HOMO2exodh081.jpg|150x150px]] || Anti-symmetric|| -0.242 || [[Image:HOMO1ENDOdh08.jpg|150x150px]]|| [[Image:HOMO2ENDOdh08.jpg|150x150px]]||Anti-symmetric || -0.242
|}

Digital repositories:

exo: http://hdl.handle.net/10042/to-5596

endo: http://hdl.handle.net/10042/to-5597

From examining table 15 it is clear to see that both the HOMO and LUMO are anti-symmetric for both the transition states. The LUMO of the maleic anhydride fragment is the π* anti-bonding orbital of the C=C bond. Within the reaction then this has electrons transferred to it and the C=C bond breaks, which leaves just a C-C bond. The LUMOs for both transition states are both anti-bonding with respect to the two fragments. The HOMOs of both transitions state represent the σ-bonding that will be formed between the termini carbons on the products and reactants. This is most likely from the overlap of the cyclohexa-1,3-diene's HOMO and the maleic anhydride's LUMO.

='''References'''=
<references/>

Rep:Mod:holbein16

2011-01-17T19:48:50Z

Dh08: /* '''Module 1: The basic techniques of molecular mechanics and semi-empirical molecular orbital methods for structural and spectroscopic evaluations''' */

='''Module 1: The basic techniques of molecular mechanics and semi-empirical molecular orbital methods for structural and spectroscopic evaluations'''=

=='''Objectives'''==

The aim of this experiment is to use molecular mechanics to predict the stereochemistry and regioselectivity for a range of molecules. In addition to this a mini project is carried out using semi-empirical and DFT molecular orbital theory to investigate the stereoselectivity of an imino Diels-Alder reaction taken from experimental literature.

== '''The Hydrogenation of Cyclopentadiene Dimer''' ==

Cyclopentadiene dimerises to produce specifically the endo dimer 2 rather than the exo dimer 1. Hydrogenation of this dimer proceeds to give initially one of the dihydro derivatives 3 or 4.

{| align=center
|+ '''Dimers of Cyclopentadiene'''
| [[Image:Molecule_1_chemdraw.jpg|thumb|200x150px|<jmol>
<jmolAppletButton>
<uploadedFileContents>1dh08.mol</uploadedFileContents>
<text>Molecule 1</text>
<color>white</color>
</jmolAppletButton>
</jmol>]]
| [[Image:Molecule_2_chemdraw.jpg|thumb|200x150px|<jmol>
<jmolAppletButton>
<uploadedFileContents>2dh08.mol</uploadedFileContents>
<text>Molecule 2</text>
<color>white</color>
</jmolAppletButton>
</jmol>]]
| [[Image:Molecule_3_chemdraw.jpg|thumb|200x150px|<jmol>
<jmolAppletButton>
<uploadedFileContents>3dh08.mol</uploadedFileContents>
<text>Molecule 3</text>
<color>white</color>
</jmolAppletButton>
</jmol>]]
| [[Image:Molecule_4_chemdraw.jpg|thumb|200x150px|<jmol>
<jmolAppletButton>
<uploadedFileContents>4dh08.mol</uploadedFileContents>
<text>Molecule 4</text>
<color>white</color>
</jmolAppletButton>
</jmol>]]
|}

'''Dimerisation of Cyclopentadiene'''

[[Image:Mechanismex1dh08.jpg|thumb|[π4+π2] Diels-Alder Cycloaddition Mechanism]]

Cyclopentadiene dimerises to form a dimer via the mechanism of the [π4+π2] Diels-Alder cycloaddition reaction (shown on the right). This can theoretically result in either the exo product (molecule 1) or the endo product (molecule 2).

The endo product is favoured and so is the only one which is formed. This is known as the endo dimer is less strained within its conformation and therefore more thermodynamically stable (thermodynamic product). This means that the exo dimer is the kinetic product by it being higher in energy. The endo product is also favoured due to it's formation being faster due to a lower energy transition state because of secondary orbital interaction.<ref name="Caramella">P. Caramella, P. Quadrelli, L. Toma., J. Am. Chem. Soc., 2002, 124, 7, pp. 1130–1131.[http://pubs.acs.org/doi/abs/10.1021/ja016622h]</ref>

{| class="wikitable" border="1" align=center
|+ Table 1:MM2 calculated total energies for molecules 1 and 2
! Molecule !! Total Energy (kcal/mol)
|-
| 1 (Exo Dimer) || 31.88
|-
| 2 (Endo Dimer) || 34.01
|}

'''Hydrogenation'''

Table 2 shows the total energy and individual contributions between molecules 3 and 4. Molecule 4 is less hindered than molecule 3 by it having a total energy of 31.1594 kcal/mol when molecule 3 has a total energy of 35.6989 kcal/mol. From this it can be deduced that molecule 4 is more thermodynamically stable and this is mainly due to molecule 4's bend value being 5.2819 kcal/mol more stable. This therefore means more bend strain was released within the hydrogenation of creating molecule 4 to that of molecule 3.

So from this data we can conclude that molecule 4 is theoretically more likely to be the product of the hydrogenation of the cyclopentadiene dimer.

{| class="wikitable" border="1" align=center
|+ Table 2:Individual contributions towards the MM2 calculated energies for molecules 3 and 4
! Relative Contributions !! Molecule 3 Energies (kcal/mol) !! Molecule 4 Energies (kcal/mol)
|-
| '''Stretch''' || 1.3040 || 1.0899
|-
| '''Bend''' || 19.8180 || 14.5361
|-
| '''Torsion''' || 10.8217 ||12.5062
|-
| '''1,4 VDW''' || 5.6307 ||4.5110
|-
| '''Total Energy''' || 35.6989 ||31.1594
|}

=='''Stereochemistry of Nucleophilic additions to a pyridinium ring (NAD+ analogue)'''==
[[Image:Angleex2dh.png|right|thumb| The dihedral angle between the carbonyl group and the aromatic ring (in yellow) in molecule 5 ]]

You cannot include the MeMgI component (Grignard reagent) for the MM2 energy minimisation due to Magnesium not being supported by ChemBio3D Ultra. However the addition of MeMgI would increase the total energy of the molecule.

The MMFF94 energy minimisation has a flaw by the calculation going to a local energy minimum and not a universal one. Therefore the conformation of the molecule after the MMFF94 calculation might not be the lowest energy conformation. To get around this problem I carried out an MMFF94 energy minimisation on molecules 5 and 7, noted down the total energy and dihedral angle (shown in the image on the right), moved multiple atoms and carried out the minimisation again. This was done a total of 6 times and the lowest energy was taken (with its respective angle).

{| class="wikitable" border="1" align=center
|+ Table 3:MMFF94 calculated total energies for molecules 5 and 7
! Molecule !! Total Energy (kcal/mol)!! Dihedral Angle
|-
| 5 || 57.4193 || 6.94 o
|-
| 7 || 98.3665 || -35.69 o
|}

[[Image:First_mechex2dh08.jpeg|right|thumb| Mechanism of methyl magnesium iodide alkylating the prolinol derivative (5)]]

Molecule 5 is an optically active derivative of Prolinol. When it is reacted with methyl magnesium iodide it gives product 6, which is shown within the mechanism on the right. The origin of stereocontrol within this reaction is determined by the coordination between the electronegative oxygen from the carbonyl group and the electropositive Mg of the Grignard reagent, which takes place within a mechanism involving a 6 electron pericyclic transition state.

Also from the results in table 3 it can be seen that the carbonyl bond is 6.94 ° above the aromatic ring. Therefore the MeMgI attacks molecule 5 from above and the addition of the methyl to the molecule would also happen above the ring due to the coordination<ref name="Shultz">A. G. Shultz, L. Flood and J. P. Springer, J. Org. Chemistry, 1986, 51, pp. 838.[http://pubs.acs.org/doi/abs/10.1021/jo00356a016]</ref>. This therefore explains the stereochemistry of molecule 6 and thus the stereocontrol within this reaction.

[[Image:Second_mechex2dh.jpg|right|thumb| Mechanism of the pyridinium ring of 7 being derivatised by aniline to form 8]]

A similar reaction of another pyridinium ring (molecule 7) reacting with aniline was also analysed for its stereocontrol (mechanism on the right).

The reason for stereocontrol within this reaction is due to the repulsions between the lone pairs on the aniline nitrogen and the oxygen on the carbonyl group. In addition to this the relative position of the carbonyl group affects the direction at which the aniline attacks molecule 7. Also steric hindrance between the carbonyl and large phenyl group influence the selectivity. From the results in table 3 it can be seen that the carbonyl bond is 35.69 ° below (-35.69 above) the aromatic ring. This means that the aniline group would attack from the top, which gives the stereospecific conformation of molecule 8.

 

<table border="0" cell padding="0" align="center">
<tr>
<td align="center">Molecule 5</td>
<td align="center">Molecule 6</td>
<td align="center">Molecule 7</td>
<td align="center">Molecule 8</td>

<tr>
<td>
<jmol>
<jmolApplet>
<title>Molecule 5</title><color>white</color>
<size>200</size>
<script>zoom 150</script>
<uploadedFileContents>Molecule_5.adh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
</td>
<td>
<jmol>
<jmolApplet>
<title>Molecule 6</title><color>white</color>
<size>200</size>
<script>zoom 150</script>
<uploadedFileContents>Molecule_6dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
</td>
<td>
<jmol>
<jmolApplet>
<title>Molecule 7</title><color>white</color>
<size>200</size>
<script>zoom 150</script>
<uploadedFileContents>Molecule_7dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
</td>
<td>
<jmol>
<jmolApplet>
<title>Molecule 8</title><color>white</color>
<size>200</size>
<script>zoom 150</script>
<uploadedFileContents>Molecule_8dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
</td>
</table>

We can improve the simple models of the MM2 or MMFF94 minimizations by using the MOPAC/PM6 method since it takes into account electronic structure and molecular orbitals, which leads to more accurate energies and conformations.

=='''Stereochemistry and Reactivity of an Intermediate in the Synthesis of Taxol'''==

A key intermediate within the total synthesis of Taxol <ref name="Elmore">S. W. Elmore and L. Paquette, Tetrahedron Letters, 1991, pp. 319.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6THS-42GDGR3-CK&_user=217827&_coverDate=01%2F14%2F1991&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=3c4182941817e7ede7083ee2dbbef935&searchtype=a]</ref> are the atropisomers (9 and 10) shown below. They show a type of stereoisomerism because the rotation about single bonds are hindered, which is due to the steric barrier of rotation being too large for interconversion between the two isomers and therefore both of the conformations are isolated.

[[Image:Reactionex3dh08.png |frame|center|The atropisomers 9 and 10 ]]

From studying table 4 below it can be deduced that molecule 10, which has the carbonyl group pointing down is thermodynamically more stable than molecule 9. This is seen in both MM2 and MMFF94 minimisations by molecule 10 being 10.0392 kcal/mol lower in energy than molecule 9 within the MM2 calculation and 22.1336 kcal/mol in the MMFF94 calculation.

{| class="wikitable" border="1" align=center
|+ Table 4:MM2 and MMFF94 calculated total energies for molecules 9 and 10
! Molecule !! MM2 Total Energy (kcal/mol)!! MMFF94 Total Energy (kcal/mol)
|-
| 9 || 54.3889 || 82.7068
|-
| 10 || 44.3497 || 60.5732
|}

<table border="0" cell padding="0" align="center">
<tr>
<td align="center">Molecule 9</td>
<td align="center">Molecule 10</td>

<tr>
<td>
<jmol>
<jmolApplet>
<title>Molecule 9</title><color>white</color>
<size>200</size>
<script>zoom 150</script>
<uploadedFileContents>Exercise3.9.dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
</td>
<td>
<jmol>
<jmolApplet>
<title>Molecule 10</title><color>white</color>
<size>200</size>
<script>zoom 150</script>
<uploadedFileContents>Exercise3.10.chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
</td>
</table>

The alkene reacts slowly within both conformations due to it being hyperstable<ref name="Maier">W.F. Maier, P.V.R. Schleyer, J. Am. Chem. Soc., 1981, 103, pp. 1895.[http://pubs.acs.org/doi/abs/10.1021/ja00398a003]</ref>. This is due to it being a hyperstable alkene (or known as a hyperstable olefin) and occurs when the olefin strain energy is negative. This energy is normally positive within alkenes and is negative here because the interactions between the hydrogens on the alkene are stabilising the double bond, which is known as the 1,3-effect (gecko effect).

=='''Modelling Using Semi-empirical Molecular Orbital Theory'''==

<jmol>
<jmolAppletButton>
<uploadedFileContents>HOMO -1.mol</uploadedFileContents>
<text>Molecule 12</text>
<color>white</color>
</jmolAppletButton>
</jmol> (9-chloro-1,4,5,8-tetrahydro-4a,8a-methanonaphthalene) was drawn on ChemBio3D Ultra and the conformation was minimised by the MM2 calculation, which gave the total energy of '''17.8967 kcal/mol'''. Molecule 12 was then again minimised using the MOPAC/PM6 method.

The addition of dichlorocabene to molecule 12 (shown below) demonstrates high π-selectivity by the mono-adduct (syn-trichloride) and the di-adduct products having the ratio 72:23<ref name="Halton">B. Halton, S. G. G. Russell. J. Org. Chem., Vol.56,No.19, 1991, pp. 5553–5556.[http://pubs.acs.org/doi/abs/10.1021/jo00019a015]</ref>.

[[Image:Reaction_schemedh08a.jpg|center|600x250px]]

From studying the HOMO orbital below it can be seen that the largest amount of electron density is on the syn-alkene, which makes it more nucleophilic than the alternative site site of attack, the electron poor anti-alkene. Since dichlorocarbene is an electrophile it will therefore attack the syn-alkene and not the anti-alkene. In addition to this factor there is a stabilising anti-periplanar interaction amongst the anti-alkene π orbital (HOMO-1) and C-Cl σ* orbital (LUMO+1) which leads to the anti-alkene π orbital to lower its energy relative to the syn-alkene π orbital (HOMO) by 0.08eV<ref name="Halton">B. Halton, S. G. G. Russell. J. Org. Chem., Vol.56,No.19, 1991, pp. 5553–5556.[http://pubs.acs.org/doi/abs/10.1021/jo00019a015]</ref>. From this it can be deduced that the anti-alkene π orbital is less reactive by it being thermodynamically more stable to that of the syn-alkene π orbital and therefore explains the regioselectivity above.

{| align=center
|+ '''Molecular Orbitals of molecule 12'''
| [[Image:HOMO_-1.1dh08.jpg|thumb|200x150px|'''HOMO-1''']]
| [[Image:HOMO1dh08.jpg|thumb|200x150px|'''HOMO''']]
| [[Image:LUMO1dh08.jpg|thumb|200x150px|'''LUMO''']]
| [[Image:LUMO+1.1dh08.jpg|thumb|200x150px|'''LUMO+1''']]
| [[Image:LUMO+2.1dh08.jpg|thumb|200x150px|'''LUMO+2''']]
|}

Guassview 5.0 was used to calculate the IR spectra of various different derivatives of molecule 12, which is shown below in table 5. The first observation to be made is that the wavenumber for the C-Cl bond stretch within molecule 12 and the monohydrogenated molecule 12 are 770.90 and 779.80 cm-1, which corresponds to the literature value<ref name="IR">IR Correlation Table.[http://www2.ups.edu/faculty/hanson/Spectroscopy/IR/IRfrequencies.html]</ref> of 600-800 cm-1.

The wavenumbers of the C=C bonds are higher than that of the literature values<ref name="IR">IR Correlation Table.[http://www2.ups.edu/faculty/hanson/Spectroscopy/IR/IRfrequencies.html]</ref>(1620-1680 cm-1) which suggests that the C=C bonds are stronger and more stable in the molecule 12 derivatives since higher wavenumbers mean the bonds are more stable. This extra stability will originate from the rigid structure of the bicyclic compound and the chlorine may donate electron density to the π bonds.

From studying the the IR of molecule 12 one would find that the syn C=C stretch is lower to that of the anti C=C stretch, which would therefore mean that the syn bond is less stable and therefore more reactive than the anti and therefore seems to contradict the evidence from looking at the molecular orbitals. However when we were studying the molecular orbitals we were just taking the π aspect of the bond into account and not the σ contribution. So the overall stability of the syn C=C bond is lower to that of the anti C=C bond, where the de-stability will be due the σ contribution or any other destabilising interactions within the molecule.
In addition to this the frequencies of the syn C=C stretch within molecule 12 and the monohydrogenated molecule 12 are very similar to each other, which means that the anti C=C bond does not influence the stability of the syn C=C drastically.

From modifying the substituents on the anti alkene from the =C-H group to =C-OH, =C-CN, =C-BH2 and =C-SiH3 groups the effects on the Cl-C and C=C frequencies could be investigated. From this we can initially see that the syn C=C stretch and C-Cl stretch was not greatly altered by any of the substituents being added. However the anti C=C bond does change significantly with the various substituents.

The largest change is with the BH2 group and this will be due to it acting as Lewis acid and accepting electrons from the π bond to go into its low lying unoccupied p-orbital which would weaken the alkene bond. This will be the same reason to why the anti alkene bonds stability is lowered with the silyl group but the affect will be less due to the electrons being promoted to silicon's d-orbitals, which are higher in energy and more diffuse, thus less weakening. The cyano substituent also de-stabilises the anti C=C due to it again withdrawing electrons by the nitrogen by being more electronegative than carbon. The hydroxy group however stabilises the the anti alkene by oxygen donating electrons via resonance to the C=C bond and therefore strengthening it.

{| class="wikitable" border="1" align=center
|+ Table 5: IR vibrations of various derivatives of molecule 12
! '''Molecule'''
|-
| <jmol>
<jmolAppletButton>
<uploadedFileContents>HOMO -1.mol</uploadedFileContents>
<text>Molecule 12</text>
<color>white</color>
</jmolAppletButton>
</jmol> || '''C-Cl Stretch''' ||'''Anti C=C Stretch''' ||'''Syn C=C Stretch'''
|-
| '''Frequency (cm-1)''' || 770.90 ||1737.06 ||1757.35
|-
| '''Intensity '''|| 3.9367 ||4.1981||60.5732
|-
| <jmol>
<jmolAppletButton>
<uploadedFileContents>Jmol2dh08.mol</uploadedFileContents>
<text>Monohydrogenated in anti position</text>
<color>white</color>
</jmolAppletButton>
</jmol> || '''C-Cl Stretch''' ||'''Anti C=C Stretch''' ||'''Syn C=C Stretch'''
|-
| '''Frequency (cm-1)''' || 779.80 || n/a ||1753.80
|-
| '''Intensity ''' || 21.4817 ||n/a||5.0431
|-
| <jmol>
<jmolAppletButton>
<uploadedFileContents>OHdh08.mol</uploadedFileContents>
<text>Hydroxy group</text>
<color>white</color>
</jmolAppletButton>
</jmol>|| '''C-Cl Stretch''' || '''Anti C=C Stretch'''||'''Syn C=C Stretch'''
|-
| '''Frequency (cm-1)''' || 765.29 || 1753.03|| 1757.76
|-
| '''Intensity ''' || 6.6520 ||59.0175 ||37.6681
|-
| <jmol>
<jmolAppletButton>
<uploadedFileContents>CNdh08.mol</uploadedFileContents>
<text>Cyano group</text>
<color>white</color>
</jmolAppletButton>
</jmol>|| '''C-Cl Stretch''' || '''Anti C=C Stretch'''||'''Syn C=C Stretch'''
|-
| '''Frequency (cm-1)''' || 765.79 || 1706.31|| 1756.54
|-
| '''Intensity ''' || 8.0676 ||11.4291 ||4.8241
|-
| <jmol>
<jmolAppletButton>
<uploadedFileContents>BH2dh08.mol</uploadedFileContents>
<text>BH2 group</text>
<color>white</color>
</jmolAppletButton>
</jmol>|| '''C-Cl Stretch''' || '''Anti C=C Stretch'''||'''Syn C=C Stretch'''
|-
| '''Frequency (cm-1)''' || 759.04 || 1657.21|| 1756.55
|-
| '''Intensity ''' || 3.8281||131.9383 ||4.5261
|-
| <jmol>
<jmolAppletButton>
<uploadedFileContents>SiH3Dh08.mol</uploadedFileContents>
<text>Silyl group</text>
<color>white</color>
</jmolAppletButton>
</jmol>|| '''C-Cl Stretch''' || '''Anti C=C Stretch'''||'''Syn C=C Stretch'''
|-
| '''Frequency (cm-1)''' || 763.822 || 1690.33|| 1756.24
|-
|'''Intensity ''' || 17.4369 ||19.038 ||5.5053
|}

 

=='''Structure based Mini project using DFT-based Molecular orbital methods: Indium trichloride (InCl3) catalyzed imino Diels-Alder reactions'''==

The mini project will investigate the indium trichloride (InCl3) catalyzed imino Diels-Alder reaction<ref name="miniproject"> G. Babu, P. T. Perumal, Vol. 54, 1998, pp. 1627-1638.[http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6THR-3SH5D6B-1W-3&_cdi=5289&_user=217827&_pii=S0040402097103702&_origin=search&_coverDate=02%2F19%2F1998&_sk=999459991&view=c&wchp=dGLzVzz-zSkzV&md5=d9e4d7632c2ad0bd17a2ff45e3d9309e&ie=/sdarticle.pdf]</ref> between N-benzylideneaniline and cyclohex-2-enone, which produces two different stereoisomers.

The aim of this project is to differentiate between the two stereoisomers by using computational calculations of various spectroscopy techniques. From the spectra we will be able to make comparisons between the calculated and experimental data. In addition to this molecular modelling calculations will be made to try and explain the regioselectivity and the ratios of the synthesised stereoisomers for the reaction.

The reaction involved <jmol>
<jmolAppletButton>
<uploadedFileContents>N-benzylideneanilinedh08.mol</uploadedFileContents>
<text>N-benzylideneaniline</text>
<color>white</color>
</jmolAppletButton>
</jmol> reacting with <jmol>
<jmolAppletButton>
<uploadedFileContents>Cyclohex-2-enone1.mol</uploadedFileContents>
<text>cyclohex-2-enone</text>
<color>white</color>
</jmolAppletButton>
</jmol> at room temperature for 24 hours, which produced molecule 13 and 14 in a hetero Diels-Alder reaction, however within the literature they envisaged the formation of phenanthridinone derivatives 15 and 16. This happened due to chemioselectivity within the first step of the mechanism by the indium trichloride (InCl3) coordinating strongly with the cyclohexadienolate ion. This lead to the cyclohex-2-enone becoming electron rich and therefore reacting as the diene and the imine acting as the dienophile, which was the opposite to the proposed roles of both reactants.

[[Image:Reaction_scheme_main4dh08.jpg |left|thumb|700x600px|'''Reaction scheme<ref name="miniproject"> G. Babu, P. T. Perumal, Vol. 54, 1998, pp. 1627-1638.[http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6THR-3SH5D6B-1W-3&_cdi=5289&_user=217827&_pii=S0040402097103702&_origin=search&_coverDate=02%2F19%2F1998&_sk=999459991&view=c&wchp=dGLzVzz-zSkzV&md5=d9e4d7632c2ad0bd17a2ff45e3d9309e&ie=/sdarticle.pdf]</ref>''']]

[[Image:Mechanismmpdh08a.jpg|right|thumb|300px|'''Mechanism for molecules 13 and 14''' ]]

<table border="0" cell padding="0" align="center">
<tr>
<td align="left"></td>
<td align="left">Product 13</td>
<td align="left">Product 14</td>
<td align="left"></td>

<tr>
<td>
[[Image:PRODUCT_6_IMAGEdh08.jpg|200x200px|]]
</td>
<td>
<jmol>
<jmolApplet>
<title>Product 6</title><color>white</color>
<size>250</size>
<script>zoom 150</script>
<uploadedFileContents>Product6dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
</td>
<td>
<jmol>
<jmolApplet>
<title>Product 7</title><color>white</color>
<size>250</size>
<script>zoom 150</script>
<uploadedFileContents>Product7dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
</td>
<td>
[[Image:Product_7_imagedh08.jpg|200x200px]]
</td>
</table>

From this reaction two other possible regioisomers were theoretically possible however none were formed due to how the mechanism took place. The image below shows the other possible mechanism which would lead to the other 2 regioisomers for this reaction. However the 1,2-alkene bond on the cyclohexadienolate ion is highly delta negative due to the O- donating a significant amount of electron density to it and therefore the electrons within the C=N π bond attacking 1,2-alkene is not likely. This argument is further strengthened by the 1,2-alkene having a larger amount of electron density in the HOMO compared to the 5,6-alkene, which is the other possible site the cyclohexadienolate ion could attack the imine from. A final factor to why the alternative reaction does not take place is due to the carbon within the C=N bond having more electron density in the LUMO to that of the nitrogen. Thus makes the carbon a more likely place for the electrons to attack.

{|align=center
|+
| [[Image:HOMO_enolate_iondh08.jpg|thumb|200x150px|'''HOMO of the cyclohexadienolate ion''' (OH used since O- did not give reliable results)]]
| [[Image:LUMO_iminedh08.jpg|thumb|200x150px|'''LUMO of the imine reactant]]
| [[Image:Regio_isomermechdh08.jpg|right|thumb|300x300px|'''Mechanism for (13) and (14) in comparison to mechanism for regioisomer]]
|}

The ratio of molecule 13 to 14 is 69:31<ref name="miniproject"> G. Babu, P. T. Perumal, Vol. 54, 1998, pp. 1627-1638.[http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6THR-3SH5D6B-1W-3&_cdi=5289&_user=217827&_pii=S0040402097103702&_origin=search&_coverDate=02%2F19%2F1998&_sk=999459991&view=c&wchp=dGLzVzz-zSkzV&md5=d9e4d7632c2ad0bd17a2ff45e3d9309e&ie=/sdarticle.pdf]</ref>. From studying this it was deduced that the molecules stabilities could play a vital role into an explanation of why equal amounts of products were not synthesised. However by looking at table 6 and 7 this is obviously not the case. Therefore the reason will most likely be that the transition state of molecule 13 is more thermodynamically stable at room temperature and so 13 is favoured over 14.

{| class="wikitable" border="1" align=left
|+ Table 6:Individual contributions towards the MM2 calculated energies for molecules 13 and 14
! Relative Contributions !! Molecule 13 Energies (kcal/mol) !! Molecule 14 Energies (kcal/mol)
|-
| '''Stretch''' || 2.0824 || 2.1202
|-
| '''Bend''' || 9.2284 || 9.0755
|-
| '''Torsion''' || -5.0755 || -5.2468
|-
| '''1,4 VDW''' || 21.4885 || 21.4109
|-
| '''Total Energy''' || 27.5287 || 27.1443
|}

{| class="wikitable" border="1" align=right
|+ Table 7:MMFF94 calculated total energies for molecules 13 and 14
! Molecule !! Total Energies (kcal/mol)
|-
| '''Molecule 13''' || 39.7093
|-
| '''Molecule 14''' || 41.1574
|}

 
 
 
 
 
 
 

From analysing the 13C NMR data below it can be deduced that it is not possible to clearly distinguish between molecule 13 and 14 by using 13C NMR spectroscopy. This will also be the case with 1H NMR since the chemical shifts will be slightly different and therefore unnecessary to analyse. However it is possible to compare it to other experimental values and so distinguish which NMR belongs to which isomer, however this means relying on other academics experimental data and this is not recommended or ideal.

From table 8 and 9 below the simulated 13C NMR spectra can be compared to that of the experimental NMR and therefore does support the structural assignment in the paper. In spite of this though the simulated spectra did not totally match the experimental due to various reasons. The main factor is that some of the carbon peaks overlapped with other carbons with similar chemical shifts and since you cannot carry out integration on 13C NMR the experimental data did not give the amount of carbons responsible for each peak. This meant that in both experimental cases only 15 peaks were recorded however in the simulated cases 19 were calculated (since chemical formula is C19H14NO). So below some experimental peaks have been assigned multiple carbon atoms from the simulated spectra which are most likely contributing to them.

By studying the graphs showing the the difference in the calculated and experimental chemical shifts it it clear that the majority of the calculated peaks have a lower chemical shift in both cases. This will most likely be due to other conformers/chemicals within the experimental NMR sample and that the calculations are approximations and therefore not perfect.

{| class="wikitable" border="1" align=center
|+ Table 8:Calculated and experimental 13C NMR for molecule 13<ref name="nmr13"> Digital repository NMR prediction for molecule 13.[http://hdl.handle.net/10042/to-5251]</ref>
! Carbon number !! Calculated chemical shift (ppm) !! Experimental chemical shift<ref name="miniproject"> G. Babu, P. T. Perumal, Vol. 54, 1998, pp. 1627-1638.[http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6THR-3SH5D6B-1W-3&_cdi=5289&_user=217827&_pii=S0040402097103702&_origin=search&_coverDate=02%2F19%2F1998&_sk=999459991&view=c&wchp=dGLzVzz-zSkzV&md5=d9e4d7632c2ad0bd17a2ff45e3d9309e&ie=/sdarticle.pdf]</ref> (ppm)
|-
| 3 || 210.3 || 213.6
|-
| 8 || 143.7 || 148.1
|-
| 7 || 137.8 ||140.0
|-
| 17 || 125.7 ||130.0
|-
| 12, 15, 10 || 125.4, 125.3, 124.9 ||129.3
|-
| 11 || 123.7 || 127.8
|-
| 9, 13 || 123.0, 122.6 || 125.1
|-
| 16 || 113.5 ||117.6
|-
| 18, 14 || 110.9, 107.7 ||113.0
|-
| 5 || 63.7 ||62.3
|-
| 4 || 52.5 || 50.9
|-
| 2 || 47.9 || 48.1
|-
| 1 || 41.9 ||42.2
|-
| 19 || 27.3 ||25.9
|-
| 21 || 18.6 ||16.3
|}

{| class="wikitable" border="1" align=center
|+ Table 9:Calculated and experimental 13C NMR for molecule 14<ref name="nmr14"> Digital repository NMR prediction for molecule 14.[http://hdl.handle.net/10042/to-5250]</ref>
! Carbon number !! Calculated chemical shift (ppm) !! Experimental chemical shift<ref name="miniproject"> G. Babu, P. T. Perumal, Vol. 54, 1998, pp. 1627-1638.[http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6THR-3SH5D6B-1W-3&_cdi=5289&_user=217827&_pii=S0040402097103702&_origin=search&_coverDate=02%2F19%2F1998&_sk=999459991&view=c&wchp=dGLzVzz-zSkzV&md5=d9e4d7632c2ad0bd17a2ff45e3d9309e&ie=/sdarticle.pdf]</ref> (ppm)
|-
| 3 || 208.3 || 211.8
|-
| 8 || 143.8 || 147.7
|-
| 7 || 138.9 ||142.1
|-
| 17 || 125.8 ||129.2
|-
| 12, 15, 10 || 125.3, 125.2, 125.1 ||128.9
|-
| 11 || 123.7 || 127.5
|-
| 9, 13 || 122.4, 121.9 || 125.5
|-
| 16 || 113.4 ||117.6
|-
| 18, 14 || 110.8, 107.8 ||113.2
|-
| 5 || 66.9 ||65.8
|-
| 4 || 53.2 || 52.1
|-
| 2 || 48.6 || 48.4
|-
| 1 || 46.2 ||45.8
|-
| 19 || 25.5 ||29.6
|-
| 21 || 23.7 || 22.6
|}

{|align=center
|+
| [[Image:Gaussview_image13dh08.jpg|thumb|350x350px|'''Carbon labels for molecules 13 and 14''' ]]
| [[Image:NMRSPECTRA6DH08.jpg|thumb|350x350px|'''Calculated 13C NMR Spectrum for molecule 13''' ]]
| [[Image:NMRspectradh08.jpg|thumb|350x350px|'''Calculated 13C NMR Spectrum for molecule 14''' ]]
|}

{|align=center
|+ '''Difference in calculated and experimental chemical shifts for 13C NMR (calculated - experimental)'''
| [[Image:Product_13_NMR_SD.jpg|thumb|450x450px|'''Molecule 13''' ]]
| [[Image:Product_14_NMR_SD.jpg|thumb|450x450px|'''Molecule 14''' ]]
|}

Again it is not possible to distinguish between the two isomers by looking at the IR spectra. However once again it is possible to compare the calculated spectra to the experimental in order to see if it confirms the structural assignment in the paper. Within table 10 and 11 it can be seen that some calculated peaks are exactly the same to the experimental peaks. Although there are other peaks that do not match up and this will be to some bug within the calculation or not a pure product within the experimental.

{| class="wikitable" border="1" align=center
|+ Table 10:Calculated and experimental IR frequencies for molecule 13
! Functional group !! Calculated frequency (cm-1) !! Experimental frequency<ref name="miniproject"> G. Babu, P. T. Perumal, Vol. 54, 1998, pp. 1627-1638.[http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6THR-3SH5D6B-1W-3&_cdi=5289&_user=217827&_pii=S0040402097103702&_origin=search&_coverDate=02%2F19%2F1998&_sk=999459991&view=c&wchp=dGLzVzz-zSkzV&md5=d9e4d7632c2ad0bd17a2ff45e3d9309e&ie=/sdarticle.pdf]</ref> (cm-1) || Literature value<ref name="IR">IR Correlation Table.[http://www2.ups.edu/faculty/hanson/Spectroscopy/IR/IRfrequencies.html]</ref> (cm-1)
|-
| C=C aromatic || 1492.26|| 1497 || 1400-1600
|-
| C=O carbonyl || 1830.22 || 1721 || 1670 - 1820
|-
| C-H Alkane || 3003.66 || 2936 || 2850 - 3000
|-
| OH or NH by-products || too large a frequency || 3382 ||3200 - 3600(OH) or 3300 - 3500(NH)
|}

{| class="wikitable" border="1" align=center
|+ Table 11:Calculated and experimental IR frequencies for molecule 14
! Functional group !! Calculated frequency (cm-1) !! Experimental frequency<ref name="miniproject"> G. Babu, P. T. Perumal, Vol. 54, 1998, pp. 1627-1638.[http://www.sciencedirect.com/science?_ob=MImg&_imagekey=B6THR-3SH5D6B-1W-3&_cdi=5289&_user=217827&_pii=S0040402097103702&_origin=search&_coverDate=02%2F19%2F1998&_sk=999459991&view=c&wchp=dGLzVzz-zSkzV&md5=d9e4d7632c2ad0bd17a2ff45e3d9309e&ie=/sdarticle.pdf]</ref> (cm-1) || Literature value<ref name="IR">IR Correlation Table.[http://www2.ups.edu/faculty/hanson/Spectroscopy/IR/IRfrequencies.html]</ref> (cm-1)
|-
| C=C aromatic || 1492.38|| 1491 || 1400-1600
|-
| C=O carbonyl || 1835.92 || 1722 || 1670 - 1820
|-
| C-H Alkane || 2992.01 || 2928 || 2850 - 3000
|-
| OH or NH by-products || too large a frequency || 3392 ||3200 - 3600(OH) or 3300 - 3500(NH)
|}

{|align=center
|+
| [[Image:IR_spectra13dh08.jpg|thumb|350x350px|'''Calculated IR spectra for molecule 13''' ]]
| [[Image:IRspectra14dh08.jpg|thumb|350x350px|'''Calculated IR spectra for molecule 14''' ]]
|}

'''Spectroscopy techniques that will differentiate the isomers'''

[[Image:NOESY_imagedh08.jpg|right|thumb|300x300px| '''The distance between Ha and Hb in isomers 13 and 14 is different''' ]]

Nuclear Overhauser Effect Spectroscopy (NOESY) would differentiate between the two isomers. This would be possible due to it being able to measure the distance between two hydrogens within the structure. An example of how this could differentiate between the isomers 13 and 14 is shown on the right.

Another spectroscopic technique would be X-ray crystallography since again it would give you the angles and distances between the carbons and hydrogens within the structure and this was how the structure was ascertained within the literature.

 
 
 
 

=References=
<references/>

Rep:Mod:hampluscheese

2010-11-12T13:23:30Z

Dh08: /* '''cis-butadiene and ethylene: reagents''' */

='''Module 3:Physical Computational Chemistry Lab'''=

Douglas Hunt

=='''Objectives'''==

The aim of this experiment is to is to use various levels of theory (HF/3-21G, DFT/B3LYP/6-31G (d) and AM1/semi-empirical) on Gaussian to calculate various energies, molecular orbitals, geometrical parameters, reaction coordinate pathways and low and imaginary frequencies for a range of molecules and transition states. This will include using different optimisation methods throughout and will lead to a range of findings in various areas of chemical interest.

=='''The Cope Rearrangement'''==

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangement] is a [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement |3,3|-sigmatropic rearrangement] that takes place through a single [http://en.wikipedia.org/wiki/Pericyclic pericyclic], concerted [http://en.wikipedia.org/wiki/Transition_state transition state]. Within this rearrangement a σ bond is formed in concert with another being broken within a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] π system.

The Cope rearrangement we will be studying will be that of 1,5-hexadiene, which is shown below.

[[Image:Cope_rearrangement_schemedh08.jpg|300x300px|center]]

The objective is to locate the low-energy minima and transition structures on the C6H10 potential energy surface and to determine the preferred reaction mechanism by using [http://en.wikipedia.org/wiki/GAUSSIAN Gaussian], with the [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)] methods.

==='''Optimisation of 1,5-hexadiene Conformers'''===

Rough <jmol>
<jmolAppletButton>
<uploadedFileContents>Anti_2_drawndh08.mol</uploadedFileContents>
<text>anti</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche3_drawndh08.mol</uploadedFileContents>
<text>gauche</text>
<color>white</color>
</jmolAppletButton>
</jmol> conformers of 1,5-hexadiene were drawn on Gaussview and their structures were tidied by the clean function. They were then further optimised by the HF/3-21G method and were identified which conformer they were from their point group and appendix 1 on the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3 physical lab page].

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 1: Optimisation and comparison of anti and gauche 1,5-hexadiene conformers using HF/3-21G methods
| ''' Conformer''' || '''Optimised Structure''' || '''Energy (Atomic Units)''' || '''[http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1 Energies (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (Anti2 - Gauche3)''' || '''Point group'''
|-
| '''Anti2''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Anti_2_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692535 || -231.69254 ||-6.08 x 105 || 1.2606 x 10-4 a. u|| Ci
|-
| '''Gauche3''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Gauche3_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692661 || -231.69266 || -6.08 x 105 || 3.31 x 10-1 KJ mol-1 || C1
|}

From studying table 1 it can be seen that the gauche conformer is 0.33 KJ mol-1 more stable than the anti conformation. One would assume that the anti2 conformer would be more stable due to [http://en.wikipedia.org/wiki/Steric steric] factors/repulsion within the gauche3 conformation. However within the gauche3 conformer there is a favourable donation of [http://en.wikipedia.org/wiki/Electron_density electron density] from the C=C π orbital to the C-H σ* orbital from the adjacent proton, which stabilises the conformer<ref name="ref 1">B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446.[http://www.informaworld.com/smpp/content~db=all?content=10.1080/00268970110081412]</ref>.

 

The anti2 conformer was then optimised using the DFT/B3LYP/6-31G* (6-31G (d)) method.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 2: Comparison of the anti2 conformer optimisations carried out at different calculation methods
| ''' Method''' || '''Structure and bond distances (Å)''' || '''Dihedral angle (C1 to C4)''' || '''Energy (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (HF/3-21G - DFT/B3LYP/6-31G*)'''
|-
| HF/3-21G || [[Image:Anti2_opt1dh08.jpg|300x300px]] || 114.6 o || -231.692535 || -6.08 x 105 || 2.919171 a. u
|-
| DFT/B3LYP/6-31G* || [[Image:Anti2_opt2DH08.jpg|300x300px]] || 118.8 o || -234.611716 || -6.16 x 105 || 7.66 x 103 KJ mol-1
|}

From examining table 2 at first it is possible to see that the bond lengths are very similar to each other and that of the literature bond lengths<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref> (C1=C2: 1.340 Å, C2-C3: 1.508 Å and C3-C4: 1.538 Å). It is also possible to see that the C=C bond length is more accurate for the DFT/B3LYP/6-31G* method by it being 0.01 Å closer to the literature value. Another main difference in terms of geometry is that the DFT/B3LYP/6-31G* method gives a larger [http://en.wikipedia.org/wiki/Dihedral_angle dihedral angle] (and most likely more accurate) of 118.8o compared to that of 114.6o.

In terms of energy the DFT/B3LYP/6-31G* method produces a conformer of a higher stability by it possessing a lower energy with a difference of 7.66 x 103 KJ mol-1.

==='''Frequency analysis of 1,5-hexadiene Conformers'''===

[[Image:Vibrations_anti2dh08.jpg|right|300x300px]]

For the frequency analysis the DFT/B3LYP/6-31G* method was used.

Frequency analysis is essential within this calculation due to it being the second derivative of the potential energy surface. This means that if all the frequencies are all positive the system is at a minimum, if one of them is negative the optimised system is at a transition state, and if more than one are negative then the calculation has failed to find a critical point, which means the optimisation has failed. This therefore gives us a method of seeing whether the calculation has worked.

From looking to the image on the right it can deduced that the optimisation has taken place successfully by there being no negative vibrations and therefore the conformation is at a minimum.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 3: Energies under the thermochemistry section at 298.15 K and 0 K within the frequency logfiles'''
| '''Energies''' || '''298.15 K (atomic units)''' ||| '''0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (KJ mol-1)''' || '''Discussion'''
|-
| Sum of electronic and zero-point energies || -234.469177 || -234.466673 || -2.504 x 10-3 || -6.57 || This is the potential energy at 0 K, which includes the zero-point vibrational energy (E = Eelec + ZPE).

|-
| Sum of electronic and thermal energies || -234.461835 || -234.461085 || -7.5 x 10-4|| -1.97 || This is the energy at 298.15 K (0 K for 0 K calculation) and 1 atm of pressure, which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans).

|-
| Sum of electronic and thermal enthalpies || -234.460891 || -234.460141 || -7.5 x 10-4 || -1.97 || This contains an additional correction for RT (H = E + RT), which is especially important when looking at dissociation reactions.
|-
| Sum of electronic and thermal free energies || -234.500739 || -234.495931 || 4.808 x 10-3 || -12.62 || This includes the entropic contribution to the free energy (G = H - TS).
|}

{|align=right
|+
| [[Image:Anti2_freq_IRdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 298.15 K''']]
| [[Image:Anti2_freq_0Kdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 0 K''']]
|}

From looking at table 3 above it can be deduced that the anti2 conformer is more stable at room temperature to that of 0 K with respect to all the types of energies. The spectra also seem to not change much between the two temperatures but the main change is the change in the sum of electronic and thermal free energies, which is heavily dependent of temperature by it including the entropic contribution to the free energy (G = H - '''T'''S).

 
 
 
 

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

Within the Cope Rearrangement there are two possible transition states, the chair (C2h) and the boat (C2v), which are shown within [https://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_2 appendix two ].

==='''Comparison of optimisation methods for the chair transition state'''===

An <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2hdh08.mol</uploadedFileContents>
<text>allyl fragment</text>
<color>white</color>
</jmolAppletButton>
</jmol>, CH2CHCH2 was drawn on Gaussview and was optimized through the HF/3-21G method. After this the original allyl fragment was copied and pasted into a new window and placed exactly above the first fragment with a bond distance of 2.20 Å between the terminal end carbons. This therefore produced the <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2h_fulldh08.mol</uploadedFileContents>
<text>chair transition structure.</text>
<color>white</color>
</jmolAppletButton>
</jmol>

Gaussian was then set up for an optimisation for the transition state with the additional key words "Opt=NoEigen", the optimisation set to TS (Berny), the force constants set to "once" and using the HF/3-21G method.

After this the "chair" transition state was optimised an alternative way by freezing the coordinates of the two sets of terminal carbons (where the bonds form/break during the rearrangement) within the Redundant Coord Editor on Gauusview.
Following this a second calculation was run where the freeze coordinates option was changed to "Derivative" and the force constants were set to "never".

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 4: Comparison of chair transition state optimisation methods
| || '''Optimised through a TS (Berny)''' || '''Optimised through a frozen coordinate method'''
|-
| '''Optimised TS Structure''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>C2h_optfreq_TS_Bdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| '''Terminal C-C bond length (Å)''' || 2.02 || 2.02
|-
|'''Energy (atomic units)''' || -231.61932247 || -231.61932190
|-
| '''ΔE, TS (Berny) - frozen coordinate''' || 5.7 x 10-7 a.u || 1.50 x 10-3 KJ mol-1
|-
| '''Imaginary frequency (cm-1)''' || -818 || -818
|-
| '''Animation''' ||[[Image:818dh08TSB.gif|350x350px]] || [[Image:818cm-1Fdh08.gif]]
|}

The data in table 4 shows that both methods give the same results and they both give the same transition state. They both produced the same energy with a negligible energy difference between them and the bond lengths between the terminal carbons are both 2.02 Å. In addition to this the imaginary frequencies were both -818cm-1 and within the animations above it is possible to see that when one C-C bond is forming the other is breaking. This means it is a asynchronous bond formation, which is expected for the Cope Rearrangement since it is a pericyclic reaction.

The only difference between the two optimisations is the way in which the transition states were calculated. This is due to the TS (Berny) method calculating the force constant matrix within the first optimisation step and is then being recalculated as the optimisation carries on. When the frozen coordinate method freezes the reaction coordinate in order for the rest of the structure to be optimised around it, which does not require the whole hessian (force constant matrix) to be calculated but only that the reaction coordinate is differentiated (second step). The main advantage of this, is that you do not need to calculate the entire force constant matrix and just need to differentiate the reaction coordinate in order to obtain a good enough guess for the initial force constant matrix. The frozen coordinate method will therefore take less time to compute the transition state, which can be very expensive within certain systems. However to carry out this method one needs to have an idea of how far apart the terminal carbons are.

==='''Boat transition state'''===

===='''QST2 optimisation'''====

The optimisation for the boat transition state had a different method and this was the QST2 method. This calculation interpolates between two structures (product and reactant) and tries to find a transition state between them.

The reactants and products were numbered in the same way. Therefore both 1,5-hexadiene molecules had their numbering manually changed for the product molecule so it corresponded to the numbering of the reactant if it rearranged. This is illustrated in the reaction scheme below.

[[Image:Boat_rerrangmentdh08.jpg|300x300px|center]]

The chk file for the optimized anti2 conformer was opened and copied into a new window. A second window was opened within the new window and the same anti2 conformer was copied into the new window. The numbering was changed within the second molecule so the two looked like they do in the image below.

[[Image:TS_(QST2)_numbering_1dh08.jpg|400x400px|center]]

The job type was set as "Opt+Freq" and "TS (QST2)" was chosen to optimise the transition state. However this failed and produced
a structure a bit like a <jmol>
<jmolAppletButton>
<uploadedFileContents>Trialone_BPdh08.mol</uploadedFileContents>
<text>chair transition structure </text>
<color>white</color>
</jmolAppletButton>
</jmol> but more dissociated. This is because it interpolated between the two structures and translated the top allyl fragment and therefore did not consider the possibility of a rotation around the central bonds.

To get around this the C-C-C-C dihedral angle (C2-C3-C4-C5) was set to 0 o and the C-C-C angle (C2-C3-C4 and C3-C4-C5 for the product molecule) was set to 100 o for both molecules. This gave rise to the two structures below.

[[Image:TS_(QST2)_numbering_2dh08.jpg|400x400px|center]]

After this the QST2 calculation was run again and the results in table 5 were produced. The imaginary frequency at -839 cm-1 shows that while one bond is forming the other one is breaking, which is again an asynchronous bond formation. It is also possible to see that the boat transition state has a longer terminal C-C bond length by 0.12 Å to that of the chair.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5: Data for the boat transition state from the QST2 optimisation method
| Optimised TS Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.14
|-
|Energy (atomic units) || -231.60280151
|-
| Energy (KJ mol-1) || -6.08 x 105
|-
| Imaginary frequency (cm-1) || -839
|-
| Animation ||[[Image:-839dh08.gif]]
|}

===='''Intrinsic Reaction Coordinate'''====

It is not possible to predict which product the chair and boat transition state will lead to. To solve this problem within Gaussian one can use the IRC method, which allows one to follow the minimum energy path from a transition structure down to its local minimum on a potential energy surface. This creates a succession of points by taking small geometry steps in the direction to where the gradient of the energy surface is at its steepest until the minimum in energy is reached.

The chk file for the optimized chair transition structure was opened and the job type selected was "IRC", the reaction coordinate was computed in the forward direction (since molecule is symmetric) with 50 points along the IRC. This was done with two type of force constants being calculated, once and always.

Within the "once" method the output showed no bond formation process by the 26th step and therefore the calculation was incomplete, since a complete 1,5-hexadiene structure was supposed to be formed. From looking at the IRC pathway it can be seen that the minimum on the potential energy surface was not reached at that point, since the points on the pathway did not form an asymptote and the gradient did not go close enough to zero.

For the "always" method the computation of force constants were carried out at every step. This therefore lets the IRC calculation distinguish whether it is moving along the minimum energy pathway while going down the potential energy surface. While this is happening it can make changes to the pathway if necessary. Within the "always" case the bond forming arrives at 1,5-hexadiene and this is backed up by the IRC pathway reaching a minimum within its energy by the gradient going close to zero and the pathway reaches an asymptote.

The structure produced through the always method had an energy of -231.69199434 atomic units, which is very close to the energy of the gauche2 conformer in the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 appendix 1]. This suggests that the reaction takes place through the chair transition state, which later produces the gauche2 conformer of 1,5-hexadiene.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 6: Comparison of IRC calculations for the chair structure
| || '''Calculate force constants once''' || '''Calculate force constants always'''
|-
| '''Number of points along IRC''' || 26 || 47
|-
| '''Structure at the last point of IRC''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Once_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Always_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|'''Energy (atomic units)''' || -231.68906923 || -231.69199434
|-
| '''ΔE, once - always''' || 2.92511 x 10-3 a.u || 7.68 KJ mol-1
|-
| '''IRC pathway''' || [[Image:One_pathwaydh08.jpg|400x400px]] || [[Image:Always_pathwaydh08.jpg|400x400px]]
|-
| '''IRC gradient''' || [[Image:One_gradientdh08.jpg|400x400px]]|| [[Image:Always_gradientdh08.jpg|400x400px]]
|}

 
 
 

===='''Comparison of activation energies for the transition states'''====

The HF/3-21G optimised chair and boat transition structures were both reoptimised using the B3LYP/6-31G* (B3LYP/6-31G (d)) method. This gives a more accurate minimisation however this method takes longer to run. The values for the zero-point energies and thermal energies were found within the thermochemistry section of the log file.

The values for the HF/3-21G reactant (anti2) optimisation were calculated and the B3LYP/6-31G* values were taken from the "Frequency analysis of 1,5-hexadiene Conformers" earlier in this module.

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 7: Energy data for chair, boat and the anti2 conformation from the HF/3-21G and B3LYP/6-31G* methods of optimisation '''
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466677
| align="center" | -231.461320
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450936
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402342
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539524
| align="center" | -231.532562
| align="center" | -234.611716
| align="center" | -234.469177
| align="center" | -234.461835
|}
Digital repositories

Chair:
HF/3-21G: http://hdl.handle.net/10042/to-5569
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5568

Boat:
HF/3-21G: http://hdl.handle.net/10042/to-5570
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5571

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 8: Summary of activation energies for the chair and boat conformations (ΔE = chair TS or boat TS minus anti2)'''
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| align="center" | 1 a.u = 627.509 kcal mol-1
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE, chair (a.u)'''
| align="center" | 0.072847
| align="center" | 0.071242
| align="center" | 0.054248
| align="center" | 0.052827
| align="center" | -
|-
| '''ΔE, chair (kcal mol-1)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE, boat (a. u)'''
| align="center" | 0.088588
| align="center" | 0.087259
| align="center" | 0.066835
| align="center" | 0.065827
| align="center" | -
|-
| '''ΔE, boat (kcal mol-1)'''
| align="center" | 55.59
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.31
| align="center" | 44.7 ± 2.0
|}

By examining table 8 it is clear to see that the B3LYP/6-31G* method gives values which are much closer to the experimental ones. This is due to the B3LYP/6-31G* giving a higher level of optimisation than the HF/3-21G method. Table 8 also shows that the chair transition state has a lower activation energy to that of the boat transition state by 8.16 kcal mol-1. This suggests that the Cope rearrangement mechanism would most likely proceed through the chair transition state to that of the boat. Therefore the reaction would take place from the anti2 conformer to the chair transition state and then onto the gauche2 conformer within the Cope rearrangement.

It is also possible to see that the calculated energies are very close to the literature values with the chair conformation being closer to its literature value to that of the boat.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9: Geometric parameters for the optimised chair and boat structures '''
| || ''' HF/3-21G chair TS''' || ''' B3LYP/6-31G* chair TS''' || '''HF/3-21G boat TS''' || '''B3LYP/6-31G* boat TS'''
|-
| Optimised Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Chair_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Boat_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.02 (C5-C10 or C3-C14)|| 1.97 (C5-C10 or C3-C14) || 2.14 (C3-C4 or C1-C6) || 2.21 (C3-C4 or C1-C6)
|-
| Fragment angle || 120 o (C3-C1-C5) || 120 o (C3-C1-C5) || 122 o (C4-C5-C6) || 122 o (C4-C5-C6)
|-
| Dihedral angle || 55 o (C1-C3-C14-C11)|| 54 o (C1-C3-C14-C11)|| 0 o (C2-C1-C6-C5) || 0 o (C2-C1-C6-C5)
|-
| Numbering of carbons || [[Image:Numbering_chairdh08.jpg|200x200px]]|| [[Image:Numbering_chairdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]]
|}

From studying table 9 it can be seen that the C-C bond breaking/forming length has changed between the two different types of optimisation methods. However the angles between the bonds remains unchanged.

=='''The Diels Alder Cycloaddition'''==

The Diels-Alder reaction is a cycloaddition between a conjugated diene and a dienophile (alkene system), which forms a cyclohexene system. It is often referred to as a [4s+2s] cycloaddition and proceeds through a single cyclic, concerted transition state where 2 σ bonds are formed in between the terminal carbons of the 2 п systems. This is shown in the reaction scheme below where ethene is reacting with cis-butadiene.

[[Image:Reaction_scheme2dh08.jpg|300x300px|center]]

The π orbitals of the dienophile form new σ bonds with the π orbitals of the diene. The number of π electrons involved determine whether the reaction proceeds in a concerted stereospecific fashion (allowed) or not (forbidden). The HOMO/LUMO of one fragment interacts with the HOMO/LUMO of the other fragment to form two new bonding and anti-bonding MOs. The nodal properties allow one to predict how the reaction will occur.

Semi-Empirical/AM1 method and the B3LYP/6-31G* method of optimisation will be used on Gaussian to determine particular properties of the reactants to predict the way in which the reaction will proceed.

==='''cis-butadiene and ethylene: reagents'''===

<jmol>
<jmolAppletButton>
<uploadedFileContents>Cis_butadienedh08.mol</uploadedFileContents>
<text>Cis-butadiene</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Ethenedh08.mol</uploadedFileContents>
<text>ethylene</text>
<color>white</color>
</jmolAppletButton>
</jmol> were optimised separately using the AM1 semi-empirical method on Gaussview.The cis-butadiene had an energy of 0.04879725 a. u (1.28 x 102 KJ mol-1) and ethylene had an energy of 0.02619027 a. u (68.76 KJ mol-1).

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10: Frontier molecular orbitals of cis-butadiene and ethylene '''
| '''Reagent''' || '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| cis-butadiene || LUMO || [[Image:LUMO1_cis_butadieneDH08.jpg|100x100px]] || [[Image:LUMO2_cis_butadienedh08.jpg|100x100px]]|| Symmetric || +0.017
|-
| cis-butadiene || HOMO || [[Image:HOMO1_cis_butadienedh08.jpg|100x100px]] || [[Image:HOMO2_cis_butadienedh08.jpg|100x100px]]|| Anti-symmetric || -0.343
|-
| ethylene || LUMO || [[Image:LUMO1_ethenedh08.jpg|100x100px]] || [[Image:LUMO2_ethenedh08.jpg|100x100px]]|| Anti-symmetric || +0.052
|-
| ethylene || HOMO || [[Image:HOMO1_ethenedh08.jpg|100x100px]] || [[Image:HOMO2_ethenedh08.jpg|100x100px]]|| Symmetric || -0.387
|}

 
The symmetries of the LUMO and HOMO orbitals for cis-butadiene are opposite of those for the ethylene. This therefore means that the only interactions that can take place are those between ethylene's LUMO and cis-butadiene's HOMO and in addition to this the interaction between cis-butadiene's LUMO and ethylene's HOMO. This is because orbital symmetry has to be conserved (the conservation of orbital symmetry states that the symmetry of an new orbital formed by the overlap of the 2 original orbitals must be the same as the symmetry of the initial orbitals).

==='''cis-butadiene and ethylene: transition state'''===

A <jmol>
<jmolAppletButton>
<uploadedFileContents>Bicyclic_startingdh08.mol</uploadedFileContents>
<text>bicyclo system</text>
<color>white</color>
</jmolAppletButton>
</jmol> was drawn and one of the -CH2-CH2- fragments were removed. Three C=C bonds were added and the previous one was taken away so a <jmol>
<jmolAppletButton>
<uploadedFileContents>Frozen_coorddh08.mol</uploadedFileContents>
<text>guess transition state</text>
<color>white</color>
</jmolAppletButton>
</jmol> was produced.

After this an optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while bonds between the termini of the fragments were frozen at 2.1Å. Following this the resulting file had two separate optimisation and frequency calculations run on it using the TS (Berny) with one using the semi-empirical/AM1 method and the other using the DFT/B3LYP/6-31G* level of theory. Both of these optimisations had the bonds unfrozen and the "derivative" option was chosen instead.

The DFT/B3LYP/6-31G* level of theory was used since it gave more accurate results in the calculations before and the energies it produces are consistent with the rest of the report (semi-empirical/AM1 method gives very different values).

The QST2 optimisation was not used since it required numbering the atoms, which would of taken a lot of time.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11: Comparison of two methods of optimsations for the transition state '''
| || '''semi-empirical/AM1 method''' || '''DFT/B3LYP/6-31G* method'''
|-
| Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>SE_AM1_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>DFT_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Energy (a. u) || 0.11165608 || -234.54389573
|-
| Bond lengths || [[Image:DistancesSEdh08.jpg|150x150px]]|| [[Image:DistancesDFTdh08.jpg|150x150px]]
|-
|Imaginary frequency (cm-1) || -957 || -525
|-
|Animation || [[Image:-957dh08.gif]] || [[Image:-525dh08.gif]]
|-
| Lowest real frequency (cm-1) || 148 || 136
|-
|Animation || [[Image: 148dh08.gif]] || [[Image: 136dh08.gif]]
|}

The C-C bond forming length between the terminal carbons is 0.15 Å larger for the DFT/B3LYP/6-31G* level of theory and this is because this method gives carries out the calculation to a greater degree of accuracy. This larger bond length leads a lower potential energy gradient to the transition state, which gives a lower total energy for the transition state. Therefore the given values below will be from the DFT/B3LYP/6-31G* and not the semi-empirical/AM1 method.

A typical sp3 C-C bond length is 1.54 Å and a typical sp2 C=C bond length is 1.34 Å<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref>. Within the optimised transition molecule above the C-C bond lengths are 1.40 Å and the C=C bond length is 1.38 Å. This suggests that the π-bonds are breaking in order for the C-C bonds to form and additionally to this a π-bond is forming in between the two sp3 carbons.

The Van der Waals radius of a C-atom is 1.72 Å<ref name="ref 3">M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer, D. G. Truhlar, 1995, 346, pp. 63-69.[http://pubs.acs.org/doi/abs/10.1021/jp8111556]</ref>. The partly formed C-C length is 2.27 Å, which is shorter than two times the the van der Waals radius of a carbon atom. Therefore the carbon atoms are within a close enough distance to each other to share electron density.

From looking at the animations for the imaginary frequency above it can be seen that the formation of the two bonds between the termini of the reactants is synchronous. This is expected of the Diels-Alder reaction since the mechanism is concerted. However the lowest real frequency is asynchronous with respect to the two bonds forming between the termini.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12: Frontier molecular orbitals of the transition state '''
| '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| LUMO || [[Image:LUMO1TSdh08.jpg|100x100px]] || [[Image:LUMO2TSdh08.jpg|100x100px]]|| Symmetric || +0.023
|-
| HOMO || [[Image:HOMO1TSdh08.jpg|100x100px]] || [[Image:HOMO2TSdh08.jpg|100x100px]]|| Anti-symmetric || -0.323
|}

From examining the images from table 12 above it can be deduced that the LUMO of the transition state originates from the cis-butadiene's LUMO and the ethylene's HOMO, which represents the σ*-antibonding orbitals. However the HOMO of the transition state originates from the cis-butadiene's HOMO and the ethylene's LUMO, which represents the σ-bonding orbitals within the two single C-C bonds that are formed within the reaction.

==='''cis-butadiene and ethylene: activation energy'''===

The cis-butadiene and ethylene were re-optimised at the DFT/B3LYP/6-31G* level of theory so the activation energy of the reaction could be calculated.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 13: Calculation for the activation energy for the reaction'''
| '''molecule''' || '''DFT/B3LYP/6-31G* energy (a. u)''' || '''Calculations''' || '''Energy '''
|-
| cis-butadiene || -155.98595591 || transition state || -234.54389573 a. u
|-
| ethylene || -78.58745828 || ΔE, activation energy || 0.02951846 a. u
|-
| cis-butadiene and ethylene combined || -234.57341419 || ΔE, activation energy || 77.50 KJ mol-1
|}

The calculated activation energy is 77.50 KJ mol-1, which is of the right magnitude when compared to the literature value (115 ± 8 KJ mol-1 at 0 K<ref name="ref 5">V. Guner, K. S. Khuong, A. G. Leach, P. S. Lee, M. D. Bartberger, K. N. Houk, J. Phys. Chem. A, 2003, 107 (51), pp. 11445–11459.[http://pubs.acs.org/doi/abs/10.1021/jp035501w]</ref>) but there is still an error of 37.50 KJ mol-1. This shows that Gaussview is not perfect but still gives an idea of the activation energy required for the reaction to take place.

==='''Cyclohexa-1,3-diene reaction with maleic anhydride'''===

The reaction scheme below shows the stereoselective Diels-Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene. The reaction is kinetically controlled with the major product being the endo product, which will be explained later on in the exercise.

[[Image:Reaction_schemedh081.jpg|center|500x500px]]

The <jmol>
<jmolAppletButton>
<uploadedFileContents>FrozenTS22dh08.mol</uploadedFileContents>
<text>exo</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_frozenTS22dh08.mol</uploadedFileContents>
<text>endo</text>
<color>white</color>
</jmolAppletButton>
</jmol> transition states were drawn on Gaussview and then the optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while the distance of the forming bonds were frozen at 2.1Å between the respective carbons. After this the structures changed slightly (<jmol>
<jmolAppletButton>
<uploadedFileContents>TS2_afteropt1_exo1dh08.mol</uploadedFileContents>
<text>exo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_TS2_afteropt1dh08.mol</uploadedFileContents>
<text>endo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol>) and the resulting files had an optimisation and frequency calculation run on it using the TA (Berny) with using the semi-empirical/AM1 level of theory while the unfrozen option as changed to "derivative". After this the endo and exo optimised transition state structures were re-optimised as they were before but using the DFT/B3LYP/6-31G* level of theory instead with the additional keyword "Opt=NoEigen".

{| class="wikitable" border="1" align="center"
|+ '''Table 14: Comparison of the exo and endo transition states '''
! rowspan="2"|-!! colspan="2"|''exo'' !! colspan="2"|''endo''
|-
| Semi-empirical/AM1 || DFT/B3LYP/6-31G* || Semi-empirical/AM1 || DFT/B3LYP/6-31G*
|-
| align="center"|Structure ||align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>After_opt2_exodh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>Endo_TS2_afteropt2dh081.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| align="center"| Energy (a. u) ||align="center"| -0.05041975 ||align="center"| -612.67931095 ||align="center"| -0.05150470 ||align="center"| -612.68339684
|-
| C-C bond forming length (Å) ||align="center"| 2.17 ||align="center"| 2.29 ||align="center"| 2.16 ||align="center"| 2.27
|-align="center"
| Imaginary frequency (cm-1) ||align="center"| -812 ||align="center"| -448 ||align="center"| -806 ||align="center"| -447
|-align="center"
| Animation ||colspan="2"|[[Image:-812_SEdh08.gif]] || colspan="2"|[[Image:-806_cm-1dh08.gif]]
|}

Again the C-C bond forming length between the termini carbons is larger (by 0.11 and 0.12 Å) for the DFT/B3LYP/6-31G* level of theory and this is due the same reasons talked about above. However the C-C bond forming length between the termini carbons are very similar in both the endo and exo transition states (0.02 Å difference).

From looking at the animations for the imaginary frequency above it can also be seen that the formation of the two bonds between the carbons in this case as well is synchronous. Again this is expected of the Diels-Alder reaction since it is concerted.

{| align=center
|+ '''Strain diagrams for the endo and exo transition states'''
| [[Image:Exostraindh08.jpg|thumb|200x200px|'''Exo TS''']]
| [[Image:Strainendodh08.jpg|thumb|200x200px|'''Endo TS''']]
|}

From studying the energies in table 14 it can be seen that the endo transition state is more stable than the exo by 4.08589 x 10-3 a. u (10.73 KJ mol-1). However by looking at the strain of both transition states one would expect the exo TS to be more stable. This is because the distance between the cyclohexa-1,3-diene ring and the -(C=O)-O-(C=O)- maleic anhydride fragment is 2.99 Å in the endo TS and 3.03 Å in the exo TS. Thus the endo form would have more strain due to steric hindrance.

However within the exo TS the vicinal hydrogens on the CH2-CH2 (carbon numbers 18 and 21) fragment on the cyclohexa-1,3-diene ring are found below the plane and next to the maleic anhydride. This leads to steric hindrance is this is where the strain originates from within the exo TS. Therefore this disfavours the exo form, which makes the endo more stable.

Another factor making the endo TS more favourable is secondary orbital overlap. These orbitals contribute to the bonds forming and they interact with each other through space. Within the endo TS below there are 2 more orbitals interacting (secondary orbitals) than within the exo TS. This therefore makes the endo TS more stable and therefore more favoured.

[[Image:Secondary_orbital_shizzledh08.jpg|500x500px|center]]

{| class="wikitable" border="1" align="center"
|+ '''Table 15: Frontier molecular orbitals of the the transition state for the cyclohexa-1,3-diene reaction '''
! rowspan="2"|Molecular Orbital !! colspan="4"|''exo'' !! colspan="4"|''endo''
|- align="center"
| '''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''||'''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''
|- align="center"
| LUMO || [[Image:LUMO1exoh081.jpg|150x150px]]|| [[Image:LUMO2exodh081.jpg|150x150px]]|| Anti-symmetric || -0.078 || [[Image:LUMO1ENDOdh08.jpg|150x150px]]||[[Image:LUMO2ENDOdh08.jpg|150x150px]] || Anti-symmetric || -0.067
|- align="center"
| HOMO || [[Image:HOMO1exodh081.jpg|150x150px]]||[[Image:HOMO2exodh081.jpg|150x150px]] || Anti-symmetric|| -0.242 || [[Image:HOMO1ENDOdh08.jpg|150x150px]]|| [[Image:HOMO2ENDOdh08.jpg|150x150px]]||Anti-symmetric || -0.242
|}

Digital repositories:

exo: http://hdl.handle.net/10042/to-5596

endo: http://hdl.handle.net/10042/to-5597

From examining table 15 it is clear to see that both the HOMO and LUMO are anti-symmetric for both the transition states. The LUMO of the maleic anhydride fragment is the π* anti-bonding orbital of the C=C bond. Within the reaction then this has electrons transferred to it and the C=C bond breaks, which leaves just a C-C bond. The LUMOs for both transition states are both anti-bonding with respect to the two fragments. The HOMOs of both transitions state represent the σ-bonding that will be formed between the termini carbons on the products and reactants. This is most likely from the overlap of the cyclohexa-1,3-diene's HOMO and the maleic anhydride's LUMO.

='''References'''=
<references/>

Rep:Mod:hampluscheese

2010-11-12T13:22:55Z

Dh08: /* '''The Diels Alder Cycloaddition''' */

='''Module 3:Physical Computational Chemistry Lab'''=

Douglas Hunt

=='''Objectives'''==

The aim of this experiment is to is to use various levels of theory (HF/3-21G, DFT/B3LYP/6-31G (d) and AM1/semi-empirical) on Gaussian to calculate various energies, molecular orbitals, geometrical parameters, reaction coordinate pathways and low and imaginary frequencies for a range of molecules and transition states. This will include using different optimisation methods throughout and will lead to a range of findings in various areas of chemical interest.

=='''The Cope Rearrangement'''==

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangement] is a [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement |3,3|-sigmatropic rearrangement] that takes place through a single [http://en.wikipedia.org/wiki/Pericyclic pericyclic], concerted [http://en.wikipedia.org/wiki/Transition_state transition state]. Within this rearrangement a σ bond is formed in concert with another being broken within a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] π system.

The Cope rearrangement we will be studying will be that of 1,5-hexadiene, which is shown below.

[[Image:Cope_rearrangement_schemedh08.jpg|300x300px|center]]

The objective is to locate the low-energy minima and transition structures on the C6H10 potential energy surface and to determine the preferred reaction mechanism by using [http://en.wikipedia.org/wiki/GAUSSIAN Gaussian], with the [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)] methods.

==='''Optimisation of 1,5-hexadiene Conformers'''===

Rough <jmol>
<jmolAppletButton>
<uploadedFileContents>Anti_2_drawndh08.mol</uploadedFileContents>
<text>anti</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche3_drawndh08.mol</uploadedFileContents>
<text>gauche</text>
<color>white</color>
</jmolAppletButton>
</jmol> conformers of 1,5-hexadiene were drawn on Gaussview and their structures were tidied by the clean function. They were then further optimised by the HF/3-21G method and were identified which conformer they were from their point group and appendix 1 on the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3 physical lab page].

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 1: Optimisation and comparison of anti and gauche 1,5-hexadiene conformers using HF/3-21G methods
| ''' Conformer''' || '''Optimised Structure''' || '''Energy (Atomic Units)''' || '''[http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1 Energies (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (Anti2 - Gauche3)''' || '''Point group'''
|-
| '''Anti2''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Anti_2_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692535 || -231.69254 ||-6.08 x 105 || 1.2606 x 10-4 a. u|| Ci
|-
| '''Gauche3''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Gauche3_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692661 || -231.69266 || -6.08 x 105 || 3.31 x 10-1 KJ mol-1 || C1
|}

From studying table 1 it can be seen that the gauche conformer is 0.33 KJ mol-1 more stable than the anti conformation. One would assume that the anti2 conformer would be more stable due to [http://en.wikipedia.org/wiki/Steric steric] factors/repulsion within the gauche3 conformation. However within the gauche3 conformer there is a favourable donation of [http://en.wikipedia.org/wiki/Electron_density electron density] from the C=C π orbital to the C-H σ* orbital from the adjacent proton, which stabilises the conformer<ref name="ref 1">B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446.[http://www.informaworld.com/smpp/content~db=all?content=10.1080/00268970110081412]</ref>.

 

The anti2 conformer was then optimised using the DFT/B3LYP/6-31G* (6-31G (d)) method.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 2: Comparison of the anti2 conformer optimisations carried out at different calculation methods
| ''' Method''' || '''Structure and bond distances (Å)''' || '''Dihedral angle (C1 to C4)''' || '''Energy (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (HF/3-21G - DFT/B3LYP/6-31G*)'''
|-
| HF/3-21G || [[Image:Anti2_opt1dh08.jpg|300x300px]] || 114.6 o || -231.692535 || -6.08 x 105 || 2.919171 a. u
|-
| DFT/B3LYP/6-31G* || [[Image:Anti2_opt2DH08.jpg|300x300px]] || 118.8 o || -234.611716 || -6.16 x 105 || 7.66 x 103 KJ mol-1
|}

From examining table 2 at first it is possible to see that the bond lengths are very similar to each other and that of the literature bond lengths<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref> (C1=C2: 1.340 Å, C2-C3: 1.508 Å and C3-C4: 1.538 Å). It is also possible to see that the C=C bond length is more accurate for the DFT/B3LYP/6-31G* method by it being 0.01 Å closer to the literature value. Another main difference in terms of geometry is that the DFT/B3LYP/6-31G* method gives a larger [http://en.wikipedia.org/wiki/Dihedral_angle dihedral angle] (and most likely more accurate) of 118.8o compared to that of 114.6o.

In terms of energy the DFT/B3LYP/6-31G* method produces a conformer of a higher stability by it possessing a lower energy with a difference of 7.66 x 103 KJ mol-1.

==='''Frequency analysis of 1,5-hexadiene Conformers'''===

[[Image:Vibrations_anti2dh08.jpg|right|300x300px]]

For the frequency analysis the DFT/B3LYP/6-31G* method was used.

Frequency analysis is essential within this calculation due to it being the second derivative of the potential energy surface. This means that if all the frequencies are all positive the system is at a minimum, if one of them is negative the optimised system is at a transition state, and if more than one are negative then the calculation has failed to find a critical point, which means the optimisation has failed. This therefore gives us a method of seeing whether the calculation has worked.

From looking to the image on the right it can deduced that the optimisation has taken place successfully by there being no negative vibrations and therefore the conformation is at a minimum.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 3: Energies under the thermochemistry section at 298.15 K and 0 K within the frequency logfiles'''
| '''Energies''' || '''298.15 K (atomic units)''' ||| '''0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (KJ mol-1)''' || '''Discussion'''
|-
| Sum of electronic and zero-point energies || -234.469177 || -234.466673 || -2.504 x 10-3 || -6.57 || This is the potential energy at 0 K, which includes the zero-point vibrational energy (E = Eelec + ZPE).

|-
| Sum of electronic and thermal energies || -234.461835 || -234.461085 || -7.5 x 10-4|| -1.97 || This is the energy at 298.15 K (0 K for 0 K calculation) and 1 atm of pressure, which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans).

|-
| Sum of electronic and thermal enthalpies || -234.460891 || -234.460141 || -7.5 x 10-4 || -1.97 || This contains an additional correction for RT (H = E + RT), which is especially important when looking at dissociation reactions.
|-
| Sum of electronic and thermal free energies || -234.500739 || -234.495931 || 4.808 x 10-3 || -12.62 || This includes the entropic contribution to the free energy (G = H - TS).
|}

{|align=right
|+
| [[Image:Anti2_freq_IRdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 298.15 K''']]
| [[Image:Anti2_freq_0Kdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 0 K''']]
|}

From looking at table 3 above it can be deduced that the anti2 conformer is more stable at room temperature to that of 0 K with respect to all the types of energies. The spectra also seem to not change much between the two temperatures but the main change is the change in the sum of electronic and thermal free energies, which is heavily dependent of temperature by it including the entropic contribution to the free energy (G = H - '''T'''S).

 
 
 
 

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

Within the Cope Rearrangement there are two possible transition states, the chair (C2h) and the boat (C2v), which are shown within [https://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_2 appendix two ].

==='''Comparison of optimisation methods for the chair transition state'''===

An <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2hdh08.mol</uploadedFileContents>
<text>allyl fragment</text>
<color>white</color>
</jmolAppletButton>
</jmol>, CH2CHCH2 was drawn on Gaussview and was optimized through the HF/3-21G method. After this the original allyl fragment was copied and pasted into a new window and placed exactly above the first fragment with a bond distance of 2.20 Å between the terminal end carbons. This therefore produced the <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2h_fulldh08.mol</uploadedFileContents>
<text>chair transition structure.</text>
<color>white</color>
</jmolAppletButton>
</jmol>

Gaussian was then set up for an optimisation for the transition state with the additional key words "Opt=NoEigen", the optimisation set to TS (Berny), the force constants set to "once" and using the HF/3-21G method.

After this the "chair" transition state was optimised an alternative way by freezing the coordinates of the two sets of terminal carbons (where the bonds form/break during the rearrangement) within the Redundant Coord Editor on Gauusview.
Following this a second calculation was run where the freeze coordinates option was changed to "Derivative" and the force constants were set to "never".

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 4: Comparison of chair transition state optimisation methods
| || '''Optimised through a TS (Berny)''' || '''Optimised through a frozen coordinate method'''
|-
| '''Optimised TS Structure''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>C2h_optfreq_TS_Bdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| '''Terminal C-C bond length (Å)''' || 2.02 || 2.02
|-
|'''Energy (atomic units)''' || -231.61932247 || -231.61932190
|-
| '''ΔE, TS (Berny) - frozen coordinate''' || 5.7 x 10-7 a.u || 1.50 x 10-3 KJ mol-1
|-
| '''Imaginary frequency (cm-1)''' || -818 || -818
|-
| '''Animation''' ||[[Image:818dh08TSB.gif|350x350px]] || [[Image:818cm-1Fdh08.gif]]
|}

The data in table 4 shows that both methods give the same results and they both give the same transition state. They both produced the same energy with a negligible energy difference between them and the bond lengths between the terminal carbons are both 2.02 Å. In addition to this the imaginary frequencies were both -818cm-1 and within the animations above it is possible to see that when one C-C bond is forming the other is breaking. This means it is a asynchronous bond formation, which is expected for the Cope Rearrangement since it is a pericyclic reaction.

The only difference between the two optimisations is the way in which the transition states were calculated. This is due to the TS (Berny) method calculating the force constant matrix within the first optimisation step and is then being recalculated as the optimisation carries on. When the frozen coordinate method freezes the reaction coordinate in order for the rest of the structure to be optimised around it, which does not require the whole hessian (force constant matrix) to be calculated but only that the reaction coordinate is differentiated (second step). The main advantage of this, is that you do not need to calculate the entire force constant matrix and just need to differentiate the reaction coordinate in order to obtain a good enough guess for the initial force constant matrix. The frozen coordinate method will therefore take less time to compute the transition state, which can be very expensive within certain systems. However to carry out this method one needs to have an idea of how far apart the terminal carbons are.

==='''Boat transition state'''===

===='''QST2 optimisation'''====

The optimisation for the boat transition state had a different method and this was the QST2 method. This calculation interpolates between two structures (product and reactant) and tries to find a transition state between them.

The reactants and products were numbered in the same way. Therefore both 1,5-hexadiene molecules had their numbering manually changed for the product molecule so it corresponded to the numbering of the reactant if it rearranged. This is illustrated in the reaction scheme below.

[[Image:Boat_rerrangmentdh08.jpg|300x300px|center]]

The chk file for the optimized anti2 conformer was opened and copied into a new window. A second window was opened within the new window and the same anti2 conformer was copied into the new window. The numbering was changed within the second molecule so the two looked like they do in the image below.

[[Image:TS_(QST2)_numbering_1dh08.jpg|400x400px|center]]

The job type was set as "Opt+Freq" and "TS (QST2)" was chosen to optimise the transition state. However this failed and produced
a structure a bit like a <jmol>
<jmolAppletButton>
<uploadedFileContents>Trialone_BPdh08.mol</uploadedFileContents>
<text>chair transition structure </text>
<color>white</color>
</jmolAppletButton>
</jmol> but more dissociated. This is because it interpolated between the two structures and translated the top allyl fragment and therefore did not consider the possibility of a rotation around the central bonds.

To get around this the C-C-C-C dihedral angle (C2-C3-C4-C5) was set to 0 o and the C-C-C angle (C2-C3-C4 and C3-C4-C5 for the product molecule) was set to 100 o for both molecules. This gave rise to the two structures below.

[[Image:TS_(QST2)_numbering_2dh08.jpg|400x400px|center]]

After this the QST2 calculation was run again and the results in table 5 were produced. The imaginary frequency at -839 cm-1 shows that while one bond is forming the other one is breaking, which is again an asynchronous bond formation. It is also possible to see that the boat transition state has a longer terminal C-C bond length by 0.12 Å to that of the chair.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5: Data for the boat transition state from the QST2 optimisation method
| Optimised TS Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.14
|-
|Energy (atomic units) || -231.60280151
|-
| Energy (KJ mol-1) || -6.08 x 105
|-
| Imaginary frequency (cm-1) || -839
|-
| Animation ||[[Image:-839dh08.gif]]
|}

===='''Intrinsic Reaction Coordinate'''====

It is not possible to predict which product the chair and boat transition state will lead to. To solve this problem within Gaussian one can use the IRC method, which allows one to follow the minimum energy path from a transition structure down to its local minimum on a potential energy surface. This creates a succession of points by taking small geometry steps in the direction to where the gradient of the energy surface is at its steepest until the minimum in energy is reached.

The chk file for the optimized chair transition structure was opened and the job type selected was "IRC", the reaction coordinate was computed in the forward direction (since molecule is symmetric) with 50 points along the IRC. This was done with two type of force constants being calculated, once and always.

Within the "once" method the output showed no bond formation process by the 26th step and therefore the calculation was incomplete, since a complete 1,5-hexadiene structure was supposed to be formed. From looking at the IRC pathway it can be seen that the minimum on the potential energy surface was not reached at that point, since the points on the pathway did not form an asymptote and the gradient did not go close enough to zero.

For the "always" method the computation of force constants were carried out at every step. This therefore lets the IRC calculation distinguish whether it is moving along the minimum energy pathway while going down the potential energy surface. While this is happening it can make changes to the pathway if necessary. Within the "always" case the bond forming arrives at 1,5-hexadiene and this is backed up by the IRC pathway reaching a minimum within its energy by the gradient going close to zero and the pathway reaches an asymptote.

The structure produced through the always method had an energy of -231.69199434 atomic units, which is very close to the energy of the gauche2 conformer in the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 appendix 1]. This suggests that the reaction takes place through the chair transition state, which later produces the gauche2 conformer of 1,5-hexadiene.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 6: Comparison of IRC calculations for the chair structure
| || '''Calculate force constants once''' || '''Calculate force constants always'''
|-
| '''Number of points along IRC''' || 26 || 47
|-
| '''Structure at the last point of IRC''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Once_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Always_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|'''Energy (atomic units)''' || -231.68906923 || -231.69199434
|-
| '''ΔE, once - always''' || 2.92511 x 10-3 a.u || 7.68 KJ mol-1
|-
| '''IRC pathway''' || [[Image:One_pathwaydh08.jpg|400x400px]] || [[Image:Always_pathwaydh08.jpg|400x400px]]
|-
| '''IRC gradient''' || [[Image:One_gradientdh08.jpg|400x400px]]|| [[Image:Always_gradientdh08.jpg|400x400px]]
|}

 
 
 

===='''Comparison of activation energies for the transition states'''====

The HF/3-21G optimised chair and boat transition structures were both reoptimised using the B3LYP/6-31G* (B3LYP/6-31G (d)) method. This gives a more accurate minimisation however this method takes longer to run. The values for the zero-point energies and thermal energies were found within the thermochemistry section of the log file.

The values for the HF/3-21G reactant (anti2) optimisation were calculated and the B3LYP/6-31G* values were taken from the "Frequency analysis of 1,5-hexadiene Conformers" earlier in this module.

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 7: Energy data for chair, boat and the anti2 conformation from the HF/3-21G and B3LYP/6-31G* methods of optimisation '''
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466677
| align="center" | -231.461320
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450936
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402342
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539524
| align="center" | -231.532562
| align="center" | -234.611716
| align="center" | -234.469177
| align="center" | -234.461835
|}
Digital repositories

Chair:
HF/3-21G: http://hdl.handle.net/10042/to-5569
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5568

Boat:
HF/3-21G: http://hdl.handle.net/10042/to-5570
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5571

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 8: Summary of activation energies for the chair and boat conformations (ΔE = chair TS or boat TS minus anti2)'''
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| align="center" | 1 a.u = 627.509 kcal mol-1
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE, chair (a.u)'''
| align="center" | 0.072847
| align="center" | 0.071242
| align="center" | 0.054248
| align="center" | 0.052827
| align="center" | -
|-
| '''ΔE, chair (kcal mol-1)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE, boat (a. u)'''
| align="center" | 0.088588
| align="center" | 0.087259
| align="center" | 0.066835
| align="center" | 0.065827
| align="center" | -
|-
| '''ΔE, boat (kcal mol-1)'''
| align="center" | 55.59
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.31
| align="center" | 44.7 ± 2.0
|}

By examining table 8 it is clear to see that the B3LYP/6-31G* method gives values which are much closer to the experimental ones. This is due to the B3LYP/6-31G* giving a higher level of optimisation than the HF/3-21G method. Table 8 also shows that the chair transition state has a lower activation energy to that of the boat transition state by 8.16 kcal mol-1. This suggests that the Cope rearrangement mechanism would most likely proceed through the chair transition state to that of the boat. Therefore the reaction would take place from the anti2 conformer to the chair transition state and then onto the gauche2 conformer within the Cope rearrangement.

It is also possible to see that the calculated energies are very close to the literature values with the chair conformation being closer to its literature value to that of the boat.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9: Geometric parameters for the optimised chair and boat structures '''
| || ''' HF/3-21G chair TS''' || ''' B3LYP/6-31G* chair TS''' || '''HF/3-21G boat TS''' || '''B3LYP/6-31G* boat TS'''
|-
| Optimised Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Chair_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Boat_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.02 (C5-C10 or C3-C14)|| 1.97 (C5-C10 or C3-C14) || 2.14 (C3-C4 or C1-C6) || 2.21 (C3-C4 or C1-C6)
|-
| Fragment angle || 120 o (C3-C1-C5) || 120 o (C3-C1-C5) || 122 o (C4-C5-C6) || 122 o (C4-C5-C6)
|-
| Dihedral angle || 55 o (C1-C3-C14-C11)|| 54 o (C1-C3-C14-C11)|| 0 o (C2-C1-C6-C5) || 0 o (C2-C1-C6-C5)
|-
| Numbering of carbons || [[Image:Numbering_chairdh08.jpg|200x200px]]|| [[Image:Numbering_chairdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]]
|}

From studying table 9 it can be seen that the C-C bond breaking/forming length has changed between the two different types of optimisation methods. However the angles between the bonds remains unchanged.

=='''The Diels Alder Cycloaddition'''==

The Diels-Alder reaction is a cycloaddition between a conjugated diene and a dienophile (alkene system), which forms a cyclohexene system. It is often referred to as a [4s+2s] cycloaddition and proceeds through a single cyclic, concerted transition state where 2 σ bonds are formed in between the terminal carbons of the 2 п systems. This is shown in the reaction scheme below where ethene is reacting with cis-butadiene.

[[Image:Reaction_scheme2dh08.jpg|300x300px|center]]

The π orbitals of the dienophile form new σ bonds with the π orbitals of the diene. The number of π electrons involved determine whether the reaction proceeds in a concerted stereospecific fashion (allowed) or not (forbidden). The HOMO/LUMO of one fragment interacts with the HOMO/LUMO of the other fragment to form two new bonding and anti-bonding MOs. The nodal properties allow one to predict how the reaction will occur.

Semi-Empirical/AM1 method and the B3LYP/6-31G* method of optimisation will be used on Gaussian to determine particular properties of the reactants to predict the way in which the reaction will proceed.

==='''cis-butadiene and ethylene: reagents'''===

<jmol>
<jmolAppletButton>
<uploadedFileContents>Cis_butadienedh08.mol</uploadedFileContents>
<text>Cis-butadiene</text>
<color>whi-te</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Ethenedh08.mol</uploadedFileContents>
<text>ethylene</text>
<color>white</color>
</jmolAppletButton>
</jmol> were optimised separately using the AM1 semi-empirical method on Gaussview.The cis-butadiene had an energy of 0.04879725 a. u (1.28 x 102 KJ mol-1) and ethylene had an energy of 0.02619027 a. u (68.76 KJ mol-1).

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10: Frontier molecular orbitals of cis-butadiene and ethylene '''
| '''Reagent''' || '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| cis-butadiene || LUMO || [[Image:LUMO1_cis_butadieneDH08.jpg|100x100px]] || [[Image:LUMO2_cis_butadienedh08.jpg|100x100px]]|| Symmetric || +0.017
|-
| cis-butadiene || HOMO || [[Image:HOMO1_cis_butadienedh08.jpg|100x100px]] || [[Image:HOMO2_cis_butadienedh08.jpg|100x100px]]|| Anti-symmetric || -0.343
|-
| ethylene || LUMO || [[Image:LUMO1_ethenedh08.jpg|100x100px]] || [[Image:LUMO2_ethenedh08.jpg|100x100px]]|| Anti-symmetric || +0.052
|-
| ethylene || HOMO || [[Image:HOMO1_ethenedh08.jpg|100x100px]] || [[Image:HOMO2_ethenedh08.jpg|100x100px]]|| Symmetric || -0.387
|}

 
The symmetries of the LUMO and HOMO orbitals for cis-butadiene are opposite of those for the ethylene. This therefore means that the only interactions that can take place are those between ethylene's LUMO and cis-butadiene's HOMO and in addition to this the interaction between cis-butadiene's LUMO and ethylene's HOMO. This is because orbital symmetry has to be conserved (the conservation of orbital symmetry states that the symmetry of an new orbital formed by the overlap of the 2 original orbitals must be the same as the symmetry of the initial orbitals).

==='''cis-butadiene and ethylene: transition state'''===

A <jmol>
<jmolAppletButton>
<uploadedFileContents>Bicyclic_startingdh08.mol</uploadedFileContents>
<text>bicyclo system</text>
<color>white</color>
</jmolAppletButton>
</jmol> was drawn and one of the -CH2-CH2- fragments were removed. Three C=C bonds were added and the previous one was taken away so a <jmol>
<jmolAppletButton>
<uploadedFileContents>Frozen_coorddh08.mol</uploadedFileContents>
<text>guess transition state</text>
<color>white</color>
</jmolAppletButton>
</jmol> was produced.

After this an optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while bonds between the termini of the fragments were frozen at 2.1Å. Following this the resulting file had two separate optimisation and frequency calculations run on it using the TS (Berny) with one using the semi-empirical/AM1 method and the other using the DFT/B3LYP/6-31G* level of theory. Both of these optimisations had the bonds unfrozen and the "derivative" option was chosen instead.

The DFT/B3LYP/6-31G* level of theory was used since it gave more accurate results in the calculations before and the energies it produces are consistent with the rest of the report (semi-empirical/AM1 method gives very different values).

The QST2 optimisation was not used since it required numbering the atoms, which would of taken a lot of time.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11: Comparison of two methods of optimsations for the transition state '''
| || '''semi-empirical/AM1 method''' || '''DFT/B3LYP/6-31G* method'''
|-
| Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>SE_AM1_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>DFT_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Energy (a. u) || 0.11165608 || -234.54389573
|-
| Bond lengths || [[Image:DistancesSEdh08.jpg|150x150px]]|| [[Image:DistancesDFTdh08.jpg|150x150px]]
|-
|Imaginary frequency (cm-1) || -957 || -525
|-
|Animation || [[Image:-957dh08.gif]] || [[Image:-525dh08.gif]]
|-
| Lowest real frequency (cm-1) || 148 || 136
|-
|Animation || [[Image: 148dh08.gif]] || [[Image: 136dh08.gif]]
|}

The C-C bond forming length between the terminal carbons is 0.15 Å larger for the DFT/B3LYP/6-31G* level of theory and this is because this method gives carries out the calculation to a greater degree of accuracy. This larger bond length leads a lower potential energy gradient to the transition state, which gives a lower total energy for the transition state. Therefore the given values below will be from the DFT/B3LYP/6-31G* and not the semi-empirical/AM1 method.

A typical sp3 C-C bond length is 1.54 Å and a typical sp2 C=C bond length is 1.34 Å<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref>. Within the optimised transition molecule above the C-C bond lengths are 1.40 Å and the C=C bond length is 1.38 Å. This suggests that the π-bonds are breaking in order for the C-C bonds to form and additionally to this a π-bond is forming in between the two sp3 carbons.

The Van der Waals radius of a C-atom is 1.72 Å<ref name="ref 3">M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer, D. G. Truhlar, 1995, 346, pp. 63-69.[http://pubs.acs.org/doi/abs/10.1021/jp8111556]</ref>. The partly formed C-C length is 2.27 Å, which is shorter than two times the the van der Waals radius of a carbon atom. Therefore the carbon atoms are within a close enough distance to each other to share electron density.

From looking at the animations for the imaginary frequency above it can be seen that the formation of the two bonds between the termini of the reactants is synchronous. This is expected of the Diels-Alder reaction since the mechanism is concerted. However the lowest real frequency is asynchronous with respect to the two bonds forming between the termini.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12: Frontier molecular orbitals of the transition state '''
| '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| LUMO || [[Image:LUMO1TSdh08.jpg|100x100px]] || [[Image:LUMO2TSdh08.jpg|100x100px]]|| Symmetric || +0.023
|-
| HOMO || [[Image:HOMO1TSdh08.jpg|100x100px]] || [[Image:HOMO2TSdh08.jpg|100x100px]]|| Anti-symmetric || -0.323
|}

From examining the images from table 12 above it can be deduced that the LUMO of the transition state originates from the cis-butadiene's LUMO and the ethylene's HOMO, which represents the σ*-antibonding orbitals. However the HOMO of the transition state originates from the cis-butadiene's HOMO and the ethylene's LUMO, which represents the σ-bonding orbitals within the two single C-C bonds that are formed within the reaction.

==='''cis-butadiene and ethylene: activation energy'''===

The cis-butadiene and ethylene were re-optimised at the DFT/B3LYP/6-31G* level of theory so the activation energy of the reaction could be calculated.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 13: Calculation for the activation energy for the reaction'''
| '''molecule''' || '''DFT/B3LYP/6-31G* energy (a. u)''' || '''Calculations''' || '''Energy '''
|-
| cis-butadiene || -155.98595591 || transition state || -234.54389573 a. u
|-
| ethylene || -78.58745828 || ΔE, activation energy || 0.02951846 a. u
|-
| cis-butadiene and ethylene combined || -234.57341419 || ΔE, activation energy || 77.50 KJ mol-1
|}

The calculated activation energy is 77.50 KJ mol-1, which is of the right magnitude when compared to the literature value (115 ± 8 KJ mol-1 at 0 K<ref name="ref 5">V. Guner, K. S. Khuong, A. G. Leach, P. S. Lee, M. D. Bartberger, K. N. Houk, J. Phys. Chem. A, 2003, 107 (51), pp. 11445–11459.[http://pubs.acs.org/doi/abs/10.1021/jp035501w]</ref>) but there is still an error of 37.50 KJ mol-1. This shows that Gaussview is not perfect but still gives an idea of the activation energy required for the reaction to take place.

==='''Cyclohexa-1,3-diene reaction with maleic anhydride'''===

The reaction scheme below shows the stereoselective Diels-Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene. The reaction is kinetically controlled with the major product being the endo product, which will be explained later on in the exercise.

[[Image:Reaction_schemedh081.jpg|center|500x500px]]

The <jmol>
<jmolAppletButton>
<uploadedFileContents>FrozenTS22dh08.mol</uploadedFileContents>
<text>exo</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_frozenTS22dh08.mol</uploadedFileContents>
<text>endo</text>
<color>white</color>
</jmolAppletButton>
</jmol> transition states were drawn on Gaussview and then the optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while the distance of the forming bonds were frozen at 2.1Å between the respective carbons. After this the structures changed slightly (<jmol>
<jmolAppletButton>
<uploadedFileContents>TS2_afteropt1_exo1dh08.mol</uploadedFileContents>
<text>exo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_TS2_afteropt1dh08.mol</uploadedFileContents>
<text>endo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol>) and the resulting files had an optimisation and frequency calculation run on it using the TA (Berny) with using the semi-empirical/AM1 level of theory while the unfrozen option as changed to "derivative". After this the endo and exo optimised transition state structures were re-optimised as they were before but using the DFT/B3LYP/6-31G* level of theory instead with the additional keyword "Opt=NoEigen".

{| class="wikitable" border="1" align="center"
|+ '''Table 14: Comparison of the exo and endo transition states '''
! rowspan="2"|-!! colspan="2"|''exo'' !! colspan="2"|''endo''
|-
| Semi-empirical/AM1 || DFT/B3LYP/6-31G* || Semi-empirical/AM1 || DFT/B3LYP/6-31G*
|-
| align="center"|Structure ||align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>After_opt2_exodh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>Endo_TS2_afteropt2dh081.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| align="center"| Energy (a. u) ||align="center"| -0.05041975 ||align="center"| -612.67931095 ||align="center"| -0.05150470 ||align="center"| -612.68339684
|-
| C-C bond forming length (Å) ||align="center"| 2.17 ||align="center"| 2.29 ||align="center"| 2.16 ||align="center"| 2.27
|-align="center"
| Imaginary frequency (cm-1) ||align="center"| -812 ||align="center"| -448 ||align="center"| -806 ||align="center"| -447
|-align="center"
| Animation ||colspan="2"|[[Image:-812_SEdh08.gif]] || colspan="2"|[[Image:-806_cm-1dh08.gif]]
|}

Again the C-C bond forming length between the termini carbons is larger (by 0.11 and 0.12 Å) for the DFT/B3LYP/6-31G* level of theory and this is due the same reasons talked about above. However the C-C bond forming length between the termini carbons are very similar in both the endo and exo transition states (0.02 Å difference).

From looking at the animations for the imaginary frequency above it can also be seen that the formation of the two bonds between the carbons in this case as well is synchronous. Again this is expected of the Diels-Alder reaction since it is concerted.

{| align=center
|+ '''Strain diagrams for the endo and exo transition states'''
| [[Image:Exostraindh08.jpg|thumb|200x200px|'''Exo TS''']]
| [[Image:Strainendodh08.jpg|thumb|200x200px|'''Endo TS''']]
|}

From studying the energies in table 14 it can be seen that the endo transition state is more stable than the exo by 4.08589 x 10-3 a. u (10.73 KJ mol-1). However by looking at the strain of both transition states one would expect the exo TS to be more stable. This is because the distance between the cyclohexa-1,3-diene ring and the -(C=O)-O-(C=O)- maleic anhydride fragment is 2.99 Å in the endo TS and 3.03 Å in the exo TS. Thus the endo form would have more strain due to steric hindrance.

However within the exo TS the vicinal hydrogens on the CH2-CH2 (carbon numbers 18 and 21) fragment on the cyclohexa-1,3-diene ring are found below the plane and next to the maleic anhydride. This leads to steric hindrance is this is where the strain originates from within the exo TS. Therefore this disfavours the exo form, which makes the endo more stable.

Another factor making the endo TS more favourable is secondary orbital overlap. These orbitals contribute to the bonds forming and they interact with each other through space. Within the endo TS below there are 2 more orbitals interacting (secondary orbitals) than within the exo TS. This therefore makes the endo TS more stable and therefore more favoured.

[[Image:Secondary_orbital_shizzledh08.jpg|500x500px|center]]

{| class="wikitable" border="1" align="center"
|+ '''Table 15: Frontier molecular orbitals of the the transition state for the cyclohexa-1,3-diene reaction '''
! rowspan="2"|Molecular Orbital !! colspan="4"|''exo'' !! colspan="4"|''endo''
|- align="center"
| '''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''||'''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''
|- align="center"
| LUMO || [[Image:LUMO1exoh081.jpg|150x150px]]|| [[Image:LUMO2exodh081.jpg|150x150px]]|| Anti-symmetric || -0.078 || [[Image:LUMO1ENDOdh08.jpg|150x150px]]||[[Image:LUMO2ENDOdh08.jpg|150x150px]] || Anti-symmetric || -0.067
|- align="center"
| HOMO || [[Image:HOMO1exodh081.jpg|150x150px]]||[[Image:HOMO2exodh081.jpg|150x150px]] || Anti-symmetric|| -0.242 || [[Image:HOMO1ENDOdh08.jpg|150x150px]]|| [[Image:HOMO2ENDOdh08.jpg|150x150px]]||Anti-symmetric || -0.242
|}

Digital repositories:

exo: http://hdl.handle.net/10042/to-5596

endo: http://hdl.handle.net/10042/to-5597

From examining table 15 it is clear to see that both the HOMO and LUMO are anti-symmetric for both the transition states. The LUMO of the maleic anhydride fragment is the π* anti-bonding orbital of the C=C bond. Within the reaction then this has electrons transferred to it and the C=C bond breaks, which leaves just a C-C bond. The LUMOs for both transition states are both anti-bonding with respect to the two fragments. The HOMOs of both transitions state represent the σ-bonding that will be formed between the termini carbons on the products and reactants. This is most likely from the overlap of the cyclohexa-1,3-diene's HOMO and the maleic anhydride's LUMO.

='''References'''=
<references/>

Rep:Mod:hampluscheese

2010-11-12T13:19:07Z

Dh08: /* '''Comparison of activation energies for the transition states''' */

='''Module 3:Physical Computational Chemistry Lab'''=

Douglas Hunt

=='''Objectives'''==

The aim of this experiment is to is to use various levels of theory (HF/3-21G, DFT/B3LYP/6-31G (d) and AM1/semi-empirical) on Gaussian to calculate various energies, molecular orbitals, geometrical parameters, reaction coordinate pathways and low and imaginary frequencies for a range of molecules and transition states. This will include using different optimisation methods throughout and will lead to a range of findings in various areas of chemical interest.

=='''The Cope Rearrangement'''==

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangement] is a [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement |3,3|-sigmatropic rearrangement] that takes place through a single [http://en.wikipedia.org/wiki/Pericyclic pericyclic], concerted [http://en.wikipedia.org/wiki/Transition_state transition state]. Within this rearrangement a σ bond is formed in concert with another being broken within a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] π system.

The Cope rearrangement we will be studying will be that of 1,5-hexadiene, which is shown below.

[[Image:Cope_rearrangement_schemedh08.jpg|300x300px|center]]

The objective is to locate the low-energy minima and transition structures on the C6H10 potential energy surface and to determine the preferred reaction mechanism by using [http://en.wikipedia.org/wiki/GAUSSIAN Gaussian], with the [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)] methods.

==='''Optimisation of 1,5-hexadiene Conformers'''===

Rough <jmol>
<jmolAppletButton>
<uploadedFileContents>Anti_2_drawndh08.mol</uploadedFileContents>
<text>anti</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche3_drawndh08.mol</uploadedFileContents>
<text>gauche</text>
<color>white</color>
</jmolAppletButton>
</jmol> conformers of 1,5-hexadiene were drawn on Gaussview and their structures were tidied by the clean function. They were then further optimised by the HF/3-21G method and were identified which conformer they were from their point group and appendix 1 on the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3 physical lab page].

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 1: Optimisation and comparison of anti and gauche 1,5-hexadiene conformers using HF/3-21G methods
| ''' Conformer''' || '''Optimised Structure''' || '''Energy (Atomic Units)''' || '''[http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1 Energies (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (Anti2 - Gauche3)''' || '''Point group'''
|-
| '''Anti2''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Anti_2_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692535 || -231.69254 ||-6.08 x 105 || 1.2606 x 10-4 a. u|| Ci
|-
| '''Gauche3''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Gauche3_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692661 || -231.69266 || -6.08 x 105 || 3.31 x 10-1 KJ mol-1 || C1
|}

From studying table 1 it can be seen that the gauche conformer is 0.33 KJ mol-1 more stable than the anti conformation. One would assume that the anti2 conformer would be more stable due to [http://en.wikipedia.org/wiki/Steric steric] factors/repulsion within the gauche3 conformation. However within the gauche3 conformer there is a favourable donation of [http://en.wikipedia.org/wiki/Electron_density electron density] from the C=C π orbital to the C-H σ* orbital from the adjacent proton, which stabilises the conformer<ref name="ref 1">B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446.[http://www.informaworld.com/smpp/content~db=all?content=10.1080/00268970110081412]</ref>.

 

The anti2 conformer was then optimised using the DFT/B3LYP/6-31G* (6-31G (d)) method.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 2: Comparison of the anti2 conformer optimisations carried out at different calculation methods
| ''' Method''' || '''Structure and bond distances (Å)''' || '''Dihedral angle (C1 to C4)''' || '''Energy (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (HF/3-21G - DFT/B3LYP/6-31G*)'''
|-
| HF/3-21G || [[Image:Anti2_opt1dh08.jpg|300x300px]] || 114.6 o || -231.692535 || -6.08 x 105 || 2.919171 a. u
|-
| DFT/B3LYP/6-31G* || [[Image:Anti2_opt2DH08.jpg|300x300px]] || 118.8 o || -234.611716 || -6.16 x 105 || 7.66 x 103 KJ mol-1
|}

From examining table 2 at first it is possible to see that the bond lengths are very similar to each other and that of the literature bond lengths<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref> (C1=C2: 1.340 Å, C2-C3: 1.508 Å and C3-C4: 1.538 Å). It is also possible to see that the C=C bond length is more accurate for the DFT/B3LYP/6-31G* method by it being 0.01 Å closer to the literature value. Another main difference in terms of geometry is that the DFT/B3LYP/6-31G* method gives a larger [http://en.wikipedia.org/wiki/Dihedral_angle dihedral angle] (and most likely more accurate) of 118.8o compared to that of 114.6o.

In terms of energy the DFT/B3LYP/6-31G* method produces a conformer of a higher stability by it possessing a lower energy with a difference of 7.66 x 103 KJ mol-1.

==='''Frequency analysis of 1,5-hexadiene Conformers'''===

[[Image:Vibrations_anti2dh08.jpg|right|300x300px]]

For the frequency analysis the DFT/B3LYP/6-31G* method was used.

Frequency analysis is essential within this calculation due to it being the second derivative of the potential energy surface. This means that if all the frequencies are all positive the system is at a minimum, if one of them is negative the optimised system is at a transition state, and if more than one are negative then the calculation has failed to find a critical point, which means the optimisation has failed. This therefore gives us a method of seeing whether the calculation has worked.

From looking to the image on the right it can deduced that the optimisation has taken place successfully by there being no negative vibrations and therefore the conformation is at a minimum.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 3: Energies under the thermochemistry section at 298.15 K and 0 K within the frequency logfiles'''
| '''Energies''' || '''298.15 K (atomic units)''' ||| '''0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (KJ mol-1)''' || '''Discussion'''
|-
| Sum of electronic and zero-point energies || -234.469177 || -234.466673 || -2.504 x 10-3 || -6.57 || This is the potential energy at 0 K, which includes the zero-point vibrational energy (E = Eelec + ZPE).

|-
| Sum of electronic and thermal energies || -234.461835 || -234.461085 || -7.5 x 10-4|| -1.97 || This is the energy at 298.15 K (0 K for 0 K calculation) and 1 atm of pressure, which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans).

|-
| Sum of electronic and thermal enthalpies || -234.460891 || -234.460141 || -7.5 x 10-4 || -1.97 || This contains an additional correction for RT (H = E + RT), which is especially important when looking at dissociation reactions.
|-
| Sum of electronic and thermal free energies || -234.500739 || -234.495931 || 4.808 x 10-3 || -12.62 || This includes the entropic contribution to the free energy (G = H - TS).
|}

{|align=right
|+
| [[Image:Anti2_freq_IRdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 298.15 K''']]
| [[Image:Anti2_freq_0Kdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 0 K''']]
|}

From looking at table 3 above it can be deduced that the anti2 conformer is more stable at room temperature to that of 0 K with respect to all the types of energies. The spectra also seem to not change much between the two temperatures but the main change is the change in the sum of electronic and thermal free energies, which is heavily dependent of temperature by it including the entropic contribution to the free energy (G = H - '''T'''S).

 
 
 
 

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

Within the Cope Rearrangement there are two possible transition states, the chair (C2h) and the boat (C2v), which are shown within [https://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_2 appendix two ].

==='''Comparison of optimisation methods for the chair transition state'''===

An <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2hdh08.mol</uploadedFileContents>
<text>allyl fragment</text>
<color>white</color>
</jmolAppletButton>
</jmol>, CH2CHCH2 was drawn on Gaussview and was optimized through the HF/3-21G method. After this the original allyl fragment was copied and pasted into a new window and placed exactly above the first fragment with a bond distance of 2.20 Å between the terminal end carbons. This therefore produced the <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2h_fulldh08.mol</uploadedFileContents>
<text>chair transition structure.</text>
<color>white</color>
</jmolAppletButton>
</jmol>

Gaussian was then set up for an optimisation for the transition state with the additional key words "Opt=NoEigen", the optimisation set to TS (Berny), the force constants set to "once" and using the HF/3-21G method.

After this the "chair" transition state was optimised an alternative way by freezing the coordinates of the two sets of terminal carbons (where the bonds form/break during the rearrangement) within the Redundant Coord Editor on Gauusview.
Following this a second calculation was run where the freeze coordinates option was changed to "Derivative" and the force constants were set to "never".

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 4: Comparison of chair transition state optimisation methods
| || '''Optimised through a TS (Berny)''' || '''Optimised through a frozen coordinate method'''
|-
| '''Optimised TS Structure''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>C2h_optfreq_TS_Bdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| '''Terminal C-C bond length (Å)''' || 2.02 || 2.02
|-
|'''Energy (atomic units)''' || -231.61932247 || -231.61932190
|-
| '''ΔE, TS (Berny) - frozen coordinate''' || 5.7 x 10-7 a.u || 1.50 x 10-3 KJ mol-1
|-
| '''Imaginary frequency (cm-1)''' || -818 || -818
|-
| '''Animation''' ||[[Image:818dh08TSB.gif|350x350px]] || [[Image:818cm-1Fdh08.gif]]
|}

The data in table 4 shows that both methods give the same results and they both give the same transition state. They both produced the same energy with a negligible energy difference between them and the bond lengths between the terminal carbons are both 2.02 Å. In addition to this the imaginary frequencies were both -818cm-1 and within the animations above it is possible to see that when one C-C bond is forming the other is breaking. This means it is a asynchronous bond formation, which is expected for the Cope Rearrangement since it is a pericyclic reaction.

The only difference between the two optimisations is the way in which the transition states were calculated. This is due to the TS (Berny) method calculating the force constant matrix within the first optimisation step and is then being recalculated as the optimisation carries on. When the frozen coordinate method freezes the reaction coordinate in order for the rest of the structure to be optimised around it, which does not require the whole hessian (force constant matrix) to be calculated but only that the reaction coordinate is differentiated (second step). The main advantage of this, is that you do not need to calculate the entire force constant matrix and just need to differentiate the reaction coordinate in order to obtain a good enough guess for the initial force constant matrix. The frozen coordinate method will therefore take less time to compute the transition state, which can be very expensive within certain systems. However to carry out this method one needs to have an idea of how far apart the terminal carbons are.

==='''Boat transition state'''===

===='''QST2 optimisation'''====

The optimisation for the boat transition state had a different method and this was the QST2 method. This calculation interpolates between two structures (product and reactant) and tries to find a transition state between them.

The reactants and products were numbered in the same way. Therefore both 1,5-hexadiene molecules had their numbering manually changed for the product molecule so it corresponded to the numbering of the reactant if it rearranged. This is illustrated in the reaction scheme below.

[[Image:Boat_rerrangmentdh08.jpg|300x300px|center]]

The chk file for the optimized anti2 conformer was opened and copied into a new window. A second window was opened within the new window and the same anti2 conformer was copied into the new window. The numbering was changed within the second molecule so the two looked like they do in the image below.

[[Image:TS_(QST2)_numbering_1dh08.jpg|400x400px|center]]

The job type was set as "Opt+Freq" and "TS (QST2)" was chosen to optimise the transition state. However this failed and produced
a structure a bit like a <jmol>
<jmolAppletButton>
<uploadedFileContents>Trialone_BPdh08.mol</uploadedFileContents>
<text>chair transition structure </text>
<color>white</color>
</jmolAppletButton>
</jmol> but more dissociated. This is because it interpolated between the two structures and translated the top allyl fragment and therefore did not consider the possibility of a rotation around the central bonds.

To get around this the C-C-C-C dihedral angle (C2-C3-C4-C5) was set to 0 o and the C-C-C angle (C2-C3-C4 and C3-C4-C5 for the product molecule) was set to 100 o for both molecules. This gave rise to the two structures below.

[[Image:TS_(QST2)_numbering_2dh08.jpg|400x400px|center]]

After this the QST2 calculation was run again and the results in table 5 were produced. The imaginary frequency at -839 cm-1 shows that while one bond is forming the other one is breaking, which is again an asynchronous bond formation. It is also possible to see that the boat transition state has a longer terminal C-C bond length by 0.12 Å to that of the chair.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5: Data for the boat transition state from the QST2 optimisation method
| Optimised TS Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.14
|-
|Energy (atomic units) || -231.60280151
|-
| Energy (KJ mol-1) || -6.08 x 105
|-
| Imaginary frequency (cm-1) || -839
|-
| Animation ||[[Image:-839dh08.gif]]
|}

===='''Intrinsic Reaction Coordinate'''====

It is not possible to predict which product the chair and boat transition state will lead to. To solve this problem within Gaussian one can use the IRC method, which allows one to follow the minimum energy path from a transition structure down to its local minimum on a potential energy surface. This creates a succession of points by taking small geometry steps in the direction to where the gradient of the energy surface is at its steepest until the minimum in energy is reached.

The chk file for the optimized chair transition structure was opened and the job type selected was "IRC", the reaction coordinate was computed in the forward direction (since molecule is symmetric) with 50 points along the IRC. This was done with two type of force constants being calculated, once and always.

Within the "once" method the output showed no bond formation process by the 26th step and therefore the calculation was incomplete, since a complete 1,5-hexadiene structure was supposed to be formed. From looking at the IRC pathway it can be seen that the minimum on the potential energy surface was not reached at that point, since the points on the pathway did not form an asymptote and the gradient did not go close enough to zero.

For the "always" method the computation of force constants were carried out at every step. This therefore lets the IRC calculation distinguish whether it is moving along the minimum energy pathway while going down the potential energy surface. While this is happening it can make changes to the pathway if necessary. Within the "always" case the bond forming arrives at 1,5-hexadiene and this is backed up by the IRC pathway reaching a minimum within its energy by the gradient going close to zero and the pathway reaches an asymptote.

The structure produced through the always method had an energy of -231.69199434 atomic units, which is very close to the energy of the gauche2 conformer in the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 appendix 1]. This suggests that the reaction takes place through the chair transition state, which later produces the gauche2 conformer of 1,5-hexadiene.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 6: Comparison of IRC calculations for the chair structure
| || '''Calculate force constants once''' || '''Calculate force constants always'''
|-
| '''Number of points along IRC''' || 26 || 47
|-
| '''Structure at the last point of IRC''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Once_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Always_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|'''Energy (atomic units)''' || -231.68906923 || -231.69199434
|-
| '''ΔE, once - always''' || 2.92511 x 10-3 a.u || 7.68 KJ mol-1
|-
| '''IRC pathway''' || [[Image:One_pathwaydh08.jpg|400x400px]] || [[Image:Always_pathwaydh08.jpg|400x400px]]
|-
| '''IRC gradient''' || [[Image:One_gradientdh08.jpg|400x400px]]|| [[Image:Always_gradientdh08.jpg|400x400px]]
|}

 
 
 

===='''Comparison of activation energies for the transition states'''====

The HF/3-21G optimised chair and boat transition structures were both reoptimised using the B3LYP/6-31G* (B3LYP/6-31G (d)) method. This gives a more accurate minimisation however this method takes longer to run. The values for the zero-point energies and thermal energies were found within the thermochemistry section of the log file.

The values for the HF/3-21G reactant (anti2) optimisation were calculated and the B3LYP/6-31G* values were taken from the "Frequency analysis of 1,5-hexadiene Conformers" earlier in this module.

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 7: Energy data for chair, boat and the anti2 conformation from the HF/3-21G and B3LYP/6-31G* methods of optimisation '''
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466677
| align="center" | -231.461320
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450936
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402342
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539524
| align="center" | -231.532562
| align="center" | -234.611716
| align="center" | -234.469177
| align="center" | -234.461835
|}
Digital repositories

Chair:
HF/3-21G: http://hdl.handle.net/10042/to-5569
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5568

Boat:
HF/3-21G: http://hdl.handle.net/10042/to-5570
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5571

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 8: Summary of activation energies for the chair and boat conformations (ΔE = chair TS or boat TS minus anti2)'''
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| align="center" | 1 a.u = 627.509 kcal mol-1
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE, chair (a.u)'''
| align="center" | 0.072847
| align="center" | 0.071242
| align="center" | 0.054248
| align="center" | 0.052827
| align="center" | -
|-
| '''ΔE, chair (kcal mol-1)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE, boat (a. u)'''
| align="center" | 0.088588
| align="center" | 0.087259
| align="center" | 0.066835
| align="center" | 0.065827
| align="center" | -
|-
| '''ΔE, boat (kcal mol-1)'''
| align="center" | 55.59
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.31
| align="center" | 44.7 ± 2.0
|}

By examining table 8 it is clear to see that the B3LYP/6-31G* method gives values which are much closer to the experimental ones. This is due to the B3LYP/6-31G* giving a higher level of optimisation than the HF/3-21G method. Table 8 also shows that the chair transition state has a lower activation energy to that of the boat transition state by 8.16 kcal mol-1. This suggests that the Cope rearrangement mechanism would most likely proceed through the chair transition state to that of the boat. Therefore the reaction would take place from the anti2 conformer to the chair transition state and then onto the gauche2 conformer within the Cope rearrangement.

It is also possible to see that the calculated energies are very close to the literature values with the chair conformation being closer to its literature value to that of the boat.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9: Geometric parameters for the optimised chair and boat structures '''
| || ''' HF/3-21G chair TS''' || ''' B3LYP/6-31G* chair TS''' || '''HF/3-21G boat TS''' || '''B3LYP/6-31G* boat TS'''
|-
| Optimised Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Chair_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Boat_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.02 (C5-C10 or C3-C14)|| 1.97 (C5-C10 or C3-C14) || 2.14 (C3-C4 or C1-C6) || 2.21 (C3-C4 or C1-C6)
|-
| Fragment angle || 120 o (C3-C1-C5) || 120 o (C3-C1-C5) || 122 o (C4-C5-C6) || 122 o (C4-C5-C6)
|-
| Dihedral angle || 55 o (C1-C3-C14-C11)|| 54 o (C1-C3-C14-C11)|| 0 o (C2-C1-C6-C5) || 0 o (C2-C1-C6-C5)
|-
| Numbering of carbons || [[Image:Numbering_chairdh08.jpg|200x200px]]|| [[Image:Numbering_chairdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]]
|}

From studying table 9 it can be seen that the C-C bond breaking/forming length has changed between the two different types of optimisation methods. However the angles between the bonds remains unchanged.

=='''The Diels Alder Cycloaddition'''==

The Diels-Alder reaction is a cycloaddition between a conjugated diene and a dienophile (alkene system), which forms a cyclohexene system. It is often referred to as a [4s+2s] cycloaddition and proceeds through a single cyclic, concerted transition state where 2 σ bonds are formed in between the termini carbons of the 2 п systems. This is shown in the reaction scheme below where ethene is reacting with cis-butadiene.

[[Image:Reaction_scheme2dh08.jpg|300x300px|center]]

The π orbitals of the dienophile form new σ bonds with the π orbitals of the diene. The number of π electrons involved determine whether the reaction proceeds in a concerted stereospecific fashion (allowed) or not (forbidden). The HOMO/LUMO of one fragment interacts with the HOMO/LUMO of the other fragment to form two new bonding and anti-bonding MOs. The nodal properties allow one to predict how the reaction will occur.

Semi-Empirical/AM1 method and the B3LYP/6-31G* method of optimisation will be used on Gaussian to determine particular properites of the reactants to predict the way in which the reaction will proceed.

==='''cis-butadiene and ethylene: reagents'''===

<jmol>
<jmolAppletButton>
<uploadedFileContents>Cis_butadienedh08.mol</uploadedFileContents>
<text>Cis-butadiene</text>
<color>whi-te</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Ethenedh08.mol</uploadedFileContents>
<text>ethylene</text>
<color>white</color>
</jmolAppletButton>
</jmol> were optimised separately using the AM1 semi-empirical method on Gaussview.The cis-butadiene had an energy of 0.04879725 a. u (1.28 x 102 KJ mol-1) and ethylene had an energy of 0.02619027 a. u (68.76 KJ mol-1).

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10: Frontier molecular orbitals of cis-butadiene and ethylene '''
| '''Reagent''' || '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| cis-butadiene || LUMO || [[Image:LUMO1_cis_butadieneDH08.jpg|100x100px]] || [[Image:LUMO2_cis_butadienedh08.jpg|100x100px]]|| Symmetric || +0.017
|-
| cis-butadiene || HOMO || [[Image:HOMO1_cis_butadienedh08.jpg|100x100px]] || [[Image:HOMO2_cis_butadienedh08.jpg|100x100px]]|| Anti-symmetric || -0.343
|-
| ethylene || LUMO || [[Image:LUMO1_ethenedh08.jpg|100x100px]] || [[Image:LUMO2_ethenedh08.jpg|100x100px]]|| Anti-symmetric || +0.052
|-
| ethylene || HOMO || [[Image:HOMO1_ethenedh08.jpg|100x100px]] || [[Image:HOMO2_ethenedh08.jpg|100x100px]]|| Symmetric || -0.387
|}

 
The symmetries of the LUMO and HOMO orbitals for cis-butadiene are opposite of those for the ethylene. This therefore means that the only interactions that can take place are those between ethylene's LUMO and cis-butadiene's HOMO and in addition to this the interaction between cis-butadiene's LUMO and ethylene's HOMO. This is because orbital symmetry has to be conserved (the conservation of orbital symmetry states that the symmetry of an new orbital formed by the overlap of the 2 original orbitals must be the same as the symmetry of the initial orbitals).

==='''cis-butadiene and ethylene: transition state'''===

A <jmol>
<jmolAppletButton>
<uploadedFileContents>Bicyclic_startingdh08.mol</uploadedFileContents>
<text>bicyclo system</text>
<color>white</color>
</jmolAppletButton>
</jmol> was drawn and one of the -CH2-CH2- fragments were removed. Three C=C bonds were added and the previous one was taken away so a <jmol>
<jmolAppletButton>
<uploadedFileContents>Frozen_coorddh08.mol</uploadedFileContents>
<text>guess transition state</text>
<color>white</color>
</jmolAppletButton>
</jmol> was produced.

After this an optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while bonds between the termini of the fragments were frozen at 2.1Å. Following this the resulting file had two separate optimisation and frequency calculations run on it using the TS (Berny) with one using the semi-empirical/AM1 method and the other using the DFT/B3LYP/6-31G* level of theory. Both of these optimisations had the bonds unfrozen and the "derivative" option was chosen instead.

The DFT/B3LYP/6-31G* level of theory was used since it gave more accurate results in the calculations before and the energies it produces are consistent with the rest of the report (semi-empirical/AM1 method gives very different values).

The QST2 optimisation was not used since it required numbering the atoms, which will take a lot of time.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11: Comparison of two methods of optimsations for the transition state '''
| || '''semi-empirical/AM1 method''' || '''DFT/B3LYP/6-31G* method'''
|-
| Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>SE_AM1_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>DFT_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Energy (a. u) || 0.11165608 || -234.54389573
|-
| Bond lengths || [[Image:DistancesSEdh08.jpg|150x150px]]|| [[Image:DistancesDFTdh08.jpg|150x150px]]
|-
|Imaginary frequency (cm-1) || -957 || -525
|-
|Animation || [[Image:-957dh08.gif]] || [[Image:-525dh08.gif]]
|-
| Lowest real frequency (cm-1) || 148 || 136
|-
|Animation || [[Image: 148dh08.gif]] || [[Image: 136dh08.gif]]
|}

The C-C bond forming length between the termini carbons is 0.15 Å larger for the DFT/B3LYP/6-31G* level of theory and this is because this method gives carries out the calculation to a greater degree of accuracy. This larger bond length leads a lower potential energy gradient to the transition state, which gives a lower total energy for the transition state. Therefore the given values below will be the DFT/B3LYP/6-31G* and not the semi-empirical/AM1 method.

A typical sp3 C-C bond length is 1.54 Å and a typical sp2 C=C bond length is 1.34 Å<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref>. Within the optimised transition molecule above the C-C bond lengths are 1.40 Å and the C=C bond length is 1.38 Å. This suggests that the π-bonds are breaking in order for the C-C bonds to form and additionally to this a π-bond is forming in between the two sp3 carbons.

The Van der Waals radius of a C-atom is 1.72 Å<ref name="ref 3">M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer, D. G. Truhlar, 1995, 346, pp. 63-69.[http://pubs.acs.org/doi/abs/10.1021/jp8111556]</ref>. The partly formed C-C length is 2.27 Å, which is shorter than two times the the van der Waals radius of a carbon atom. Therefore the carbon atoms are within a close enough distance to each other to share electron density and yet the bond has not formed at this point.

From looking at the animations for the imaginary frequency above it can be seen that the formation of the two bonds between the termini of the reactants is synchronous. This is expected of the Diels-Alder reaction since the mechanism is concerted. However the lowest real frequency is asynchronous with respect to the two bonds forming between the termini.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12: Frontier molecular orbitals of the transition state '''
| '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| LUMO || [[Image:LUMO1TSdh08.jpg|100x100px]] || [[Image:LUMO2TSdh08.jpg|100x100px]]|| Symmetric || +0.023
|-
| HOMO || [[Image:HOMO1TSdh08.jpg|100x100px]] || [[Image:HOMO2TSdh08.jpg|100x100px]]|| Anti-symmetric || -0.323
|}

From examining the images from table 12 above it can be deduced that the LUMO of the transition state originates from the cis-butadiene's LUMO and the ethylene's HOMO, which represents the σ*-antibonding orbitals. However the HOMO of the transition state originates from the cis-butadiene's HOMO and the ethylene's LUMO, which represents the σ-bonding orbitals within the two single C-C bonds that are formed within the reaction.

==='''cis-butadiene and ethylene: activation energy'''===

The cis-butadiene and ethylene were re-optimised at the DFT/B3LYP/6-31G* level of theory so the activation energy of the reaction could be calculated.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 13: Calculation for the activation energy for the reaction'''
| '''molecule''' || '''DFT/B3LYP/6-31G* energy (a. u)''' || '''Calculations''' || '''Energy '''
|-
| cis-butadiene || -155.98595591 || transition state || -234.54389573 a. u
|-
| ethylene || -78.58745828 || ΔE, activation energy || 0.02951846 a. u
|-
| cis-butadiene and ethylene combined || -234.57341419 || ΔE, activation energy || 77.50 KJ mol-1
|}

The calculated activation energy is 77.50 KJ mol-1, which is of the right magnitude when compared to the literature value (115 ± 8 KJ mol-1 at 0 K<ref name="ref 5">V. Guner, K. S. Khuong, A. G. Leach, P. S. Lee, M. D. Bartberger, K. N. Houk, J. Phys. Chem. A, 2003, 107 (51), pp. 11445–11459.[http://pubs.acs.org/doi/abs/10.1021/jp035501w]</ref>) but there is still an error of 37.50 KJ mol-1. This shows that Gaussview is not perfect but still gives an idea of the activation energy required for the reaction to take place.

==='''Cyclohexa-1,3-diene reaction with maleic anhydride'''===

The reaction scheme below shows the stereoselective Diels-Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene. The reaction is kinetically controlled with the major product being the endo product, which will be explained later on in the exercise.

[[Image:Reaction_schemedh081.jpg|center|500x500px]]

The <jmol>
<jmolAppletButton>
<uploadedFileContents>FrozenTS22dh08.mol</uploadedFileContents>
<text>exo</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_frozenTS22dh08.mol</uploadedFileContents>
<text>endo</text>
<color>white</color>
</jmolAppletButton>
</jmol> transition states were drawn on Gaussview and then the optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while the distance of the forming bonds were frozen at 2.1Å between the respective carbons. After this the structures changed slightly (<jmol>
<jmolAppletButton>
<uploadedFileContents>TS2_afteropt1_exo1dh08.mol</uploadedFileContents>
<text>exo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_TS2_afteropt1dh08.mol</uploadedFileContents>
<text>endo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol>) and the resulting files had an optimisation and frequency calculation run on it using the TA (Berny) with using the semi-empirical/AM1 level of theory while the unfrozen option as changed to "derivative". After this the endo and exo optimised transition state structures were re-optimised as they were before but using the DFT/B3LYP/6-31G* level of theory instead with the additional keyword "Opt=NoEigen".

{| class="wikitable" border="1" align="center"
|+ '''Table 14: Comparison of the exo and endo transition states '''
! rowspan="2"|-!! colspan="2"|''exo'' !! colspan="2"|''endo''
|-
| Semi-empirical/AM1 || DFT/B3LYP/6-31G* || Semi-empirical/AM1 || DFT/B3LYP/6-31G*
|-
| align="center"|Structure ||align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>After_opt2_exodh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>Endo_TS2_afteropt2dh081.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| align="center"| Energy (a. u) ||align="center"| -0.05041975 ||align="center"| -612.67931095 ||align="center"| -0.05150470 ||align="center"| -612.68339684
|-
| C-C bond forming length (Å) ||align="center"| 2.17 ||align="center"| 2.29 ||align="center"| 2.16 ||align="center"| 2.27
|-align="center"
| Imaginary frequency (cm-1) ||align="center"| -812 ||align="center"| -448 ||align="center"| -806 ||align="center"| -447
|-align="center"
| Animation ||colspan="2"|[[Image:-812_SEdh08.gif]] || colspan="2"|[[Image:-806_cm-1dh08.gif]]
|}

Again the C-C bond forming length between the termini carbons is larger (by 0.11 and 0.12 Å) for the DFT/B3LYP/6-31G* level of theory and this is due the same reasons talked about above. However the C-C bond forming length between the termini carbons are very similar in both the endo and exo transition states (0.02 Å difference).

From looking at the animations for the imaginary frequency above it can also be seen that the formation of the two bonds between the carbons in this case as well is synchronous. Again this is expected of the Diels-Alder reaction since it is concerted.

{| align=center
|+ '''Strain diagrams for the endo and exo transition states'''
| [[Image:Exostraindh08.jpg|thumb|200x200px|'''Exo TS''']]
| [[Image:Strainendodh08.jpg|thumb|200x200px|'''Endo TS''']]
|}

From studying the energies in table 14 it can be seen that the endo transition state is more stable than the exo by 4.08589 x 10-3 a. u (10.73 KJ mol-1). However by looking at the strain of both transition states one would expect the exo TS to be more stable. This is because the distance between the cyclohexa-1,3-diene ring and the -(C=O)-O-(C=O)- maleic anhydride fragment is 2.99 Å in the endo TS and 3.03 Å in the exo TS. Thus the endo form would have more strain due to steric hindrance.

However within the exo TS the vicinal hydrogens on the CH2-CH2 (carbon numbers 18 and 21) fragment on the cyclohexa-1,3-diene ring are found below the plane and next to the maleic anhydride. This leads to steric hindrance is this is where the strain originates from within the exo TS. Therefore this disfavours the exo form, which makes the endo more stable.

Another factor making the endo TS more favourable is secondary orbital overlap. These orbitals contribute to the bonds forming and they interact with each other through space. Within the endo TS below there are 2 more orbitals interacting (secondary orbitals) than within the exo TS. This therefore makes the endo TS more stable and therefore more favoured.

[[Image:Secondary_orbital_shizzledh08.jpg|500x500px|center]]

{| class="wikitable" border="1" align="center"
|+ '''Table 15: Frontier molecular orbitals of the the transition state for the cyclohexa-1,3-diene reaction '''
! rowspan="2"|Molecular Orbital !! colspan="4"|''exo'' !! colspan="4"|''endo''
|- align="center"
| '''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''||'''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''
|- align="center"
| LUMO || [[Image:LUMO1exoh081.jpg|150x150px]]|| [[Image:LUMO2exodh081.jpg|150x150px]]|| Anti-symmetric || -0.078 || [[Image:LUMO1ENDOdh08.jpg|150x150px]]||[[Image:LUMO2ENDOdh08.jpg|150x150px]] || Anti-symmetric || -0.067
|- align="center"
| HOMO || [[Image:HOMO1exodh081.jpg|150x150px]]||[[Image:HOMO2exodh081.jpg|150x150px]] || Anti-symmetric|| -0.242 || [[Image:HOMO1ENDOdh08.jpg|150x150px]]|| [[Image:HOMO2ENDOdh08.jpg|150x150px]]||Anti-symmetric || -0.242
|}

Digital repositories:

exo: http://hdl.handle.net/10042/to-5596

endo: http://hdl.handle.net/10042/to-5597

From examining table 15 it is clear to see that both the HOMO and LUMO are anti-symmetric for both the transition states. The LUMO of the maleic anhydride fragment is the π* anti-bonding orbital of the C=C bond. Within the reaction then this has electrons transferred to it and the C=C bond breaks, which leaves just a C-C bond. The LUMOs for both transition states are both anti-bonding with respect to the two fragments. The HOMOs of both transitions state represent the σ-bonding that will be formed between the termini carbons on the products and reactants. This is most likely from the overlap of the cyclohexa-1,3-diene's HOMO and the maleic anhydride's LUMO.

='''References'''=
<references/>

Rep:Mod:hampluscheese

2010-11-12T13:14:32Z

Dh08: /* '''Comparison of optimisation methods for the chair transition state''' */

='''Module 3:Physical Computational Chemistry Lab'''=

Douglas Hunt

=='''Objectives'''==

The aim of this experiment is to is to use various levels of theory (HF/3-21G, DFT/B3LYP/6-31G (d) and AM1/semi-empirical) on Gaussian to calculate various energies, molecular orbitals, geometrical parameters, reaction coordinate pathways and low and imaginary frequencies for a range of molecules and transition states. This will include using different optimisation methods throughout and will lead to a range of findings in various areas of chemical interest.

=='''The Cope Rearrangement'''==

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangement] is a [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement |3,3|-sigmatropic rearrangement] that takes place through a single [http://en.wikipedia.org/wiki/Pericyclic pericyclic], concerted [http://en.wikipedia.org/wiki/Transition_state transition state]. Within this rearrangement a σ bond is formed in concert with another being broken within a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] π system.

The Cope rearrangement we will be studying will be that of 1,5-hexadiene, which is shown below.

[[Image:Cope_rearrangement_schemedh08.jpg|300x300px|center]]

The objective is to locate the low-energy minima and transition structures on the C6H10 potential energy surface and to determine the preferred reaction mechanism by using [http://en.wikipedia.org/wiki/GAUSSIAN Gaussian], with the [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)] methods.

==='''Optimisation of 1,5-hexadiene Conformers'''===

Rough <jmol>
<jmolAppletButton>
<uploadedFileContents>Anti_2_drawndh08.mol</uploadedFileContents>
<text>anti</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche3_drawndh08.mol</uploadedFileContents>
<text>gauche</text>
<color>white</color>
</jmolAppletButton>
</jmol> conformers of 1,5-hexadiene were drawn on Gaussview and their structures were tidied by the clean function. They were then further optimised by the HF/3-21G method and were identified which conformer they were from their point group and appendix 1 on the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3 physical lab page].

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 1: Optimisation and comparison of anti and gauche 1,5-hexadiene conformers using HF/3-21G methods
| ''' Conformer''' || '''Optimised Structure''' || '''Energy (Atomic Units)''' || '''[http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1 Energies (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (Anti2 - Gauche3)''' || '''Point group'''
|-
| '''Anti2''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Anti_2_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692535 || -231.69254 ||-6.08 x 105 || 1.2606 x 10-4 a. u|| Ci
|-
| '''Gauche3''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Gauche3_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692661 || -231.69266 || -6.08 x 105 || 3.31 x 10-1 KJ mol-1 || C1
|}

From studying table 1 it can be seen that the gauche conformer is 0.33 KJ mol-1 more stable than the anti conformation. One would assume that the anti2 conformer would be more stable due to [http://en.wikipedia.org/wiki/Steric steric] factors/repulsion within the gauche3 conformation. However within the gauche3 conformer there is a favourable donation of [http://en.wikipedia.org/wiki/Electron_density electron density] from the C=C π orbital to the C-H σ* orbital from the adjacent proton, which stabilises the conformer<ref name="ref 1">B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446.[http://www.informaworld.com/smpp/content~db=all?content=10.1080/00268970110081412]</ref>.

 

The anti2 conformer was then optimised using the DFT/B3LYP/6-31G* (6-31G (d)) method.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 2: Comparison of the anti2 conformer optimisations carried out at different calculation methods
| ''' Method''' || '''Structure and bond distances (Å)''' || '''Dihedral angle (C1 to C4)''' || '''Energy (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (HF/3-21G - DFT/B3LYP/6-31G*)'''
|-
| HF/3-21G || [[Image:Anti2_opt1dh08.jpg|300x300px]] || 114.6 o || -231.692535 || -6.08 x 105 || 2.919171 a. u
|-
| DFT/B3LYP/6-31G* || [[Image:Anti2_opt2DH08.jpg|300x300px]] || 118.8 o || -234.611716 || -6.16 x 105 || 7.66 x 103 KJ mol-1
|}

From examining table 2 at first it is possible to see that the bond lengths are very similar to each other and that of the literature bond lengths<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref> (C1=C2: 1.340 Å, C2-C3: 1.508 Å and C3-C4: 1.538 Å). It is also possible to see that the C=C bond length is more accurate for the DFT/B3LYP/6-31G* method by it being 0.01 Å closer to the literature value. Another main difference in terms of geometry is that the DFT/B3LYP/6-31G* method gives a larger [http://en.wikipedia.org/wiki/Dihedral_angle dihedral angle] (and most likely more accurate) of 118.8o compared to that of 114.6o.

In terms of energy the DFT/B3LYP/6-31G* method produces a conformer of a higher stability by it possessing a lower energy with a difference of 7.66 x 103 KJ mol-1.

==='''Frequency analysis of 1,5-hexadiene Conformers'''===

[[Image:Vibrations_anti2dh08.jpg|right|300x300px]]

For the frequency analysis the DFT/B3LYP/6-31G* method was used.

Frequency analysis is essential within this calculation due to it being the second derivative of the potential energy surface. This means that if all the frequencies are all positive the system is at a minimum, if one of them is negative the optimised system is at a transition state, and if more than one are negative then the calculation has failed to find a critical point, which means the optimisation has failed. This therefore gives us a method of seeing whether the calculation has worked.

From looking to the image on the right it can deduced that the optimisation has taken place successfully by there being no negative vibrations and therefore the conformation is at a minimum.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 3: Energies under the thermochemistry section at 298.15 K and 0 K within the frequency logfiles'''
| '''Energies''' || '''298.15 K (atomic units)''' ||| '''0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (KJ mol-1)''' || '''Discussion'''
|-
| Sum of electronic and zero-point energies || -234.469177 || -234.466673 || -2.504 x 10-3 || -6.57 || This is the potential energy at 0 K, which includes the zero-point vibrational energy (E = Eelec + ZPE).

|-
| Sum of electronic and thermal energies || -234.461835 || -234.461085 || -7.5 x 10-4|| -1.97 || This is the energy at 298.15 K (0 K for 0 K calculation) and 1 atm of pressure, which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans).

|-
| Sum of electronic and thermal enthalpies || -234.460891 || -234.460141 || -7.5 x 10-4 || -1.97 || This contains an additional correction for RT (H = E + RT), which is especially important when looking at dissociation reactions.
|-
| Sum of electronic and thermal free energies || -234.500739 || -234.495931 || 4.808 x 10-3 || -12.62 || This includes the entropic contribution to the free energy (G = H - TS).
|}

{|align=right
|+
| [[Image:Anti2_freq_IRdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 298.15 K''']]
| [[Image:Anti2_freq_0Kdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 0 K''']]
|}

From looking at table 3 above it can be deduced that the anti2 conformer is more stable at room temperature to that of 0 K with respect to all the types of energies. The spectra also seem to not change much between the two temperatures but the main change is the change in the sum of electronic and thermal free energies, which is heavily dependent of temperature by it including the entropic contribution to the free energy (G = H - '''T'''S).

 
 
 
 

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

Within the Cope Rearrangement there are two possible transition states, the chair (C2h) and the boat (C2v), which are shown within [https://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_2 appendix two ].

==='''Comparison of optimisation methods for the chair transition state'''===

An <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2hdh08.mol</uploadedFileContents>
<text>allyl fragment</text>
<color>white</color>
</jmolAppletButton>
</jmol>, CH2CHCH2 was drawn on Gaussview and was optimized through the HF/3-21G method. After this the original allyl fragment was copied and pasted into a new window and placed exactly above the first fragment with a bond distance of 2.20 Å between the terminal end carbons. This therefore produced the <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2h_fulldh08.mol</uploadedFileContents>
<text>chair transition structure.</text>
<color>white</color>
</jmolAppletButton>
</jmol>

Gaussian was then set up for an optimisation for the transition state with the additional key words "Opt=NoEigen", the optimisation set to TS (Berny), the force constants set to "once" and using the HF/3-21G method.

After this the "chair" transition state was optimised an alternative way by freezing the coordinates of the two sets of terminal carbons (where the bonds form/break during the rearrangement) within the Redundant Coord Editor on Gauusview.
Following this a second calculation was run where the freeze coordinates option was changed to "Derivative" and the force constants were set to "never".

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 4: Comparison of chair transition state optimisation methods
| || '''Optimised through a TS (Berny)''' || '''Optimised through a frozen coordinate method'''
|-
| '''Optimised TS Structure''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>C2h_optfreq_TS_Bdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| '''Terminal C-C bond length (Å)''' || 2.02 || 2.02
|-
|'''Energy (atomic units)''' || -231.61932247 || -231.61932190
|-
| '''ΔE, TS (Berny) - frozen coordinate''' || 5.7 x 10-7 a.u || 1.50 x 10-3 KJ mol-1
|-
| '''Imaginary frequency (cm-1)''' || -818 || -818
|-
| '''Animation''' ||[[Image:818dh08TSB.gif|350x350px]] || [[Image:818cm-1Fdh08.gif]]
|}

The data in table 4 shows that both methods give the same results and they both give the same transition state. They both produced the same energy with a negligible energy difference between them and the bond lengths between the terminal carbons are both 2.02 Å. In addition to this the imaginary frequencies were both -818cm-1 and within the animations above it is possible to see that when one C-C bond is forming the other is breaking. This means it is a asynchronous bond formation, which is expected for the Cope Rearrangement since it is a pericyclic reaction.

The only difference between the two optimisations is the way in which the transition states were calculated. This is due to the TS (Berny) method calculating the force constant matrix within the first optimisation step and is then being recalculated as the optimisation carries on. When the frozen coordinate method freezes the reaction coordinate in order for the rest of the structure to be optimised around it, which does not require the whole hessian (force constant matrix) to be calculated but only that the reaction coordinate is differentiated (second step). The main advantage of this, is that you do not need to calculate the entire force constant matrix and just need to differentiate the reaction coordinate in order to obtain a good enough guess for the initial force constant matrix. The frozen coordinate method will therefore take less time to compute the transition state, which can be very expensive within certain systems. However to carry out this method one needs to have an idea of how far apart the terminal carbons are.

==='''Boat transition state'''===

===='''QST2 optimisation'''====

The optimisation for the boat transition state had a different method and this was the QST2 method. This calculation interpolates between two structures (product and reactant) and tries to find a transition state between them.

The reactants and products were numbered in the same way. Therefore both 1,5-hexadiene molecules had their numbering manually changed for the product molecule so it corresponded to the numbering of the reactant if it rearranged. This is illustrated in the reaction scheme below.

[[Image:Boat_rerrangmentdh08.jpg|300x300px|center]]

The chk file for the optimized anti2 conformer was opened and copied into a new window. A second window was opened within the new window and the same anti2 conformer was copied into the new window. The numbering was changed within the second molecule so the two looked like they do in the image below.

[[Image:TS_(QST2)_numbering_1dh08.jpg|400x400px|center]]

The job type was set as "Opt+Freq" and "TS (QST2)" was chosen to optimise the transition state. However this failed and produced
a structure a bit like a <jmol>
<jmolAppletButton>
<uploadedFileContents>Trialone_BPdh08.mol</uploadedFileContents>
<text>chair transition structure </text>
<color>white</color>
</jmolAppletButton>
</jmol> but more dissociated. This is because it interpolated between the two structures and translated the top allyl fragment and therefore did not consider the possibility of a rotation around the central bonds.

To get around this the C-C-C-C dihedral angle (C2-C3-C4-C5) was set to 0 o and the C-C-C angle (C2-C3-C4 and C3-C4-C5 for the product molecule) was set to 100 o for both molecules. This gave rise to the two structures below.

[[Image:TS_(QST2)_numbering_2dh08.jpg|400x400px|center]]

After this the QST2 calculation was run again and the results in table 5 were produced. The imaginary frequency at -839 cm-1 shows that while one bond is forming the other one is breaking, which is again an asynchronous bond formation. It is also possible to see that the boat transition state has a longer terminal C-C bond length by 0.12 Å to that of the chair.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5: Data for the boat transition state from the QST2 optimisation method
| Optimised TS Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.14
|-
|Energy (atomic units) || -231.60280151
|-
| Energy (KJ mol-1) || -6.08 x 105
|-
| Imaginary frequency (cm-1) || -839
|-
| Animation ||[[Image:-839dh08.gif]]
|}

===='''Intrinsic Reaction Coordinate'''====

It is not possible to predict which product the chair and boat transition state will lead to. To solve this problem within Gaussian one can use the IRC method, which allows one to follow the minimum energy path from a transition structure down to its local minimum on a potential energy surface. This creates a succession of points by taking small geometry steps in the direction to where the gradient of the energy surface is at its steepest until the minimum in energy is reached.

The chk file for the optimized chair transition structure was opened and the job type selected was "IRC", the reaction coordinate was computed in the forward direction (since molecule is symmetric) with 50 points along the IRC. This was done with two type of force constants being calculated, once and always.

Within the "once" method the output showed no bond formation process by the 26th step and therefore the calculation was incomplete, since a complete 1,5-hexadiene structure was supposed to be formed. From looking at the IRC pathway it can be seen that the minimum on the potential energy surface was not reached at that point, since the points on the pathway did not form an asymptote and the gradient did not go close enough to zero.

For the "always" method the computation of force constants were carried out at every step. This therefore lets the IRC calculation distinguish whether it is moving along the minimum energy pathway while going down the potential energy surface. While this is happening it can make changes to the pathway if necessary. Within the "always" case the bond forming arrives at 1,5-hexadiene and this is backed up by the IRC pathway reaching a minimum within its energy by the gradient going close to zero and the pathway reaches an asymptote.

The structure produced through the always method had an energy of -231.69199434 atomic units, which is very close to the energy of the gauche2 conformer in the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 appendix 1]. This suggests that the reaction takes place through the chair transition state, which later produces the gauche2 conformer of 1,5-hexadiene.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 6: Comparison of IRC calculations for the chair structure
| || '''Calculate force constants once''' || '''Calculate force constants always'''
|-
| '''Number of points along IRC''' || 26 || 47
|-
| '''Structure at the last point of IRC''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Once_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Always_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|'''Energy (atomic units)''' || -231.68906923 || -231.69199434
|-
| '''ΔE, once - always''' || 2.92511 x 10-3 a.u || 7.68 KJ mol-1
|-
| '''IRC pathway''' || [[Image:One_pathwaydh08.jpg|400x400px]] || [[Image:Always_pathwaydh08.jpg|400x400px]]
|-
| '''IRC gradient''' || [[Image:One_gradientdh08.jpg|400x400px]]|| [[Image:Always_gradientdh08.jpg|400x400px]]
|}

 
 
 

===='''Comparison of activation energies for the transition states'''====

The HF/3-21G optimised chair and boat transition structures were both reoptimised using the B3LYP/6-31G* (B3LYP/6-31G (d)) method. This gives a more accurate minimisation however this method takes longer to run. The values for the zero-point energies and thermal energies were found within the thermochemistry section of the log file.

The values for the HF/3-21G reactant (anti2) optimisation were calculated and the B3LYP/6-31G* values were taken from the "Frequency analysis of 1,5-hexadiene Conformers" earlier in this module.

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 7: Energy data for chair, boat and the anti2 conformation from the HF/3-21G and B3LYP/6-31G* methods of optimisation '''
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466677
| align="center" | -231.461320
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450936
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402342
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539524
| align="center" | -231.532562
| align="center" | -234.611716
| align="center" | -234.469177
| align="center" | -234.461835
|}
Digital repositories

Chair:
HF/3-21G: http://hdl.handle.net/10042/to-5569
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5568

Boat:
HF/3-21G: http://hdl.handle.net/10042/to-5570
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5571

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 8: Summary of activation energies for the chair and boat conformations (ΔE = chair TS or boat TS minus anti2)'''
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| align="center" | 1 a.u = 627.509 kcal mol-1
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE, chair (a.u)'''
| align="center" | 0.072847
| align="center" | 0.071242
| align="center" | 0.054248
| align="center" | 0.052827
| align="center" | -
|-
| '''ΔE, chair (kcal mol-1)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE, boat (a. u)'''
| align="center" | 0.088588
| align="center" | 0.087259
| align="center" | 0.066835
| align="center" | 0.065827
| align="center" | -
|-
| '''ΔE, boat (kcal mol-1)'''
| align="center" | 55.59
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.31
| align="center" | 44.7 ± 2.0
|}

By examining table 8 it is clear to see that the B3LYP/6-31G* method gives values which are much closer to the experimental ones. This is due to the B3LYP/6-31G* giving a higher level of optimisation than the HF/3-21G method. Table 8 also shows that the chair transition state has a lower activation energy to that of the boat transition state by 8.16 kcal mol-1. This suggests that the Cope rearrangement mechanism would most likely proceed through the chair transition state to that of the boat. Therefore the reaction would take place from the anti2 conformer to the chair transition state and then onto the gauche2 conformer within the Cope rearrangement.

It is also possible to see that the calcualted energies are very close to the literature values with the chair conformation being closer to its literature value to that of the boat.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9: Geometric parameters for the optimised chair and boat structures '''
| || ''' HF/3-21G chair TS''' || ''' B3LYP/6-31G* chair TS''' || '''HF/3-21G boat TS''' || '''B3LYP/6-31G* boat TS'''
|-
| Optimised Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Chair_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Boat_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.02 (C5-C10 or C3-C14)|| 1.97 (C5-C10 or C3-C14) || 2.14 (C3-C4 or C1-C6) || 2.21 (C3-C4 or C1-C6)
|-
| Fragment angle || 120 o (C3-C1-C5) || 120 o (C3-C1-C5) || 122 o (C4-C5-C6) || 122 o (C4-C5-C6)
|-
| Dihedral angle || 55 o (C1-C3-C14-C11)|| 54 o (C1-C3-C14-C11)|| 0 o (C2-C1-C6-C5) || 0 o (C2-C1-C6-C5)
|-
| Numbering of carbons || [[Image:Numbering_chairdh08.jpg|200x200px]]|| [[Image:Numbering_chairdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]]
|}

From studying table 9 it can be seen that the C-C bond breaking/forming length has changed between the two different types of optimisation methods. However the angles between the bonds remains unchanged.

=='''The Diels Alder Cycloaddition'''==

The Diels-Alder reaction is a cycloaddition between a conjugated diene and a dienophile (alkene system), which forms a cyclohexene system. It is often referred to as a [4s+2s] cycloaddition and proceeds through a single cyclic, concerted transition state where 2 σ bonds are formed in between the termini carbons of the 2 п systems. This is shown in the reaction scheme below where ethene is reacting with cis-butadiene.

[[Image:Reaction_scheme2dh08.jpg|300x300px|center]]

The π orbitals of the dienophile form new σ bonds with the π orbitals of the diene. The number of π electrons involved determine whether the reaction proceeds in a concerted stereospecific fashion (allowed) or not (forbidden). The HOMO/LUMO of one fragment interacts with the HOMO/LUMO of the other fragment to form two new bonding and anti-bonding MOs. The nodal properties allow one to predict how the reaction will occur.

Semi-Empirical/AM1 method and the B3LYP/6-31G* method of optimisation will be used on Gaussian to determine particular properites of the reactants to predict the way in which the reaction will proceed.

==='''cis-butadiene and ethylene: reagents'''===

<jmol>
<jmolAppletButton>
<uploadedFileContents>Cis_butadienedh08.mol</uploadedFileContents>
<text>Cis-butadiene</text>
<color>whi-te</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Ethenedh08.mol</uploadedFileContents>
<text>ethylene</text>
<color>white</color>
</jmolAppletButton>
</jmol> were optimised separately using the AM1 semi-empirical method on Gaussview.The cis-butadiene had an energy of 0.04879725 a. u (1.28 x 102 KJ mol-1) and ethylene had an energy of 0.02619027 a. u (68.76 KJ mol-1).

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10: Frontier molecular orbitals of cis-butadiene and ethylene '''
| '''Reagent''' || '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| cis-butadiene || LUMO || [[Image:LUMO1_cis_butadieneDH08.jpg|100x100px]] || [[Image:LUMO2_cis_butadienedh08.jpg|100x100px]]|| Symmetric || +0.017
|-
| cis-butadiene || HOMO || [[Image:HOMO1_cis_butadienedh08.jpg|100x100px]] || [[Image:HOMO2_cis_butadienedh08.jpg|100x100px]]|| Anti-symmetric || -0.343
|-
| ethylene || LUMO || [[Image:LUMO1_ethenedh08.jpg|100x100px]] || [[Image:LUMO2_ethenedh08.jpg|100x100px]]|| Anti-symmetric || +0.052
|-
| ethylene || HOMO || [[Image:HOMO1_ethenedh08.jpg|100x100px]] || [[Image:HOMO2_ethenedh08.jpg|100x100px]]|| Symmetric || -0.387
|}

 
The symmetries of the LUMO and HOMO orbitals for cis-butadiene are opposite of those for the ethylene. This therefore means that the only interactions that can take place are those between ethylene's LUMO and cis-butadiene's HOMO and in addition to this the interaction between cis-butadiene's LUMO and ethylene's HOMO. This is because orbital symmetry has to be conserved (the conservation of orbital symmetry states that the symmetry of an new orbital formed by the overlap of the 2 original orbitals must be the same as the symmetry of the initial orbitals).

==='''cis-butadiene and ethylene: transition state'''===

A <jmol>
<jmolAppletButton>
<uploadedFileContents>Bicyclic_startingdh08.mol</uploadedFileContents>
<text>bicyclo system</text>
<color>white</color>
</jmolAppletButton>
</jmol> was drawn and one of the -CH2-CH2- fragments were removed. Three C=C bonds were added and the previous one was taken away so a <jmol>
<jmolAppletButton>
<uploadedFileContents>Frozen_coorddh08.mol</uploadedFileContents>
<text>guess transition state</text>
<color>white</color>
</jmolAppletButton>
</jmol> was produced.

After this an optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while bonds between the termini of the fragments were frozen at 2.1Å. Following this the resulting file had two separate optimisation and frequency calculations run on it using the TS (Berny) with one using the semi-empirical/AM1 method and the other using the DFT/B3LYP/6-31G* level of theory. Both of these optimisations had the bonds unfrozen and the "derivative" option was chosen instead.

The DFT/B3LYP/6-31G* level of theory was used since it gave more accurate results in the calculations before and the energies it produces are consistent with the rest of the report (semi-empirical/AM1 method gives very different values).

The QST2 optimisation was not used since it required numbering the atoms, which will take a lot of time.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11: Comparison of two methods of optimsations for the transition state '''
| || '''semi-empirical/AM1 method''' || '''DFT/B3LYP/6-31G* method'''
|-
| Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>SE_AM1_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>DFT_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Energy (a. u) || 0.11165608 || -234.54389573
|-
| Bond lengths || [[Image:DistancesSEdh08.jpg|150x150px]]|| [[Image:DistancesDFTdh08.jpg|150x150px]]
|-
|Imaginary frequency (cm-1) || -957 || -525
|-
|Animation || [[Image:-957dh08.gif]] || [[Image:-525dh08.gif]]
|-
| Lowest real frequency (cm-1) || 148 || 136
|-
|Animation || [[Image: 148dh08.gif]] || [[Image: 136dh08.gif]]
|}

The C-C bond forming length between the termini carbons is 0.15 Å larger for the DFT/B3LYP/6-31G* level of theory and this is because this method gives carries out the calculation to a greater degree of accuracy. This larger bond length leads a lower potential energy gradient to the transition state, which gives a lower total energy for the transition state. Therefore the given values below will be the DFT/B3LYP/6-31G* and not the semi-empirical/AM1 method.

A typical sp3 C-C bond length is 1.54 Å and a typical sp2 C=C bond length is 1.34 Å<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref>. Within the optimised transition molecule above the C-C bond lengths are 1.40 Å and the C=C bond length is 1.38 Å. This suggests that the π-bonds are breaking in order for the C-C bonds to form and additionally to this a π-bond is forming in between the two sp3 carbons.

The Van der Waals radius of a C-atom is 1.72 Å<ref name="ref 3">M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer, D. G. Truhlar, 1995, 346, pp. 63-69.[http://pubs.acs.org/doi/abs/10.1021/jp8111556]</ref>. The partly formed C-C length is 2.27 Å, which is shorter than two times the the van der Waals radius of a carbon atom. Therefore the carbon atoms are within a close enough distance to each other to share electron density and yet the bond has not formed at this point.

From looking at the animations for the imaginary frequency above it can be seen that the formation of the two bonds between the termini of the reactants is synchronous. This is expected of the Diels-Alder reaction since the mechanism is concerted. However the lowest real frequency is asynchronous with respect to the two bonds forming between the termini.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12: Frontier molecular orbitals of the transition state '''
| '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| LUMO || [[Image:LUMO1TSdh08.jpg|100x100px]] || [[Image:LUMO2TSdh08.jpg|100x100px]]|| Symmetric || +0.023
|-
| HOMO || [[Image:HOMO1TSdh08.jpg|100x100px]] || [[Image:HOMO2TSdh08.jpg|100x100px]]|| Anti-symmetric || -0.323
|}

From examining the images from table 12 above it can be deduced that the LUMO of the transition state originates from the cis-butadiene's LUMO and the ethylene's HOMO, which represents the σ*-antibonding orbitals. However the HOMO of the transition state originates from the cis-butadiene's HOMO and the ethylene's LUMO, which represents the σ-bonding orbitals within the two single C-C bonds that are formed within the reaction.

==='''cis-butadiene and ethylene: activation energy'''===

The cis-butadiene and ethylene were re-optimised at the DFT/B3LYP/6-31G* level of theory so the activation energy of the reaction could be calculated.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 13: Calculation for the activation energy for the reaction'''
| '''molecule''' || '''DFT/B3LYP/6-31G* energy (a. u)''' || '''Calculations''' || '''Energy '''
|-
| cis-butadiene || -155.98595591 || transition state || -234.54389573 a. u
|-
| ethylene || -78.58745828 || ΔE, activation energy || 0.02951846 a. u
|-
| cis-butadiene and ethylene combined || -234.57341419 || ΔE, activation energy || 77.50 KJ mol-1
|}

The calculated activation energy is 77.50 KJ mol-1, which is of the right magnitude when compared to the literature value (115 ± 8 KJ mol-1 at 0 K<ref name="ref 5">V. Guner, K. S. Khuong, A. G. Leach, P. S. Lee, M. D. Bartberger, K. N. Houk, J. Phys. Chem. A, 2003, 107 (51), pp. 11445–11459.[http://pubs.acs.org/doi/abs/10.1021/jp035501w]</ref>) but there is still an error of 37.50 KJ mol-1. This shows that Gaussview is not perfect but still gives an idea of the activation energy required for the reaction to take place.

==='''Cyclohexa-1,3-diene reaction with maleic anhydride'''===

The reaction scheme below shows the stereoselective Diels-Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene. The reaction is kinetically controlled with the major product being the endo product, which will be explained later on in the exercise.

[[Image:Reaction_schemedh081.jpg|center|500x500px]]

The <jmol>
<jmolAppletButton>
<uploadedFileContents>FrozenTS22dh08.mol</uploadedFileContents>
<text>exo</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_frozenTS22dh08.mol</uploadedFileContents>
<text>endo</text>
<color>white</color>
</jmolAppletButton>
</jmol> transition states were drawn on Gaussview and then the optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while the distance of the forming bonds were frozen at 2.1Å between the respective carbons. After this the structures changed slightly (<jmol>
<jmolAppletButton>
<uploadedFileContents>TS2_afteropt1_exo1dh08.mol</uploadedFileContents>
<text>exo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_TS2_afteropt1dh08.mol</uploadedFileContents>
<text>endo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol>) and the resulting files had an optimisation and frequency calculation run on it using the TA (Berny) with using the semi-empirical/AM1 level of theory while the unfrozen option as changed to "derivative". After this the endo and exo optimised transition state structures were re-optimised as they were before but using the DFT/B3LYP/6-31G* level of theory instead with the additional keyword "Opt=NoEigen".

{| class="wikitable" border="1" align="center"
|+ '''Table 14: Comparison of the exo and endo transition states '''
! rowspan="2"|-!! colspan="2"|''exo'' !! colspan="2"|''endo''
|-
| Semi-empirical/AM1 || DFT/B3LYP/6-31G* || Semi-empirical/AM1 || DFT/B3LYP/6-31G*
|-
| align="center"|Structure ||align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>After_opt2_exodh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>Endo_TS2_afteropt2dh081.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| align="center"| Energy (a. u) ||align="center"| -0.05041975 ||align="center"| -612.67931095 ||align="center"| -0.05150470 ||align="center"| -612.68339684
|-
| C-C bond forming length (Å) ||align="center"| 2.17 ||align="center"| 2.29 ||align="center"| 2.16 ||align="center"| 2.27
|-align="center"
| Imaginary frequency (cm-1) ||align="center"| -812 ||align="center"| -448 ||align="center"| -806 ||align="center"| -447
|-align="center"
| Animation ||colspan="2"|[[Image:-812_SEdh08.gif]] || colspan="2"|[[Image:-806_cm-1dh08.gif]]
|}

Again the C-C bond forming length between the termini carbons is larger (by 0.11 and 0.12 Å) for the DFT/B3LYP/6-31G* level of theory and this is due the same reasons talked about above. However the C-C bond forming length between the termini carbons are very similar in both the endo and exo transition states (0.02 Å difference).

From looking at the animations for the imaginary frequency above it can also be seen that the formation of the two bonds between the carbons in this case as well is synchronous. Again this is expected of the Diels-Alder reaction since it is concerted.

{| align=center
|+ '''Strain diagrams for the endo and exo transition states'''
| [[Image:Exostraindh08.jpg|thumb|200x200px|'''Exo TS''']]
| [[Image:Strainendodh08.jpg|thumb|200x200px|'''Endo TS''']]
|}

From studying the energies in table 14 it can be seen that the endo transition state is more stable than the exo by 4.08589 x 10-3 a. u (10.73 KJ mol-1). However by looking at the strain of both transition states one would expect the exo TS to be more stable. This is because the distance between the cyclohexa-1,3-diene ring and the -(C=O)-O-(C=O)- maleic anhydride fragment is 2.99 Å in the endo TS and 3.03 Å in the exo TS. Thus the endo form would have more strain due to steric hindrance.

However within the exo TS the vicinal hydrogens on the CH2-CH2 (carbon numbers 18 and 21) fragment on the cyclohexa-1,3-diene ring are found below the plane and next to the maleic anhydride. This leads to steric hindrance is this is where the strain originates from within the exo TS. Therefore this disfavours the exo form, which makes the endo more stable.

Another factor making the endo TS more favourable is secondary orbital overlap. These orbitals contribute to the bonds forming and they interact with each other through space. Within the endo TS below there are 2 more orbitals interacting (secondary orbitals) than within the exo TS. This therefore makes the endo TS more stable and therefore more favoured.

[[Image:Secondary_orbital_shizzledh08.jpg|500x500px|center]]

{| class="wikitable" border="1" align="center"
|+ '''Table 15: Frontier molecular orbitals of the the transition state for the cyclohexa-1,3-diene reaction '''
! rowspan="2"|Molecular Orbital !! colspan="4"|''exo'' !! colspan="4"|''endo''
|- align="center"
| '''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''||'''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''
|- align="center"
| LUMO || [[Image:LUMO1exoh081.jpg|150x150px]]|| [[Image:LUMO2exodh081.jpg|150x150px]]|| Anti-symmetric || -0.078 || [[Image:LUMO1ENDOdh08.jpg|150x150px]]||[[Image:LUMO2ENDOdh08.jpg|150x150px]] || Anti-symmetric || -0.067
|- align="center"
| HOMO || [[Image:HOMO1exodh081.jpg|150x150px]]||[[Image:HOMO2exodh081.jpg|150x150px]] || Anti-symmetric|| -0.242 || [[Image:HOMO1ENDOdh08.jpg|150x150px]]|| [[Image:HOMO2ENDOdh08.jpg|150x150px]]||Anti-symmetric || -0.242
|}

Digital repositories:

exo: http://hdl.handle.net/10042/to-5596

endo: http://hdl.handle.net/10042/to-5597

From examining table 15 it is clear to see that both the HOMO and LUMO are anti-symmetric for both the transition states. The LUMO of the maleic anhydride fragment is the π* anti-bonding orbital of the C=C bond. Within the reaction then this has electrons transferred to it and the C=C bond breaks, which leaves just a C-C bond. The LUMOs for both transition states are both anti-bonding with respect to the two fragments. The HOMOs of both transitions state represent the σ-bonding that will be formed between the termini carbons on the products and reactants. This is most likely from the overlap of the cyclohexa-1,3-diene's HOMO and the maleic anhydride's LUMO.

='''References'''=
<references/>

Rep:Mod:hampluscheese

2010-11-12T13:13:54Z

Dh08: /* '''Frequency analysis of 1,5-hexadiene Conformers''' */

='''Module 3:Physical Computational Chemistry Lab'''=

Douglas Hunt

=='''Objectives'''==

The aim of this experiment is to is to use various levels of theory (HF/3-21G, DFT/B3LYP/6-31G (d) and AM1/semi-empirical) on Gaussian to calculate various energies, molecular orbitals, geometrical parameters, reaction coordinate pathways and low and imaginary frequencies for a range of molecules and transition states. This will include using different optimisation methods throughout and will lead to a range of findings in various areas of chemical interest.

=='''The Cope Rearrangement'''==

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangement] is a [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement |3,3|-sigmatropic rearrangement] that takes place through a single [http://en.wikipedia.org/wiki/Pericyclic pericyclic], concerted [http://en.wikipedia.org/wiki/Transition_state transition state]. Within this rearrangement a σ bond is formed in concert with another being broken within a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] π system.

The Cope rearrangement we will be studying will be that of 1,5-hexadiene, which is shown below.

[[Image:Cope_rearrangement_schemedh08.jpg|300x300px|center]]

The objective is to locate the low-energy minima and transition structures on the C6H10 potential energy surface and to determine the preferred reaction mechanism by using [http://en.wikipedia.org/wiki/GAUSSIAN Gaussian], with the [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)] methods.

==='''Optimisation of 1,5-hexadiene Conformers'''===

Rough <jmol>
<jmolAppletButton>
<uploadedFileContents>Anti_2_drawndh08.mol</uploadedFileContents>
<text>anti</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche3_drawndh08.mol</uploadedFileContents>
<text>gauche</text>
<color>white</color>
</jmolAppletButton>
</jmol> conformers of 1,5-hexadiene were drawn on Gaussview and their structures were tidied by the clean function. They were then further optimised by the HF/3-21G method and were identified which conformer they were from their point group and appendix 1 on the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3 physical lab page].

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 1: Optimisation and comparison of anti and gauche 1,5-hexadiene conformers using HF/3-21G methods
| ''' Conformer''' || '''Optimised Structure''' || '''Energy (Atomic Units)''' || '''[http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1 Energies (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (Anti2 - Gauche3)''' || '''Point group'''
|-
| '''Anti2''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Anti_2_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692535 || -231.69254 ||-6.08 x 105 || 1.2606 x 10-4 a. u|| Ci
|-
| '''Gauche3''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Gauche3_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692661 || -231.69266 || -6.08 x 105 || 3.31 x 10-1 KJ mol-1 || C1
|}

From studying table 1 it can be seen that the gauche conformer is 0.33 KJ mol-1 more stable than the anti conformation. One would assume that the anti2 conformer would be more stable due to [http://en.wikipedia.org/wiki/Steric steric] factors/repulsion within the gauche3 conformation. However within the gauche3 conformer there is a favourable donation of [http://en.wikipedia.org/wiki/Electron_density electron density] from the C=C π orbital to the C-H σ* orbital from the adjacent proton, which stabilises the conformer<ref name="ref 1">B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446.[http://www.informaworld.com/smpp/content~db=all?content=10.1080/00268970110081412]</ref>.

 

The anti2 conformer was then optimised using the DFT/B3LYP/6-31G* (6-31G (d)) method.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 2: Comparison of the anti2 conformer optimisations carried out at different calculation methods
| ''' Method''' || '''Structure and bond distances (Å)''' || '''Dihedral angle (C1 to C4)''' || '''Energy (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (HF/3-21G - DFT/B3LYP/6-31G*)'''
|-
| HF/3-21G || [[Image:Anti2_opt1dh08.jpg|300x300px]] || 114.6 o || -231.692535 || -6.08 x 105 || 2.919171 a. u
|-
| DFT/B3LYP/6-31G* || [[Image:Anti2_opt2DH08.jpg|300x300px]] || 118.8 o || -234.611716 || -6.16 x 105 || 7.66 x 103 KJ mol-1
|}

From examining table 2 at first it is possible to see that the bond lengths are very similar to each other and that of the literature bond lengths<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref> (C1=C2: 1.340 Å, C2-C3: 1.508 Å and C3-C4: 1.538 Å). It is also possible to see that the C=C bond length is more accurate for the DFT/B3LYP/6-31G* method by it being 0.01 Å closer to the literature value. Another main difference in terms of geometry is that the DFT/B3LYP/6-31G* method gives a larger [http://en.wikipedia.org/wiki/Dihedral_angle dihedral angle] (and most likely more accurate) of 118.8o compared to that of 114.6o.

In terms of energy the DFT/B3LYP/6-31G* method produces a conformer of a higher stability by it possessing a lower energy with a difference of 7.66 x 103 KJ mol-1.

==='''Frequency analysis of 1,5-hexadiene Conformers'''===

[[Image:Vibrations_anti2dh08.jpg|right|300x300px]]

For the frequency analysis the DFT/B3LYP/6-31G* method was used.

Frequency analysis is essential within this calculation due to it being the second derivative of the potential energy surface. This means that if all the frequencies are all positive the system is at a minimum, if one of them is negative the optimised system is at a transition state, and if more than one are negative then the calculation has failed to find a critical point, which means the optimisation has failed. This therefore gives us a method of seeing whether the calculation has worked.

From looking to the image on the right it can deduced that the optimisation has taken place successfully by there being no negative vibrations and therefore the conformation is at a minimum.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 3: Energies under the thermochemistry section at 298.15 K and 0 K within the frequency logfiles'''
| '''Energies''' || '''298.15 K (atomic units)''' ||| '''0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (KJ mol-1)''' || '''Discussion'''
|-
| Sum of electronic and zero-point energies || -234.469177 || -234.466673 || -2.504 x 10-3 || -6.57 || This is the potential energy at 0 K, which includes the zero-point vibrational energy (E = Eelec + ZPE).

|-
| Sum of electronic and thermal energies || -234.461835 || -234.461085 || -7.5 x 10-4|| -1.97 || This is the energy at 298.15 K (0 K for 0 K calculation) and 1 atm of pressure, which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans).

|-
| Sum of electronic and thermal enthalpies || -234.460891 || -234.460141 || -7.5 x 10-4 || -1.97 || This contains an additional correction for RT (H = E + RT), which is especially important when looking at dissociation reactions.
|-
| Sum of electronic and thermal free energies || -234.500739 || -234.495931 || 4.808 x 10-3 || -12.62 || This includes the entropic contribution to the free energy (G = H - TS).
|}

{|align=right
|+
| [[Image:Anti2_freq_IRdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 298.15 K''']]
| [[Image:Anti2_freq_0Kdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 0 K''']]
|}

From looking at table 3 above it can be deduced that the anti2 conformer is more stable at room temperature to that of 0 K with respect to all the types of energies. The spectra also seem to not change much between the two temperatures but the main change is the change in the sum of electronic and thermal free energies, which is heavily dependent of temperature by it including the entropic contribution to the free energy (G = H - '''T'''S).

 
 
 
 

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

Within the Cope Rearrangement there are two possible transition states, the chair (C2h) and the boat (C2v), which are shown within [https://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_2 appendix two ].

==='''Comparison of optimisation methods for the chair transition state'''===

An <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2hdh08.mol</uploadedFileContents>
<text>allyl fragment</text>
<color>white</color>
</jmolAppletButton>
</jmol>, CH2CHCH2 was drawn on Gaussview and was optimized through the HF/3-21G method. After this the original allyl fragment was copied and pasted into a new window and place exactly above the first fragment with a bond distance of 2.20 Å between the terminal end carbons. This therefore produced the <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2h_fulldh08.mol</uploadedFileContents>
<text>chair transition structure.</text>
<color>white</color>
</jmolAppletButton>
</jmol>

Gaussian was then set up for an optimisation for the transition state with the additional key words "Opt=NoEigen", the optimisation set to TS (Berny), the force constants set to "once" and using the HF/3-21G method.

After this the "chair" transition state was optimised an alternative way by freezing the coordinates of the two sets of terminal carbons (where the bonds form/break during the rearrangement) within the Redundant Coord Editor on Gauusview.
Following this a second calculation was run where the freeze coordinates option was changed to "Derivative" and the force constants were set to "never".

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 4: Comparison of chair transition state optimisation methods
| || '''Optimised through a TS (Berny)''' || '''Optimised through a frozen coordinate method'''
|-
| '''Optimised TS Structure''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>C2h_optfreq_TS_Bdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| '''Terminal C-C bond length (Å)''' || 2.02 || 2.02
|-
|'''Energy (atomic units)''' || -231.61932247 || -231.61932190
|-
| '''ΔE, TS (Berny) - frozen coordinate''' || 5.7 x 10-7 a.u || 1.50 x 10-3 KJ mol-1
|-
| '''Imaginary frequency (cm-1)''' || -818 || -818
|-
| '''Animation''' ||[[Image:818dh08TSB.gif|350x350px]] || [[Image:818cm-1Fdh08.gif]]
|}

The data in table 4 shows that both methods give the same results and they both give the same transition state. They both produced the same energy with a negligible energy difference between them and the bond lengths between the terminal carbons are both 2.02 Å. In addition to this the imaginary frequencies were both -818cm-1 and within the animations above it is possible to see that when one C-C bond is forming the other is breaking. This means it is a asynchronous bond formation, which is expected for the Cope Rearrangement since it is a pericyclic reaction.

The only difference between the two optimisations is the way in which the transition states were calculated. This is due to the TS (Berny) method calculating the force constant matrix within the first optimisation step and is then being recalculated as the optimisation carries on. When the frozen coordinate method freezes the reaction coordinate in order for the rest of the structure to be optimised around it, which does not require the whole hessian (force constant matrix) to be calculated but only that the reaction coordinate is differentiated (second step). The main advantage of this, is that you do not need to calculate the entire force constant matrix and just need to differentiate the reaction coordinate in order to obtain a good enough guess for the initial force constant matrix. The frozen coordinate method will therefore take less time to compute the transition state, which can be very expensive within certain systems. However to carry out this method one needs to have an idea of how far apart the terminal carbons are.

==='''Boat transition state'''===

===='''QST2 optimisation'''====

The optimisation for the boat transition state had a different method and this was the QST2 method. This calculation interpolates between two structures (product and reactant) and tries to find a transition state between them.

The reactants and products were numbered in the same way. Therefore both 1,5-hexadiene molecules had their numbering manually changed for the product molecule so it corresponded to the numbering of the reactant if it rearranged. This is illustrated in the reaction scheme below.

[[Image:Boat_rerrangmentdh08.jpg|300x300px|center]]

The chk file for the optimized anti2 conformer was opened and copied into a new window. A second window was opened within the new window and the same anti2 conformer was copied into the new window. The numbering was changed within the second molecule so the two looked like they do in the image below.

[[Image:TS_(QST2)_numbering_1dh08.jpg|400x400px|center]]

The job type was set as "Opt+Freq" and "TS (QST2)" was chosen to optimise the transition state. However this failed and produced
a structure a bit like a <jmol>
<jmolAppletButton>
<uploadedFileContents>Trialone_BPdh08.mol</uploadedFileContents>
<text>chair transition structure </text>
<color>white</color>
</jmolAppletButton>
</jmol> but more dissociated. This is because it interpolated between the two structures and translated the top allyl fragment and therefore did not consider the possibility of a rotation around the central bonds.

To get around this the C-C-C-C dihedral angle (C2-C3-C4-C5) was set to 0 o and the C-C-C angle (C2-C3-C4 and C3-C4-C5 for the product molecule) was set to 100 o for both molecules. This gave rise to the two structures below.

[[Image:TS_(QST2)_numbering_2dh08.jpg|400x400px|center]]

After this the QST2 calculation was run again and the results in table 5 were produced. The imaginary frequency at -839 cm-1 shows that while one bond is forming the other one is breaking, which is again an asynchronous bond formation. It is also possible to see that the boat transition state has a longer terminal C-C bond length by 0.12 Å to that of the chair.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5: Data for the boat transition state from the QST2 optimisation method
| Optimised TS Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.14
|-
|Energy (atomic units) || -231.60280151
|-
| Energy (KJ mol-1) || -6.08 x 105
|-
| Imaginary frequency (cm-1) || -839
|-
| Animation ||[[Image:-839dh08.gif]]
|}

===='''Intrinsic Reaction Coordinate'''====

It is not possible to predict which product the chair and boat transition state will lead to. To solve this problem within Gaussian one can use the IRC method, which allows one to follow the minimum energy path from a transition structure down to its local minimum on a potential energy surface. This creates a succession of points by taking small geometry steps in the direction to where the gradient of the energy surface is at its steepest until the minimum in energy is reached.

The chk file for the optimized chair transition structure was opened and the job type selected was "IRC", the reaction coordinate was computed in the forward direction (since molecule is symmetric) with 50 points along the IRC. This was done with two type of force constants being calculated, once and always.

Within the "once" method the output showed no bond formation process by the 26th step and therefore the calculation was incomplete, since a complete 1,5-hexadiene structure was supposed to be formed. From looking at the IRC pathway it can be seen that the minimum on the potential energy surface was not reached at that point, since the points on the pathway did not form an asymptote and the gradient did not go close enough to zero.

For the "always" method the computation of force constants were carried out at every step. This therefore lets the IRC calculation distinguish whether it is moving along the minimum energy pathway while going down the potential energy surface. While this is happening it can make changes to the pathway if necessary. Within the "always" case the bond forming arrives at 1,5-hexadiene and this is backed up by the IRC pathway reaching a minimum within its energy by the gradient going close to zero and the pathway reaches an asymptote.

The structure produced through the always method had an energy of -231.69199434 atomic units, which is very close to the energy of the gauche2 conformer in the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 appendix 1]. This suggests that the reaction takes place through the chair transition state, which later produces the gauche2 conformer of 1,5-hexadiene.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 6: Comparison of IRC calculations for the chair structure
| || '''Calculate force constants once''' || '''Calculate force constants always'''
|-
| '''Number of points along IRC''' || 26 || 47
|-
| '''Structure at the last point of IRC''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Once_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Always_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|'''Energy (atomic units)''' || -231.68906923 || -231.69199434
|-
| '''ΔE, once - always''' || 2.92511 x 10-3 a.u || 7.68 KJ mol-1
|-
| '''IRC pathway''' || [[Image:One_pathwaydh08.jpg|400x400px]] || [[Image:Always_pathwaydh08.jpg|400x400px]]
|-
| '''IRC gradient''' || [[Image:One_gradientdh08.jpg|400x400px]]|| [[Image:Always_gradientdh08.jpg|400x400px]]
|}

 
 
 

===='''Comparison of activation energies for the transition states'''====

The HF/3-21G optimised chair and boat transition structures were both reoptimised using the B3LYP/6-31G* (B3LYP/6-31G (d)) method. This gives a more accurate minimisation however this method takes longer to run. The values for the zero-point energies and thermal energies were found within the thermochemistry section of the log file.

The values for the HF/3-21G reactant (anti2) optimisation were calculated and the B3LYP/6-31G* values were taken from the "Frequency analysis of 1,5-hexadiene Conformers" earlier in this module.

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 7: Energy data for chair, boat and the anti2 conformation from the HF/3-21G and B3LYP/6-31G* methods of optimisation '''
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466677
| align="center" | -231.461320
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450936
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402342
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539524
| align="center" | -231.532562
| align="center" | -234.611716
| align="center" | -234.469177
| align="center" | -234.461835
|}
Digital repositories

Chair:
HF/3-21G: http://hdl.handle.net/10042/to-5569
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5568

Boat:
HF/3-21G: http://hdl.handle.net/10042/to-5570
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5571

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 8: Summary of activation energies for the chair and boat conformations (ΔE = chair TS or boat TS minus anti2)'''
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| align="center" | 1 a.u = 627.509 kcal mol-1
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE, chair (a.u)'''
| align="center" | 0.072847
| align="center" | 0.071242
| align="center" | 0.054248
| align="center" | 0.052827
| align="center" | -
|-
| '''ΔE, chair (kcal mol-1)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE, boat (a. u)'''
| align="center" | 0.088588
| align="center" | 0.087259
| align="center" | 0.066835
| align="center" | 0.065827
| align="center" | -
|-
| '''ΔE, boat (kcal mol-1)'''
| align="center" | 55.59
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.31
| align="center" | 44.7 ± 2.0
|}

By examining table 8 it is clear to see that the B3LYP/6-31G* method gives values which are much closer to the experimental ones. This is due to the B3LYP/6-31G* giving a higher level of optimisation than the HF/3-21G method. Table 8 also shows that the chair transition state has a lower activation energy to that of the boat transition state by 8.16 kcal mol-1. This suggests that the Cope rearrangement mechanism would most likely proceed through the chair transition state to that of the boat. Therefore the reaction would take place from the anti2 conformer to the chair transition state and then onto the gauche2 conformer within the Cope rearrangement.

It is also possible to see that the calcualted energies are very close to the literature values with the chair conformation being closer to its literature value to that of the boat.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9: Geometric parameters for the optimised chair and boat structures '''
| || ''' HF/3-21G chair TS''' || ''' B3LYP/6-31G* chair TS''' || '''HF/3-21G boat TS''' || '''B3LYP/6-31G* boat TS'''
|-
| Optimised Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Chair_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Boat_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.02 (C5-C10 or C3-C14)|| 1.97 (C5-C10 or C3-C14) || 2.14 (C3-C4 or C1-C6) || 2.21 (C3-C4 or C1-C6)
|-
| Fragment angle || 120 o (C3-C1-C5) || 120 o (C3-C1-C5) || 122 o (C4-C5-C6) || 122 o (C4-C5-C6)
|-
| Dihedral angle || 55 o (C1-C3-C14-C11)|| 54 o (C1-C3-C14-C11)|| 0 o (C2-C1-C6-C5) || 0 o (C2-C1-C6-C5)
|-
| Numbering of carbons || [[Image:Numbering_chairdh08.jpg|200x200px]]|| [[Image:Numbering_chairdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]]
|}

From studying table 9 it can be seen that the C-C bond breaking/forming length has changed between the two different types of optimisation methods. However the angles between the bonds remains unchanged.

=='''The Diels Alder Cycloaddition'''==

The Diels-Alder reaction is a cycloaddition between a conjugated diene and a dienophile (alkene system), which forms a cyclohexene system. It is often referred to as a [4s+2s] cycloaddition and proceeds through a single cyclic, concerted transition state where 2 σ bonds are formed in between the termini carbons of the 2 п systems. This is shown in the reaction scheme below where ethene is reacting with cis-butadiene.

[[Image:Reaction_scheme2dh08.jpg|300x300px|center]]

The π orbitals of the dienophile form new σ bonds with the π orbitals of the diene. The number of π electrons involved determine whether the reaction proceeds in a concerted stereospecific fashion (allowed) or not (forbidden). The HOMO/LUMO of one fragment interacts with the HOMO/LUMO of the other fragment to form two new bonding and anti-bonding MOs. The nodal properties allow one to predict how the reaction will occur.

Semi-Empirical/AM1 method and the B3LYP/6-31G* method of optimisation will be used on Gaussian to determine particular properites of the reactants to predict the way in which the reaction will proceed.

==='''cis-butadiene and ethylene: reagents'''===

<jmol>
<jmolAppletButton>
<uploadedFileContents>Cis_butadienedh08.mol</uploadedFileContents>
<text>Cis-butadiene</text>
<color>whi-te</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Ethenedh08.mol</uploadedFileContents>
<text>ethylene</text>
<color>white</color>
</jmolAppletButton>
</jmol> were optimised separately using the AM1 semi-empirical method on Gaussview.The cis-butadiene had an energy of 0.04879725 a. u (1.28 x 102 KJ mol-1) and ethylene had an energy of 0.02619027 a. u (68.76 KJ mol-1).

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10: Frontier molecular orbitals of cis-butadiene and ethylene '''
| '''Reagent''' || '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| cis-butadiene || LUMO || [[Image:LUMO1_cis_butadieneDH08.jpg|100x100px]] || [[Image:LUMO2_cis_butadienedh08.jpg|100x100px]]|| Symmetric || +0.017
|-
| cis-butadiene || HOMO || [[Image:HOMO1_cis_butadienedh08.jpg|100x100px]] || [[Image:HOMO2_cis_butadienedh08.jpg|100x100px]]|| Anti-symmetric || -0.343
|-
| ethylene || LUMO || [[Image:LUMO1_ethenedh08.jpg|100x100px]] || [[Image:LUMO2_ethenedh08.jpg|100x100px]]|| Anti-symmetric || +0.052
|-
| ethylene || HOMO || [[Image:HOMO1_ethenedh08.jpg|100x100px]] || [[Image:HOMO2_ethenedh08.jpg|100x100px]]|| Symmetric || -0.387
|}

 
The symmetries of the LUMO and HOMO orbitals for cis-butadiene are opposite of those for the ethylene. This therefore means that the only interactions that can take place are those between ethylene's LUMO and cis-butadiene's HOMO and in addition to this the interaction between cis-butadiene's LUMO and ethylene's HOMO. This is because orbital symmetry has to be conserved (the conservation of orbital symmetry states that the symmetry of an new orbital formed by the overlap of the 2 original orbitals must be the same as the symmetry of the initial orbitals).

==='''cis-butadiene and ethylene: transition state'''===

A <jmol>
<jmolAppletButton>
<uploadedFileContents>Bicyclic_startingdh08.mol</uploadedFileContents>
<text>bicyclo system</text>
<color>white</color>
</jmolAppletButton>
</jmol> was drawn and one of the -CH2-CH2- fragments were removed. Three C=C bonds were added and the previous one was taken away so a <jmol>
<jmolAppletButton>
<uploadedFileContents>Frozen_coorddh08.mol</uploadedFileContents>
<text>guess transition state</text>
<color>white</color>
</jmolAppletButton>
</jmol> was produced.

After this an optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while bonds between the termini of the fragments were frozen at 2.1Å. Following this the resulting file had two separate optimisation and frequency calculations run on it using the TS (Berny) with one using the semi-empirical/AM1 method and the other using the DFT/B3LYP/6-31G* level of theory. Both of these optimisations had the bonds unfrozen and the "derivative" option was chosen instead.

The DFT/B3LYP/6-31G* level of theory was used since it gave more accurate results in the calculations before and the energies it produces are consistent with the rest of the report (semi-empirical/AM1 method gives very different values).

The QST2 optimisation was not used since it required numbering the atoms, which will take a lot of time.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11: Comparison of two methods of optimsations for the transition state '''
| || '''semi-empirical/AM1 method''' || '''DFT/B3LYP/6-31G* method'''
|-
| Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>SE_AM1_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>DFT_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Energy (a. u) || 0.11165608 || -234.54389573
|-
| Bond lengths || [[Image:DistancesSEdh08.jpg|150x150px]]|| [[Image:DistancesDFTdh08.jpg|150x150px]]
|-
|Imaginary frequency (cm-1) || -957 || -525
|-
|Animation || [[Image:-957dh08.gif]] || [[Image:-525dh08.gif]]
|-
| Lowest real frequency (cm-1) || 148 || 136
|-
|Animation || [[Image: 148dh08.gif]] || [[Image: 136dh08.gif]]
|}

The C-C bond forming length between the termini carbons is 0.15 Å larger for the DFT/B3LYP/6-31G* level of theory and this is because this method gives carries out the calculation to a greater degree of accuracy. This larger bond length leads a lower potential energy gradient to the transition state, which gives a lower total energy for the transition state. Therefore the given values below will be the DFT/B3LYP/6-31G* and not the semi-empirical/AM1 method.

A typical sp3 C-C bond length is 1.54 Å and a typical sp2 C=C bond length is 1.34 Å<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref>. Within the optimised transition molecule above the C-C bond lengths are 1.40 Å and the C=C bond length is 1.38 Å. This suggests that the π-bonds are breaking in order for the C-C bonds to form and additionally to this a π-bond is forming in between the two sp3 carbons.

The Van der Waals radius of a C-atom is 1.72 Å<ref name="ref 3">M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer, D. G. Truhlar, 1995, 346, pp. 63-69.[http://pubs.acs.org/doi/abs/10.1021/jp8111556]</ref>. The partly formed C-C length is 2.27 Å, which is shorter than two times the the van der Waals radius of a carbon atom. Therefore the carbon atoms are within a close enough distance to each other to share electron density and yet the bond has not formed at this point.

From looking at the animations for the imaginary frequency above it can be seen that the formation of the two bonds between the termini of the reactants is synchronous. This is expected of the Diels-Alder reaction since the mechanism is concerted. However the lowest real frequency is asynchronous with respect to the two bonds forming between the termini.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12: Frontier molecular orbitals of the transition state '''
| '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| LUMO || [[Image:LUMO1TSdh08.jpg|100x100px]] || [[Image:LUMO2TSdh08.jpg|100x100px]]|| Symmetric || +0.023
|-
| HOMO || [[Image:HOMO1TSdh08.jpg|100x100px]] || [[Image:HOMO2TSdh08.jpg|100x100px]]|| Anti-symmetric || -0.323
|}

From examining the images from table 12 above it can be deduced that the LUMO of the transition state originates from the cis-butadiene's LUMO and the ethylene's HOMO, which represents the σ*-antibonding orbitals. However the HOMO of the transition state originates from the cis-butadiene's HOMO and the ethylene's LUMO, which represents the σ-bonding orbitals within the two single C-C bonds that are formed within the reaction.

==='''cis-butadiene and ethylene: activation energy'''===

The cis-butadiene and ethylene were re-optimised at the DFT/B3LYP/6-31G* level of theory so the activation energy of the reaction could be calculated.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 13: Calculation for the activation energy for the reaction'''
| '''molecule''' || '''DFT/B3LYP/6-31G* energy (a. u)''' || '''Calculations''' || '''Energy '''
|-
| cis-butadiene || -155.98595591 || transition state || -234.54389573 a. u
|-
| ethylene || -78.58745828 || ΔE, activation energy || 0.02951846 a. u
|-
| cis-butadiene and ethylene combined || -234.57341419 || ΔE, activation energy || 77.50 KJ mol-1
|}

The calculated activation energy is 77.50 KJ mol-1, which is of the right magnitude when compared to the literature value (115 ± 8 KJ mol-1 at 0 K<ref name="ref 5">V. Guner, K. S. Khuong, A. G. Leach, P. S. Lee, M. D. Bartberger, K. N. Houk, J. Phys. Chem. A, 2003, 107 (51), pp. 11445–11459.[http://pubs.acs.org/doi/abs/10.1021/jp035501w]</ref>) but there is still an error of 37.50 KJ mol-1. This shows that Gaussview is not perfect but still gives an idea of the activation energy required for the reaction to take place.

==='''Cyclohexa-1,3-diene reaction with maleic anhydride'''===

The reaction scheme below shows the stereoselective Diels-Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene. The reaction is kinetically controlled with the major product being the endo product, which will be explained later on in the exercise.

[[Image:Reaction_schemedh081.jpg|center|500x500px]]

The <jmol>
<jmolAppletButton>
<uploadedFileContents>FrozenTS22dh08.mol</uploadedFileContents>
<text>exo</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_frozenTS22dh08.mol</uploadedFileContents>
<text>endo</text>
<color>white</color>
</jmolAppletButton>
</jmol> transition states were drawn on Gaussview and then the optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while the distance of the forming bonds were frozen at 2.1Å between the respective carbons. After this the structures changed slightly (<jmol>
<jmolAppletButton>
<uploadedFileContents>TS2_afteropt1_exo1dh08.mol</uploadedFileContents>
<text>exo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_TS2_afteropt1dh08.mol</uploadedFileContents>
<text>endo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol>) and the resulting files had an optimisation and frequency calculation run on it using the TA (Berny) with using the semi-empirical/AM1 level of theory while the unfrozen option as changed to "derivative". After this the endo and exo optimised transition state structures were re-optimised as they were before but using the DFT/B3LYP/6-31G* level of theory instead with the additional keyword "Opt=NoEigen".

{| class="wikitable" border="1" align="center"
|+ '''Table 14: Comparison of the exo and endo transition states '''
! rowspan="2"|-!! colspan="2"|''exo'' !! colspan="2"|''endo''
|-
| Semi-empirical/AM1 || DFT/B3LYP/6-31G* || Semi-empirical/AM1 || DFT/B3LYP/6-31G*
|-
| align="center"|Structure ||align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>After_opt2_exodh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>Endo_TS2_afteropt2dh081.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| align="center"| Energy (a. u) ||align="center"| -0.05041975 ||align="center"| -612.67931095 ||align="center"| -0.05150470 ||align="center"| -612.68339684
|-
| C-C bond forming length (Å) ||align="center"| 2.17 ||align="center"| 2.29 ||align="center"| 2.16 ||align="center"| 2.27
|-align="center"
| Imaginary frequency (cm-1) ||align="center"| -812 ||align="center"| -448 ||align="center"| -806 ||align="center"| -447
|-align="center"
| Animation ||colspan="2"|[[Image:-812_SEdh08.gif]] || colspan="2"|[[Image:-806_cm-1dh08.gif]]
|}

Again the C-C bond forming length between the termini carbons is larger (by 0.11 and 0.12 Å) for the DFT/B3LYP/6-31G* level of theory and this is due the same reasons talked about above. However the C-C bond forming length between the termini carbons are very similar in both the endo and exo transition states (0.02 Å difference).

From looking at the animations for the imaginary frequency above it can also be seen that the formation of the two bonds between the carbons in this case as well is synchronous. Again this is expected of the Diels-Alder reaction since it is concerted.

{| align=center
|+ '''Strain diagrams for the endo and exo transition states'''
| [[Image:Exostraindh08.jpg|thumb|200x200px|'''Exo TS''']]
| [[Image:Strainendodh08.jpg|thumb|200x200px|'''Endo TS''']]
|}

From studying the energies in table 14 it can be seen that the endo transition state is more stable than the exo by 4.08589 x 10-3 a. u (10.73 KJ mol-1). However by looking at the strain of both transition states one would expect the exo TS to be more stable. This is because the distance between the cyclohexa-1,3-diene ring and the -(C=O)-O-(C=O)- maleic anhydride fragment is 2.99 Å in the endo TS and 3.03 Å in the exo TS. Thus the endo form would have more strain due to steric hindrance.

However within the exo TS the vicinal hydrogens on the CH2-CH2 (carbon numbers 18 and 21) fragment on the cyclohexa-1,3-diene ring are found below the plane and next to the maleic anhydride. This leads to steric hindrance is this is where the strain originates from within the exo TS. Therefore this disfavours the exo form, which makes the endo more stable.

Another factor making the endo TS more favourable is secondary orbital overlap. These orbitals contribute to the bonds forming and they interact with each other through space. Within the endo TS below there are 2 more orbitals interacting (secondary orbitals) than within the exo TS. This therefore makes the endo TS more stable and therefore more favoured.

[[Image:Secondary_orbital_shizzledh08.jpg|500x500px|center]]

{| class="wikitable" border="1" align="center"
|+ '''Table 15: Frontier molecular orbitals of the the transition state for the cyclohexa-1,3-diene reaction '''
! rowspan="2"|Molecular Orbital !! colspan="4"|''exo'' !! colspan="4"|''endo''
|- align="center"
| '''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''||'''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''
|- align="center"
| LUMO || [[Image:LUMO1exoh081.jpg|150x150px]]|| [[Image:LUMO2exodh081.jpg|150x150px]]|| Anti-symmetric || -0.078 || [[Image:LUMO1ENDOdh08.jpg|150x150px]]||[[Image:LUMO2ENDOdh08.jpg|150x150px]] || Anti-symmetric || -0.067
|- align="center"
| HOMO || [[Image:HOMO1exodh081.jpg|150x150px]]||[[Image:HOMO2exodh081.jpg|150x150px]] || Anti-symmetric|| -0.242 || [[Image:HOMO1ENDOdh08.jpg|150x150px]]|| [[Image:HOMO2ENDOdh08.jpg|150x150px]]||Anti-symmetric || -0.242
|}

Digital repositories:

exo: http://hdl.handle.net/10042/to-5596

endo: http://hdl.handle.net/10042/to-5597

From examining table 15 it is clear to see that both the HOMO and LUMO are anti-symmetric for both the transition states. The LUMO of the maleic anhydride fragment is the π* anti-bonding orbital of the C=C bond. Within the reaction then this has electrons transferred to it and the C=C bond breaks, which leaves just a C-C bond. The LUMOs for both transition states are both anti-bonding with respect to the two fragments. The HOMOs of both transitions state represent the σ-bonding that will be formed between the termini carbons on the products and reactants. This is most likely from the overlap of the cyclohexa-1,3-diene's HOMO and the maleic anhydride's LUMO.

='''References'''=
<references/>

Rep:Mod:hampluscheese

2010-11-11T13:13:45Z

Dh08: /* '''cis-butadiene and ethylene: activation energy''' */

='''Module 3:Physical Computational Chemistry Lab'''=

Douglas Hunt

=='''Objectives'''==

The aim of this experiment is to is to use various levels of theory (HF/3-21G, DFT/B3LYP/6-31G (d) and AM1/semi-empirical) on Gaussian to calculate various energies, molecular orbitals, geometrical parameters, reaction coordinate pathways and low and imaginary frequencies for a range of molecules and transition states. This will include using different optimisation methods throughout and will lead to a range of findings in various areas of chemical interest.

=='''The Cope Rearrangement'''==

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangement] is a [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement |3,3|-sigmatropic rearrangement] that takes place through a single [http://en.wikipedia.org/wiki/Pericyclic pericyclic], concerted [http://en.wikipedia.org/wiki/Transition_state transition state]. Within this rearrangement a σ bond is formed in concert with another being broken within a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] π system.

The Cope rearrangement we will be studying will be that of 1,5-hexadiene, which is shown below.

[[Image:Cope_rearrangement_schemedh08.jpg|300x300px|center]]

The objective is to locate the low-energy minima and transition structures on the C6H10 potential energy surface and to determine the preferred reaction mechanism by using [http://en.wikipedia.org/wiki/GAUSSIAN Gaussian], with the [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)] methods.

==='''Optimisation of 1,5-hexadiene Conformers'''===

Rough <jmol>
<jmolAppletButton>
<uploadedFileContents>Anti_2_drawndh08.mol</uploadedFileContents>
<text>anti</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche3_drawndh08.mol</uploadedFileContents>
<text>gauche</text>
<color>white</color>
</jmolAppletButton>
</jmol> conformers of 1,5-hexadiene were drawn on Gaussview and their structures were tidied by the clean function. They were then further optimised by the HF/3-21G method and were identified which conformer they were from their point group and appendix 1 on the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3 physical lab page].

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 1: Optimisation and comparison of anti and gauche 1,5-hexadiene conformers using HF/3-21G methods
| ''' Conformer''' || '''Optimised Structure''' || '''Energy (Atomic Units)''' || '''[http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1 Energies (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (Anti2 - Gauche3)''' || '''Point group'''
|-
| '''Anti2''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Anti_2_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692535 || -231.69254 ||-6.08 x 105 || 1.2606 x 10-4 a. u|| Ci
|-
| '''Gauche3''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Gauche3_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692661 || -231.69266 || -6.08 x 105 || 3.31 x 10-1 KJ mol-1 || C1
|}

From studying table 1 it can be seen that the gauche conformer is 0.33 KJ mol-1 more stable than the anti conformation. One would assume that the anti2 conformer would be more stable due to [http://en.wikipedia.org/wiki/Steric steric] factors/repulsion within the gauche3 conformation. However within the gauche3 conformer there is a favourable donation of [http://en.wikipedia.org/wiki/Electron_density electron density] from the C=C π orbital to the C-H σ* orbital from the adjacent proton, which stabilises the conformer<ref name="ref 1">B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446.[http://www.informaworld.com/smpp/content~db=all?content=10.1080/00268970110081412]</ref>.

 

The anti2 conformer was then optimised using the DFT/B3LYP/6-31G* (6-31G (d)) method.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 2: Comparison of the anti2 conformer optimisations carried out at different calculation methods
| ''' Method''' || '''Structure and bond distances (Å)''' || '''Dihedral angle (C1 to C4)''' || '''Energy (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (HF/3-21G - DFT/B3LYP/6-31G*)'''
|-
| HF/3-21G || [[Image:Anti2_opt1dh08.jpg|300x300px]] || 114.6 o || -231.692535 || -6.08 x 105 || 2.919171 a. u
|-
| DFT/B3LYP/6-31G* || [[Image:Anti2_opt2DH08.jpg|300x300px]] || 118.8 o || -234.611716 || -6.16 x 105 || 7.66 x 103 KJ mol-1
|}

From examining table 2 at first it is possible to see that the bond lengths are very similar to each other and that of the literature bond lengths<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref> (C1=C2: 1.340 Å, C2-C3: 1.508 Å and C3-C4: 1.538 Å). It is also possible to see that the C=C bond length is more accurate for the DFT/B3LYP/6-31G* method by it being 0.01 Å closer to the literature value. Another main difference in terms of geometry is that the DFT/B3LYP/6-31G* method gives a larger [http://en.wikipedia.org/wiki/Dihedral_angle dihedral angle] (and most likely more accurate) of 118.8o compared to that of 114.6o.

In terms of energy the DFT/B3LYP/6-31G* method produces a conformer of a higher stability by it possessing a lower energy with a difference of 7.66 x 103 KJ mol-1.

==='''Frequency analysis of 1,5-hexadiene Conformers'''===

[[Image:Vibrations_anti2dh08.jpg|right|300x300px]]

For the frequency analysis the DFT/B3LYP/6-31G* method was used.

Frequency analysis is essential within this calculation due to it being the second derivative of the potential energy surface. This means that if all the frequencies are all positive the system is at a minimum, if one of them is negative the optimised system is at a transition state, and if more than one are negative then the calculation has failed to find a critical point, which means the optimisation has failed. This therefore gives us a method of seeing whether the calculation has worked.

From looking to the image on the right it can deduced that the optimisation has taken place successfully by there being no negative vibrations and therefore the conformation is at a minimum.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 3: Energies under the thermochemistry section at 298.15 K and 0 K within the frequency logfiles'''
| '''Energies''' || '''298.15 K (atomic units)''' ||| '''0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (KJ mol-1)''' || '''Discussion'''
|-
| Sum of electronic and zero-point energies || -234.469177 || -234.466673 || -2.504 x 10-3 || -6.57 || This is the potential energy at 0 K, which includes the zero-point vibrational energy (E = Eelec + ZPE).

|-
| Sum of electronic and thermal energies || -234.461835 || -234.461085 || -7.5 x 10-4|| -1.97 || This is the energy at 298.15 K (0 K for 0 K calculation) and 1 atm of pressure, which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans).

|-
| Sum of electronic and thermal enthalpies || -234.460891 || -234.460141 || -7.5 x 10-4 || -1.97 || This contains an additional correction for RT (H = E + RT), which is especially important when looking at dissociation reactions.
|-
| Sum of electronic and thermal free energies || -234.500739 || -234.495931 || 4.808 x 10-3 || -12.62 || This includes the entropic contribution to the free energy (G = H - TS).
|}

{|align=right
|+
| [[Image:Anti2_freq_IRdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 298.15 K''']]
| [[Image:Anti2_freq_0Kdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 0 K''']]
|}

From looking at table 3 above it can be deduced that the anti2 conformer is more stable at room temperature to that of 0 K with respect to all the types of energies. The spectra also seem to not change much between the two temperatures but the main change is the change in the sum of electronic and thermal free energies, which is heavily dependent of temperature by it including the entropic contribution to the free energy (G = H - '''T'''S)

 
 
 
 

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

Within the Cope Rearrangement there are two possible transition states, the chair (C2h) and the boat (C2v), which are shown within [https://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_2 appendix two ].

==='''Comparison of optimisation methods for the chair transition state'''===

An <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2hdh08.mol</uploadedFileContents>
<text>allyl fragment</text>
<color>white</color>
</jmolAppletButton>
</jmol>, CH2CHCH2 was drawn on Gaussview and was optimized through the HF/3-21G method. After this the original allyl fragment was copied and pasted into a new window and place exactly above the first fragment with a bond distance of 2.20 Å between the terminal end carbons. This therefore produced the <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2h_fulldh08.mol</uploadedFileContents>
<text>chair transition structure.</text>
<color>white</color>
</jmolAppletButton>
</jmol>

Gaussian was then set up for an optimisation for the transition state with the additional key words "Opt=NoEigen", the optimisation set to TS (Berny), the force constants set to "once" and using the HF/3-21G method.

After this the "chair" transition state was optimised an alternative way by freezing the coordinates of the two sets of terminal carbons (where the bonds form/break during the rearrangement) within the Redundant Coord Editor on Gauusview.
Following this a second calculation was run where the freeze coordinates option was changed to "Derivative" and the force constants were set to "never".

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 4: Comparison of chair transition state optimisation methods
| || '''Optimised through a TS (Berny)''' || '''Optimised through a frozen coordinate method'''
|-
| '''Optimised TS Structure''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>C2h_optfreq_TS_Bdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| '''Terminal C-C bond length (Å)''' || 2.02 || 2.02
|-
|'''Energy (atomic units)''' || -231.61932247 || -231.61932190
|-
| '''ΔE, TS (Berny) - frozen coordinate''' || 5.7 x 10-7 a.u || 1.50 x 10-3 KJ mol-1
|-
| '''Imaginary frequency (cm-1)''' || -818 || -818
|-
| '''Animation''' ||[[Image:818dh08TSB.gif|350x350px]] || [[Image:818cm-1Fdh08.gif]]
|}

The data in table 4 shows that both methods give the same results and they both give the same transition state. They both produced the same energy with a negligible energy difference between them and the bond lengths between the terminal carbons are both 2.02 Å. In addition to this the imaginary frequencies were both -818cm-1 and within the animations above it is possible to see that when one C-C bond is forming the other is breaking. This means it is a asynchronous bond formation, which is expected for the Cope Rearrangement since it is a pericyclic reaction.

The only difference between the two optimisations is the way in which the transition states were calculated. This is due to the TS (Berny) method calculating the force constant matrix within the first optimisation step and is then being recalculated as the optimisation carries on. When the frozen coordinate method freezes the reaction coordinate in order for the rest of the structure to be optimised around it, which does not require the whole hessian (force constant matrix) to be calculated but only that the reaction coordinate is differentiated (second step). The main advantage of this, is that you do not need to calculate the entire force constant matrix and just need to differentiate the reaction coordinate in order to obtain a good enough guess for the initial force constant matrix. The frozen coordinate method will therefore take less time to compute the transition state, which can be very expensive within certain systems. However to carry out this method one needs to have an idea of how far apart the terminal carbons are.

==='''Boat transition state'''===

===='''QST2 optimisation'''====

The optimisation for the boat transition state had a different method and this was the QST2 method. This calculation interpolates between two structures (product and reactant) and tries to find a transition state between them.

The reactants and products were numbered in the same way. Therefore both 1,5-hexadiene molecules had their numbering manually changed for the product molecule so it corresponded to the numbering of the reactant if it rearranged. This is illustrated in the reaction scheme below.

[[Image:Boat_rerrangmentdh08.jpg|300x300px|center]]

The chk file for the optimized anti2 conformer was opened and copied into a new window. A second window was opened within the new window and the same anti2 conformer was copied into the new window. The numbering was changed within the second molecule so the two looked like they do in the image below.

[[Image:TS_(QST2)_numbering_1dh08.jpg|400x400px|center]]

The job type was set as "Opt+Freq" and "TS (QST2)" was chosen to optimise the transition state. However this failed and produced
a structure a bit like a <jmol>
<jmolAppletButton>
<uploadedFileContents>Trialone_BPdh08.mol</uploadedFileContents>
<text>chair transition structure </text>
<color>white</color>
</jmolAppletButton>
</jmol> but more dissociated. This is because it interpolated between the two structures and translated the top allyl fragment and therefore did not consider the possibility of a rotation around the central bonds.

To get around this the C-C-C-C dihedral angle (C2-C3-C4-C5) was set to 0 o and the C-C-C angle (C2-C3-C4 and C3-C4-C5 for the product molecule) was set to 100 o for both molecules. This gave rise to the two structures below.

[[Image:TS_(QST2)_numbering_2dh08.jpg|400x400px|center]]

After this the QST2 calculation was run again and the results in table 5 were produced. The imaginary frequency at -839 cm-1 shows that while one bond is forming the other one is breaking, which is again an asynchronous bond formation. It is also possible to see that the boat transition state has a longer terminal C-C bond length by 0.12 Å to that of the chair.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5: Data for the boat transition state from the QST2 optimisation method
| Optimised TS Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.14
|-
|Energy (atomic units) || -231.60280151
|-
| Energy (KJ mol-1) || -6.08 x 105
|-
| Imaginary frequency (cm-1) || -839
|-
| Animation ||[[Image:-839dh08.gif]]
|}

===='''Intrinsic Reaction Coordinate'''====

It is not possible to predict which product the chair and boat transition state will lead to. To solve this problem within Gaussian one can use the IRC method, which allows one to follow the minimum energy path from a transition structure down to its local minimum on a potential energy surface. This creates a succession of points by taking small geometry steps in the direction to where the gradient of the energy surface is at its steepest until the minimum in energy is reached.

The chk file for the optimized chair transition structure was opened and the job type selected was "IRC", the reaction coordinate was computed in the forward direction (since molecule is symmetric) with 50 points along the IRC. This was done with two type of force constants being calculated, once and always.

Within the "once" method the output showed no bond formation process by the 26th step and therefore the calculation was incomplete, since a complete 1,5-hexadiene structure was supposed to be formed. From looking at the IRC pathway it can be seen that the minimum on the potential energy surface was not reached at that point, since the points on the pathway did not form an asymptote and the gradient did not go close enough to zero.

For the "always" method the computation of force constants were carried out at every step. This therefore lets the IRC calculation distinguish whether it is moving along the minimum energy pathway while going down the potential energy surface. While this is happening it can make changes to the pathway if necessary. Within the "always" case the bond forming arrives at 1,5-hexadiene and this is backed up by the IRC pathway reaching a minimum within its energy by the gradient going close to zero and the pathway reaches an asymptote.

The structure produced through the always method had an energy of -231.69199434 atomic units, which is very close to the energy of the gauche2 conformer in the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 appendix 1]. This suggests that the reaction takes place through the chair transition state, which later produces the gauche2 conformer of 1,5-hexadiene.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 6: Comparison of IRC calculations for the chair structure
| || '''Calculate force constants once''' || '''Calculate force constants always'''
|-
| '''Number of points along IRC''' || 26 || 47
|-
| '''Structure at the last point of IRC''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Once_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Always_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|'''Energy (atomic units)''' || -231.68906923 || -231.69199434
|-
| '''ΔE, once - always''' || 2.92511 x 10-3 a.u || 7.68 KJ mol-1
|-
| '''IRC pathway''' || [[Image:One_pathwaydh08.jpg|400x400px]] || [[Image:Always_pathwaydh08.jpg|400x400px]]
|-
| '''IRC gradient''' || [[Image:One_gradientdh08.jpg|400x400px]]|| [[Image:Always_gradientdh08.jpg|400x400px]]
|}

 
 
 

===='''Comparison of activation energies for the transition states'''====

The HF/3-21G optimised chair and boat transition structures were both reoptimised using the B3LYP/6-31G* (B3LYP/6-31G (d)) method. This gives a more accurate minimisation however this method takes longer to run. The values for the zero-point energies and thermal energies were found within the thermochemistry section of the log file.

The values for the HF/3-21G reactant (anti2) optimisation were calculated and the B3LYP/6-31G* values were taken from the "Frequency analysis of 1,5-hexadiene Conformers" earlier in this module.

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 7: Energy data for chair, boat and the anti2 conformation from the HF/3-21G and B3LYP/6-31G* methods of optimisation '''
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466677
| align="center" | -231.461320
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450936
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402342
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539524
| align="center" | -231.532562
| align="center" | -234.611716
| align="center" | -234.469177
| align="center" | -234.461835
|}
Digital repositories

Chair:
HF/3-21G: http://hdl.handle.net/10042/to-5569
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5568

Boat:
HF/3-21G: http://hdl.handle.net/10042/to-5570
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5571

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 8: Summary of activation energies for the chair and boat conformations (ΔE = chair TS or boat TS minus anti2)'''
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| align="center" | 1 a.u = 627.509 kcal mol-1
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE, chair (a.u)'''
| align="center" | 0.072847
| align="center" | 0.071242
| align="center" | 0.054248
| align="center" | 0.052827
| align="center" | -
|-
| '''ΔE, chair (kcal mol-1)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE, boat (a. u)'''
| align="center" | 0.088588
| align="center" | 0.087259
| align="center" | 0.066835
| align="center" | 0.065827
| align="center" | -
|-
| '''ΔE, boat (kcal mol-1)'''
| align="center" | 55.59
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.31
| align="center" | 44.7 ± 2.0
|}

By examining table 8 it is clear to see that the B3LYP/6-31G* method gives values which are much closer to the experimental ones. This is due to the B3LYP/6-31G* giving a higher level of optimisation than the HF/3-21G method. Table 8 also shows that the chair transition state has a lower activation energy to that of the boat transition state by 8.16 kcal mol-1. This suggests that the Cope rearrangement mechanism would most likely proceed through the chair transition state to that of the boat. Therefore the reaction would take place from the anti2 conformer to the chair transition state and then onto the gauche2 conformer within the Cope rearrangement.

It is also possible to see that the calcualted energies are very close to the literature values with the chair conformation being closer to its literature value to that of the boat.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9: Geometric parameters for the optimised chair and boat structures '''
| || ''' HF/3-21G chair TS''' || ''' B3LYP/6-31G* chair TS''' || '''HF/3-21G boat TS''' || '''B3LYP/6-31G* boat TS'''
|-
| Optimised Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Chair_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Boat_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.02 (C5-C10 or C3-C14)|| 1.97 (C5-C10 or C3-C14) || 2.14 (C3-C4 or C1-C6) || 2.21 (C3-C4 or C1-C6)
|-
| Fragment angle || 120 o (C3-C1-C5) || 120 o (C3-C1-C5) || 122 o (C4-C5-C6) || 122 o (C4-C5-C6)
|-
| Dihedral angle || 55 o (C1-C3-C14-C11)|| 54 o (C1-C3-C14-C11)|| 0 o (C2-C1-C6-C5) || 0 o (C2-C1-C6-C5)
|-
| Numbering of carbons || [[Image:Numbering_chairdh08.jpg|200x200px]]|| [[Image:Numbering_chairdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]]
|}

From studying table 9 it can be seen that the C-C bond breaking/forming length has changed between the two different types of optimisation methods. However the angles between the bonds remains unchanged.

=='''The Diels Alder Cycloaddition'''==

The Diels-Alder reaction is a cycloaddition between a conjugated diene and a dienophile (alkene system), which forms a cyclohexene system. It is often referred to as a [4s+2s] cycloaddition and proceeds through a single cyclic, concerted transition state where 2 σ bonds are formed in between the termini carbons of the 2 п systems. This is shown in the reaction scheme below where ethene is reacting with cis-butadiene.

[[Image:Reaction_scheme2dh08.jpg|300x300px|center]]

The π orbitals of the dienophile form new σ bonds with the π orbitals of the diene. The number of π electrons involved determine whether the reaction proceeds in a concerted stereospecific fashion (allowed) or not (forbidden). The HOMO/LUMO of one fragment interacts with the HOMO/LUMO of the other fragment to form two new bonding and anti-bonding MOs. The nodal properties allow one to predict how the reaction will occur.

Semi-Empirical/AM1 method and the B3LYP/6-31G* method of optimisation will be used on Gaussian to determine particular properites of the reactants to predict the way in which the reaction will proceed.

==='''cis-butadiene and ethylene: reagents'''===

<jmol>
<jmolAppletButton>
<uploadedFileContents>Cis_butadienedh08.mol</uploadedFileContents>
<text>Cis-butadiene</text>
<color>whi-te</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Ethenedh08.mol</uploadedFileContents>
<text>ethylene</text>
<color>white</color>
</jmolAppletButton>
</jmol> were optimised separately using the AM1 semi-empirical method on Gaussview.The cis-butadiene had an energy of 0.04879725 a. u (1.28 x 102 KJ mol-1) and ethylene had an energy of 0.02619027 a. u (68.76 KJ mol-1).

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10: Frontier molecular orbitals of cis-butadiene and ethylene '''
| '''Reagent''' || '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| cis-butadiene || LUMO || [[Image:LUMO1_cis_butadieneDH08.jpg|100x100px]] || [[Image:LUMO2_cis_butadienedh08.jpg|100x100px]]|| Symmetric || +0.017
|-
| cis-butadiene || HOMO || [[Image:HOMO1_cis_butadienedh08.jpg|100x100px]] || [[Image:HOMO2_cis_butadienedh08.jpg|100x100px]]|| Anti-symmetric || -0.343
|-
| ethylene || LUMO || [[Image:LUMO1_ethenedh08.jpg|100x100px]] || [[Image:LUMO2_ethenedh08.jpg|100x100px]]|| Anti-symmetric || +0.052
|-
| ethylene || HOMO || [[Image:HOMO1_ethenedh08.jpg|100x100px]] || [[Image:HOMO2_ethenedh08.jpg|100x100px]]|| Symmetric || -0.387
|}

 
The symmetries of the LUMO and HOMO orbitals for cis-butadiene are opposite of those for the ethylene. This therefore means that the only interactions that can take place are those between ethylene's LUMO and cis-butadiene's HOMO and in addition to this the interaction between cis-butadiene's LUMO and ethylene's HOMO. This is because orbital symmetry has to be conserved (the conservation of orbital symmetry states that the symmetry of an new orbital formed by the overlap of the 2 original orbitals must be the same as the symmetry of the initial orbitals).

==='''cis-butadiene and ethylene: transition state'''===

A <jmol>
<jmolAppletButton>
<uploadedFileContents>Bicyclic_startingdh08.mol</uploadedFileContents>
<text>bicyclo system</text>
<color>white</color>
</jmolAppletButton>
</jmol> was drawn and one of the -CH2-CH2- fragments were removed. Three C=C bonds were added and the previous one was taken away so a <jmol>
<jmolAppletButton>
<uploadedFileContents>Frozen_coorddh08.mol</uploadedFileContents>
<text>guess transition state</text>
<color>white</color>
</jmolAppletButton>
</jmol> was produced.

After this an optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while bonds between the termini of the fragments were frozen at 2.1Å. Following this the resulting file had two separate optimisation and frequency calculations run on it using the TS (Berny) with one using the semi-empirical/AM1 method and the other using the DFT/B3LYP/6-31G* level of theory. Both of these optimisations had the bonds unfrozen and the "derivative" option was chosen instead.

The DFT/B3LYP/6-31G* level of theory was used since it gave more accurate results in the calculations before and the energies it produces are consistent with the rest of the report (semi-empirical/AM1 method gives very different values).

The QST2 optimisation was not used since it required numbering the atoms, which will take a lot of time.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11: Comparison of two methods of optimsations for the transition state '''
| || '''semi-empirical/AM1 method''' || '''DFT/B3LYP/6-31G* method'''
|-
| Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>SE_AM1_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>DFT_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Energy (a. u) || 0.11165608 || -234.54389573
|-
| Bond lengths || [[Image:DistancesSEdh08.jpg|150x150px]]|| [[Image:DistancesDFTdh08.jpg|150x150px]]
|-
|Imaginary frequency (cm-1) || -957 || -525
|-
|Animation || [[Image:-957dh08.gif]] || [[Image:-525dh08.gif]]
|-
| Lowest real frequency (cm-1) || 148 || 136
|-
|Animation || [[Image: 148dh08.gif]] || [[Image: 136dh08.gif]]
|}

The C-C bond forming length between the termini carbons is 0.15 Å larger for the DFT/B3LYP/6-31G* level of theory and this is because this method gives carries out the calculation to a greater degree of accuracy. This larger bond length leads a lower potential energy gradient to the transition state, which gives a lower total energy for the transition state. Therefore the given values below will be the DFT/B3LYP/6-31G* and not the semi-empirical/AM1 method.

A typical sp3 C-C bond length is 1.54 Å and a typical sp2 C=C bond length is 1.34 Å<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref>. Within the optimised transition molecule above the C-C bond lengths are 1.40 Å and the C=C bond length is 1.38 Å. This suggests that the π-bonds are breaking in order for the C-C bonds to form and additionally to this a π-bond is forming in between the two sp3 carbons.

The Van der Waals radius of a C-atom is 1.72 Å<ref name="ref 3">M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer, D. G. Truhlar, 1995, 346, pp. 63-69.[http://pubs.acs.org/doi/abs/10.1021/jp8111556]</ref>. The partly formed C-C length is 2.27 Å, which is shorter than two times the the van der Waals radius of a carbon atom. Therefore the carbon atoms are within a close enough distance to each other to share electron density and yet the bond has not formed at this point.

From looking at the animations for the imaginary frequency above it can be seen that the formation of the two bonds between the termini of the reactants is synchronous. This is expected of the Diels-Alder reaction since the mechanism is concerted. However the lowest real frequency is asynchronous with respect to the two bonds forming between the termini.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12: Frontier molecular orbitals of the transition state '''
| '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| LUMO || [[Image:LUMO1TSdh08.jpg|100x100px]] || [[Image:LUMO2TSdh08.jpg|100x100px]]|| Symmetric || +0.023
|-
| HOMO || [[Image:HOMO1TSdh08.jpg|100x100px]] || [[Image:HOMO2TSdh08.jpg|100x100px]]|| Anti-symmetric || -0.323
|}

From examining the images from table 12 above it can be deduced that the LUMO of the transition state originates from the cis-butadiene's LUMO and the ethylene's HOMO, which represents the σ*-antibonding orbitals. However the HOMO of the transition state originates from the cis-butadiene's HOMO and the ethylene's LUMO, which represents the σ-bonding orbitals within the two single C-C bonds that are formed within the reaction.

==='''cis-butadiene and ethylene: activation energy'''===

The cis-butadiene and ethylene were re-optimised at the DFT/B3LYP/6-31G* level of theory so the activation energy of the reaction could be calculated.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 13: Calculation for the activation energy for the reaction'''
| '''molecule''' || '''DFT/B3LYP/6-31G* energy (a. u)''' || '''Calculations''' || '''Energy '''
|-
| cis-butadiene || -155.98595591 || transition state || -234.54389573 a. u
|-
| ethylene || -78.58745828 || ΔE, activation energy || 0.02951846 a. u
|-
| cis-butadiene and ethylene combined || -234.57341419 || ΔE, activation energy || 77.50 KJ mol-1
|}

The calculated activation energy is 77.50 KJ mol-1, which is of the right magnitude when compared to the literature value (115 ± 8 KJ mol-1 at 0 K<ref name="ref 5">V. Guner, K. S. Khuong, A. G. Leach, P. S. Lee, M. D. Bartberger, K. N. Houk, J. Phys. Chem. A, 2003, 107 (51), pp. 11445–11459.[http://pubs.acs.org/doi/abs/10.1021/jp035501w]</ref>) but there is still an error of 37.50 KJ mol-1. This shows that Gaussview is not perfect but still gives an idea of the activation energy required for the reaction to take place.

==='''Cyclohexa-1,3-diene reaction with maleic anhydride'''===

The reaction scheme below shows the stereoselective Diels-Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene. The reaction is kinetically controlled with the major product being the endo product, which will be explained later on in the exercise.

[[Image:Reaction_schemedh081.jpg|center|500x500px]]

The <jmol>
<jmolAppletButton>
<uploadedFileContents>FrozenTS22dh08.mol</uploadedFileContents>
<text>exo</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_frozenTS22dh08.mol</uploadedFileContents>
<text>endo</text>
<color>white</color>
</jmolAppletButton>
</jmol> transition states were drawn on Gaussview and then the optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while the distance of the forming bonds were frozen at 2.1Å between the respective carbons. After this the structures changed slightly (<jmol>
<jmolAppletButton>
<uploadedFileContents>TS2_afteropt1_exo1dh08.mol</uploadedFileContents>
<text>exo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_TS2_afteropt1dh08.mol</uploadedFileContents>
<text>endo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol>) and the resulting files had an optimisation and frequency calculation run on it using the TA (Berny) with using the semi-empirical/AM1 level of theory while the unfrozen option as changed to "derivative". After this the endo and exo optimised transition state structures were re-optimised as they were before but using the DFT/B3LYP/6-31G* level of theory instead with the additional keyword "Opt=NoEigen".

{| class="wikitable" border="1" align="center"
|+ '''Table 14: Comparison of the exo and endo transition states '''
! rowspan="2"|-!! colspan="2"|''exo'' !! colspan="2"|''endo''
|-
| Semi-empirical/AM1 || DFT/B3LYP/6-31G* || Semi-empirical/AM1 || DFT/B3LYP/6-31G*
|-
| align="center"|Structure ||align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>After_opt2_exodh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>Endo_TS2_afteropt2dh081.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| align="center"| Energy (a. u) ||align="center"| -0.05041975 ||align="center"| -612.67931095 ||align="center"| -0.05150470 ||align="center"| -612.68339684
|-
| C-C bond forming length (Å) ||align="center"| 2.17 ||align="center"| 2.29 ||align="center"| 2.16 ||align="center"| 2.27
|-align="center"
| Imaginary frequency (cm-1) ||align="center"| -812 ||align="center"| -448 ||align="center"| -806 ||align="center"| -447
|-align="center"
| Animation ||colspan="2"|[[Image:-812_SEdh08.gif]] || colspan="2"|[[Image:-806_cm-1dh08.gif]]
|}

Again the C-C bond forming length between the termini carbons is larger (by 0.11 and 0.12 Å) for the DFT/B3LYP/6-31G* level of theory and this is due the same reasons talked about above. However the C-C bond forming length between the termini carbons are very similar in both the endo and exo transition states (0.02 Å difference).

From looking at the animations for the imaginary frequency above it can also be seen that the formation of the two bonds between the carbons in this case as well is synchronous. Again this is expected of the Diels-Alder reaction since it is concerted.

{| align=center
|+ '''Strain diagrams for the endo and exo transition states'''
| [[Image:Exostraindh08.jpg|thumb|200x200px|'''Exo TS''']]
| [[Image:Strainendodh08.jpg|thumb|200x200px|'''Endo TS''']]
|}

From studying the energies in table 14 it can be seen that the endo transition state is more stable than the exo by 4.08589 x 10-3 a. u (10.73 KJ mol-1). However by looking at the strain of both transition states one would expect the exo TS to be more stable. This is because the distance between the cyclohexa-1,3-diene ring and the -(C=O)-O-(C=O)- maleic anhydride fragment is 2.99 Å in the endo TS and 3.03 Å in the exo TS. Thus the endo form would have more strain due to steric hindrance.

However within the exo TS the vicinal hydrogens on the CH2-CH2 (carbon numbers 18 and 21) fragment on the cyclohexa-1,3-diene ring are found below the plane and next to the maleic anhydride. This leads to steric hindrance is this is where the strain originates from within the exo TS. Therefore this disfavours the exo form, which makes the endo more stable.

Another factor making the endo TS more favourable is secondary orbital overlap. These orbitals contribute to the bonds forming and they interact with each other through space. Within the endo TS below there are 2 more orbitals interacting (secondary orbitals) than within the exo TS. This therefore makes the endo TS more stable and therefore more favoured.

[[Image:Secondary_orbital_shizzledh08.jpg|500x500px|center]]

{| class="wikitable" border="1" align="center"
|+ '''Table 15: Frontier molecular orbitals of the the transition state for the cyclohexa-1,3-diene reaction '''
! rowspan="2"|Molecular Orbital !! colspan="4"|''exo'' !! colspan="4"|''endo''
|- align="center"
| '''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''||'''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''
|- align="center"
| LUMO || [[Image:LUMO1exoh081.jpg|150x150px]]|| [[Image:LUMO2exodh081.jpg|150x150px]]|| Anti-symmetric || -0.078 || [[Image:LUMO1ENDOdh08.jpg|150x150px]]||[[Image:LUMO2ENDOdh08.jpg|150x150px]] || Anti-symmetric || -0.067
|- align="center"
| HOMO || [[Image:HOMO1exodh081.jpg|150x150px]]||[[Image:HOMO2exodh081.jpg|150x150px]] || Anti-symmetric|| -0.242 || [[Image:HOMO1ENDOdh08.jpg|150x150px]]|| [[Image:HOMO2ENDOdh08.jpg|150x150px]]||Anti-symmetric || -0.242
|}

Digital repositories:

exo: http://hdl.handle.net/10042/to-5596

endo: http://hdl.handle.net/10042/to-5597

From examining table 15 it is clear to see that both the HOMO and LUMO are anti-symmetric for both the transition states. The LUMO of the maleic anhydride fragment is the π* anti-bonding orbital of the C=C bond. Within the reaction then this has electrons transferred to it and the C=C bond breaks, which leaves just a C-C bond. The LUMOs for both transition states are both anti-bonding with respect to the two fragments. The HOMOs of both transitions state represent the σ-bonding that will be formed between the termini carbons on the products and reactants. This is most likely from the overlap of the cyclohexa-1,3-diene's HOMO and the maleic anhydride's LUMO.

='''References'''=
<references/>

Rep:Mod:hampluscheese

2010-11-11T13:11:11Z

Dh08: /* '''cis-butadiene and ethylene: activation energy''' */

='''Module 3:Physical Computational Chemistry Lab'''=

Douglas Hunt

=='''Objectives'''==

The aim of this experiment is to is to use various levels of theory (HF/3-21G, DFT/B3LYP/6-31G (d) and AM1/semi-empirical) on Gaussian to calculate various energies, molecular orbitals, geometrical parameters, reaction coordinate pathways and low and imaginary frequencies for a range of molecules and transition states. This will include using different optimisation methods throughout and will lead to a range of findings in various areas of chemical interest.

=='''The Cope Rearrangement'''==

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangement] is a [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement |3,3|-sigmatropic rearrangement] that takes place through a single [http://en.wikipedia.org/wiki/Pericyclic pericyclic], concerted [http://en.wikipedia.org/wiki/Transition_state transition state]. Within this rearrangement a σ bond is formed in concert with another being broken within a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] π system.

The Cope rearrangement we will be studying will be that of 1,5-hexadiene, which is shown below.

[[Image:Cope_rearrangement_schemedh08.jpg|300x300px|center]]

The objective is to locate the low-energy minima and transition structures on the C6H10 potential energy surface and to determine the preferred reaction mechanism by using [http://en.wikipedia.org/wiki/GAUSSIAN Gaussian], with the [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)] methods.

==='''Optimisation of 1,5-hexadiene Conformers'''===

Rough <jmol>
<jmolAppletButton>
<uploadedFileContents>Anti_2_drawndh08.mol</uploadedFileContents>
<text>anti</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche3_drawndh08.mol</uploadedFileContents>
<text>gauche</text>
<color>white</color>
</jmolAppletButton>
</jmol> conformers of 1,5-hexadiene were drawn on Gaussview and their structures were tidied by the clean function. They were then further optimised by the HF/3-21G method and were identified which conformer they were from their point group and appendix 1 on the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3 physical lab page].

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 1: Optimisation and comparison of anti and gauche 1,5-hexadiene conformers using HF/3-21G methods
| ''' Conformer''' || '''Optimised Structure''' || '''Energy (Atomic Units)''' || '''[http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1 Energies (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (Anti2 - Gauche3)''' || '''Point group'''
|-
| '''Anti2''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Anti_2_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692535 || -231.69254 ||-6.08 x 105 || 1.2606 x 10-4 a. u|| Ci
|-
| '''Gauche3''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Gauche3_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692661 || -231.69266 || -6.08 x 105 || 3.31 x 10-1 KJ mol-1 || C1
|}

From studying table 1 it can be seen that the gauche conformer is 0.33 KJ mol-1 more stable than the anti conformation. One would assume that the anti2 conformer would be more stable due to [http://en.wikipedia.org/wiki/Steric steric] factors/repulsion within the gauche3 conformation. However within the gauche3 conformer there is a favourable donation of [http://en.wikipedia.org/wiki/Electron_density electron density] from the C=C π orbital to the C-H σ* orbital from the adjacent proton, which stabilises the conformer<ref name="ref 1">B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446.[http://www.informaworld.com/smpp/content~db=all?content=10.1080/00268970110081412]</ref>.

 

The anti2 conformer was then optimised using the DFT/B3LYP/6-31G* (6-31G (d)) method.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 2: Comparison of the anti2 conformer optimisations carried out at different calculation methods
| ''' Method''' || '''Structure and bond distances (Å)''' || '''Dihedral angle (C1 to C4)''' || '''Energy (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (HF/3-21G - DFT/B3LYP/6-31G*)'''
|-
| HF/3-21G || [[Image:Anti2_opt1dh08.jpg|300x300px]] || 114.6 o || -231.692535 || -6.08 x 105 || 2.919171 a. u
|-
| DFT/B3LYP/6-31G* || [[Image:Anti2_opt2DH08.jpg|300x300px]] || 118.8 o || -234.611716 || -6.16 x 105 || 7.66 x 103 KJ mol-1
|}

From examining table 2 at first it is possible to see that the bond lengths are very similar to each other and that of the literature bond lengths<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref> (C1=C2: 1.340 Å, C2-C3: 1.508 Å and C3-C4: 1.538 Å). It is also possible to see that the C=C bond length is more accurate for the DFT/B3LYP/6-31G* method by it being 0.01 Å closer to the literature value. Another main difference in terms of geometry is that the DFT/B3LYP/6-31G* method gives a larger [http://en.wikipedia.org/wiki/Dihedral_angle dihedral angle] (and most likely more accurate) of 118.8o compared to that of 114.6o.

In terms of energy the DFT/B3LYP/6-31G* method produces a conformer of a higher stability by it possessing a lower energy with a difference of 7.66 x 103 KJ mol-1.

==='''Frequency analysis of 1,5-hexadiene Conformers'''===

[[Image:Vibrations_anti2dh08.jpg|right|300x300px]]

For the frequency analysis the DFT/B3LYP/6-31G* method was used.

Frequency analysis is essential within this calculation due to it being the second derivative of the potential energy surface. This means that if all the frequencies are all positive the system is at a minimum, if one of them is negative the optimised system is at a transition state, and if more than one are negative then the calculation has failed to find a critical point, which means the optimisation has failed. This therefore gives us a method of seeing whether the calculation has worked.

From looking to the image on the right it can deduced that the optimisation has taken place successfully by there being no negative vibrations and therefore the conformation is at a minimum.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 3: Energies under the thermochemistry section at 298.15 K and 0 K within the frequency logfiles'''
| '''Energies''' || '''298.15 K (atomic units)''' ||| '''0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (KJ mol-1)''' || '''Discussion'''
|-
| Sum of electronic and zero-point energies || -234.469177 || -234.466673 || -2.504 x 10-3 || -6.57 || This is the potential energy at 0 K, which includes the zero-point vibrational energy (E = Eelec + ZPE).

|-
| Sum of electronic and thermal energies || -234.461835 || -234.461085 || -7.5 x 10-4|| -1.97 || This is the energy at 298.15 K (0 K for 0 K calculation) and 1 atm of pressure, which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans).

|-
| Sum of electronic and thermal enthalpies || -234.460891 || -234.460141 || -7.5 x 10-4 || -1.97 || This contains an additional correction for RT (H = E + RT), which is especially important when looking at dissociation reactions.
|-
| Sum of electronic and thermal free energies || -234.500739 || -234.495931 || 4.808 x 10-3 || -12.62 || This includes the entropic contribution to the free energy (G = H - TS).
|}

{|align=right
|+
| [[Image:Anti2_freq_IRdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 298.15 K''']]
| [[Image:Anti2_freq_0Kdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 0 K''']]
|}

From looking at table 3 above it can be deduced that the anti2 conformer is more stable at room temperature to that of 0 K with respect to all the types of energies. The spectra also seem to not change much between the two temperatures but the main change is the change in the sum of electronic and thermal free energies, which is heavily dependent of temperature by it including the entropic contribution to the free energy (G = H - '''T'''S)

 
 
 
 

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

Within the Cope Rearrangement there are two possible transition states, the chair (C2h) and the boat (C2v), which are shown within [https://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_2 appendix two ].

==='''Comparison of optimisation methods for the chair transition state'''===

An <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2hdh08.mol</uploadedFileContents>
<text>allyl fragment</text>
<color>white</color>
</jmolAppletButton>
</jmol>, CH2CHCH2 was drawn on Gaussview and was optimized through the HF/3-21G method. After this the original allyl fragment was copied and pasted into a new window and place exactly above the first fragment with a bond distance of 2.20 Å between the terminal end carbons. This therefore produced the <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2h_fulldh08.mol</uploadedFileContents>
<text>chair transition structure.</text>
<color>white</color>
</jmolAppletButton>
</jmol>

Gaussian was then set up for an optimisation for the transition state with the additional key words "Opt=NoEigen", the optimisation set to TS (Berny), the force constants set to "once" and using the HF/3-21G method.

After this the "chair" transition state was optimised an alternative way by freezing the coordinates of the two sets of terminal carbons (where the bonds form/break during the rearrangement) within the Redundant Coord Editor on Gauusview.
Following this a second calculation was run where the freeze coordinates option was changed to "Derivative" and the force constants were set to "never".

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 4: Comparison of chair transition state optimisation methods
| || '''Optimised through a TS (Berny)''' || '''Optimised through a frozen coordinate method'''
|-
| '''Optimised TS Structure''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>C2h_optfreq_TS_Bdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| '''Terminal C-C bond length (Å)''' || 2.02 || 2.02
|-
|'''Energy (atomic units)''' || -231.61932247 || -231.61932190
|-
| '''ΔE, TS (Berny) - frozen coordinate''' || 5.7 x 10-7 a.u || 1.50 x 10-3 KJ mol-1
|-
| '''Imaginary frequency (cm-1)''' || -818 || -818
|-
| '''Animation''' ||[[Image:818dh08TSB.gif|350x350px]] || [[Image:818cm-1Fdh08.gif]]
|}

The data in table 4 shows that both methods give the same results and they both give the same transition state. They both produced the same energy with a negligible energy difference between them and the bond lengths between the terminal carbons are both 2.02 Å. In addition to this the imaginary frequencies were both -818cm-1 and within the animations above it is possible to see that when one C-C bond is forming the other is breaking. This means it is a asynchronous bond formation, which is expected for the Cope Rearrangement since it is a pericyclic reaction.

The only difference between the two optimisations is the way in which the transition states were calculated. This is due to the TS (Berny) method calculating the force constant matrix within the first optimisation step and is then being recalculated as the optimisation carries on. When the frozen coordinate method freezes the reaction coordinate in order for the rest of the structure to be optimised around it, which does not require the whole hessian (force constant matrix) to be calculated but only that the reaction coordinate is differentiated (second step). The main advantage of this, is that you do not need to calculate the entire force constant matrix and just need to differentiate the reaction coordinate in order to obtain a good enough guess for the initial force constant matrix. The frozen coordinate method will therefore take less time to compute the transition state, which can be very expensive within certain systems. However to carry out this method one needs to have an idea of how far apart the terminal carbons are.

==='''Boat transition state'''===

===='''QST2 optimisation'''====

The optimisation for the boat transition state had a different method and this was the QST2 method. This calculation interpolates between two structures (product and reactant) and tries to find a transition state between them.

The reactants and products were numbered in the same way. Therefore both 1,5-hexadiene molecules had their numbering manually changed for the product molecule so it corresponded to the numbering of the reactant if it rearranged. This is illustrated in the reaction scheme below.

[[Image:Boat_rerrangmentdh08.jpg|300x300px|center]]

The chk file for the optimized anti2 conformer was opened and copied into a new window. A second window was opened within the new window and the same anti2 conformer was copied into the new window. The numbering was changed within the second molecule so the two looked like they do in the image below.

[[Image:TS_(QST2)_numbering_1dh08.jpg|400x400px|center]]

The job type was set as "Opt+Freq" and "TS (QST2)" was chosen to optimise the transition state. However this failed and produced
a structure a bit like a <jmol>
<jmolAppletButton>
<uploadedFileContents>Trialone_BPdh08.mol</uploadedFileContents>
<text>chair transition structure </text>
<color>white</color>
</jmolAppletButton>
</jmol> but more dissociated. This is because it interpolated between the two structures and translated the top allyl fragment and therefore did not consider the possibility of a rotation around the central bonds.

To get around this the C-C-C-C dihedral angle (C2-C3-C4-C5) was set to 0 o and the C-C-C angle (C2-C3-C4 and C3-C4-C5 for the product molecule) was set to 100 o for both molecules. This gave rise to the two structures below.

[[Image:TS_(QST2)_numbering_2dh08.jpg|400x400px|center]]

After this the QST2 calculation was run again and the results in table 5 were produced. The imaginary frequency at -839 cm-1 shows that while one bond is forming the other one is breaking, which is again an asynchronous bond formation. It is also possible to see that the boat transition state has a longer terminal C-C bond length by 0.12 Å to that of the chair.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5: Data for the boat transition state from the QST2 optimisation method
| Optimised TS Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.14
|-
|Energy (atomic units) || -231.60280151
|-
| Energy (KJ mol-1) || -6.08 x 105
|-
| Imaginary frequency (cm-1) || -839
|-
| Animation ||[[Image:-839dh08.gif]]
|}

===='''Intrinsic Reaction Coordinate'''====

It is not possible to predict which product the chair and boat transition state will lead to. To solve this problem within Gaussian one can use the IRC method, which allows one to follow the minimum energy path from a transition structure down to its local minimum on a potential energy surface. This creates a succession of points by taking small geometry steps in the direction to where the gradient of the energy surface is at its steepest until the minimum in energy is reached.

The chk file for the optimized chair transition structure was opened and the job type selected was "IRC", the reaction coordinate was computed in the forward direction (since molecule is symmetric) with 50 points along the IRC. This was done with two type of force constants being calculated, once and always.

Within the "once" method the output showed no bond formation process by the 26th step and therefore the calculation was incomplete, since a complete 1,5-hexadiene structure was supposed to be formed. From looking at the IRC pathway it can be seen that the minimum on the potential energy surface was not reached at that point, since the points on the pathway did not form an asymptote and the gradient did not go close enough to zero.

For the "always" method the computation of force constants were carried out at every step. This therefore lets the IRC calculation distinguish whether it is moving along the minimum energy pathway while going down the potential energy surface. While this is happening it can make changes to the pathway if necessary. Within the "always" case the bond forming arrives at 1,5-hexadiene and this is backed up by the IRC pathway reaching a minimum within its energy by the gradient going close to zero and the pathway reaches an asymptote.

The structure produced through the always method had an energy of -231.69199434 atomic units, which is very close to the energy of the gauche2 conformer in the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 appendix 1]. This suggests that the reaction takes place through the chair transition state, which later produces the gauche2 conformer of 1,5-hexadiene.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 6: Comparison of IRC calculations for the chair structure
| || '''Calculate force constants once''' || '''Calculate force constants always'''
|-
| '''Number of points along IRC''' || 26 || 47
|-
| '''Structure at the last point of IRC''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Once_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Always_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|'''Energy (atomic units)''' || -231.68906923 || -231.69199434
|-
| '''ΔE, once - always''' || 2.92511 x 10-3 a.u || 7.68 KJ mol-1
|-
| '''IRC pathway''' || [[Image:One_pathwaydh08.jpg|400x400px]] || [[Image:Always_pathwaydh08.jpg|400x400px]]
|-
| '''IRC gradient''' || [[Image:One_gradientdh08.jpg|400x400px]]|| [[Image:Always_gradientdh08.jpg|400x400px]]
|}

 
 
 

===='''Comparison of activation energies for the transition states'''====

The HF/3-21G optimised chair and boat transition structures were both reoptimised using the B3LYP/6-31G* (B3LYP/6-31G (d)) method. This gives a more accurate minimisation however this method takes longer to run. The values for the zero-point energies and thermal energies were found within the thermochemistry section of the log file.

The values for the HF/3-21G reactant (anti2) optimisation were calculated and the B3LYP/6-31G* values were taken from the "Frequency analysis of 1,5-hexadiene Conformers" earlier in this module.

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 7: Energy data for chair, boat and the anti2 conformation from the HF/3-21G and B3LYP/6-31G* methods of optimisation '''
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466677
| align="center" | -231.461320
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450936
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402342
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539524
| align="center" | -231.532562
| align="center" | -234.611716
| align="center" | -234.469177
| align="center" | -234.461835
|}
Digital repositories

Chair:
HF/3-21G: http://hdl.handle.net/10042/to-5569
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5568

Boat:
HF/3-21G: http://hdl.handle.net/10042/to-5570
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5571

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 8: Summary of activation energies for the chair and boat conformations (ΔE = chair TS or boat TS minus anti2)'''
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| align="center" | 1 a.u = 627.509 kcal mol-1
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE, chair (a.u)'''
| align="center" | 0.072847
| align="center" | 0.071242
| align="center" | 0.054248
| align="center" | 0.052827
| align="center" | -
|-
| '''ΔE, chair (kcal mol-1)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE, boat (a. u)'''
| align="center" | 0.088588
| align="center" | 0.087259
| align="center" | 0.066835
| align="center" | 0.065827
| align="center" | -
|-
| '''ΔE, boat (kcal mol-1)'''
| align="center" | 55.59
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.31
| align="center" | 44.7 ± 2.0
|}

By examining table 8 it is clear to see that the B3LYP/6-31G* method gives values which are much closer to the experimental ones. This is due to the B3LYP/6-31G* giving a higher level of optimisation than the HF/3-21G method. Table 8 also shows that the chair transition state has a lower activation energy to that of the boat transition state by 8.16 kcal mol-1. This suggests that the Cope rearrangement mechanism would most likely proceed through the chair transition state to that of the boat. Therefore the reaction would take place from the anti2 conformer to the chair transition state and then onto the gauche2 conformer within the Cope rearrangement.

It is also possible to see that the calcualted energies are very close to the literature values with the chair conformation being closer to its literature value to that of the boat.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9: Geometric parameters for the optimised chair and boat structures '''
| || ''' HF/3-21G chair TS''' || ''' B3LYP/6-31G* chair TS''' || '''HF/3-21G boat TS''' || '''B3LYP/6-31G* boat TS'''
|-
| Optimised Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Chair_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Boat_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.02 (C5-C10 or C3-C14)|| 1.97 (C5-C10 or C3-C14) || 2.14 (C3-C4 or C1-C6) || 2.21 (C3-C4 or C1-C6)
|-
| Fragment angle || 120 o (C3-C1-C5) || 120 o (C3-C1-C5) || 122 o (C4-C5-C6) || 122 o (C4-C5-C6)
|-
| Dihedral angle || 55 o (C1-C3-C14-C11)|| 54 o (C1-C3-C14-C11)|| 0 o (C2-C1-C6-C5) || 0 o (C2-C1-C6-C5)
|-
| Numbering of carbons || [[Image:Numbering_chairdh08.jpg|200x200px]]|| [[Image:Numbering_chairdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]]
|}

From studying table 9 it can be seen that the C-C bond breaking/forming length has changed between the two different types of optimisation methods. However the angles between the bonds remains unchanged.

=='''The Diels Alder Cycloaddition'''==

The Diels-Alder reaction is a cycloaddition between a conjugated diene and a dienophile (alkene system), which forms a cyclohexene system. It is often referred to as a [4s+2s] cycloaddition and proceeds through a single cyclic, concerted transition state where 2 σ bonds are formed in between the termini carbons of the 2 п systems. This is shown in the reaction scheme below where ethene is reacting with cis-butadiene.

[[Image:Reaction_scheme2dh08.jpg|300x300px|center]]

The π orbitals of the dienophile form new σ bonds with the π orbitals of the diene. The number of π electrons involved determine whether the reaction proceeds in a concerted stereospecific fashion (allowed) or not (forbidden). The HOMO/LUMO of one fragment interacts with the HOMO/LUMO of the other fragment to form two new bonding and anti-bonding MOs. The nodal properties allow one to predict how the reaction will occur.

Semi-Empirical/AM1 method and the B3LYP/6-31G* method of optimisation will be used on Gaussian to determine particular properites of the reactants to predict the way in which the reaction will proceed.

==='''cis-butadiene and ethylene: reagents'''===

<jmol>
<jmolAppletButton>
<uploadedFileContents>Cis_butadienedh08.mol</uploadedFileContents>
<text>Cis-butadiene</text>
<color>whi-te</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Ethenedh08.mol</uploadedFileContents>
<text>ethylene</text>
<color>white</color>
</jmolAppletButton>
</jmol> were optimised separately using the AM1 semi-empirical method on Gaussview.The cis-butadiene had an energy of 0.04879725 a. u (1.28 x 102 KJ mol-1) and ethylene had an energy of 0.02619027 a. u (68.76 KJ mol-1).

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10: Frontier molecular orbitals of cis-butadiene and ethylene '''
| '''Reagent''' || '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| cis-butadiene || LUMO || [[Image:LUMO1_cis_butadieneDH08.jpg|100x100px]] || [[Image:LUMO2_cis_butadienedh08.jpg|100x100px]]|| Symmetric || +0.017
|-
| cis-butadiene || HOMO || [[Image:HOMO1_cis_butadienedh08.jpg|100x100px]] || [[Image:HOMO2_cis_butadienedh08.jpg|100x100px]]|| Anti-symmetric || -0.343
|-
| ethylene || LUMO || [[Image:LUMO1_ethenedh08.jpg|100x100px]] || [[Image:LUMO2_ethenedh08.jpg|100x100px]]|| Anti-symmetric || +0.052
|-
| ethylene || HOMO || [[Image:HOMO1_ethenedh08.jpg|100x100px]] || [[Image:HOMO2_ethenedh08.jpg|100x100px]]|| Symmetric || -0.387
|}

 
The symmetries of the LUMO and HOMO orbitals for cis-butadiene are opposite of those for the ethylene. This therefore means that the only interactions that can take place are those between ethylene's LUMO and cis-butadiene's HOMO and in addition to this the interaction between cis-butadiene's LUMO and ethylene's HOMO. This is because orbital symmetry has to be conserved (the conservation of orbital symmetry states that the symmetry of an new orbital formed by the overlap of the 2 original orbitals must be the same as the symmetry of the initial orbitals).

==='''cis-butadiene and ethylene: transition state'''===

A <jmol>
<jmolAppletButton>
<uploadedFileContents>Bicyclic_startingdh08.mol</uploadedFileContents>
<text>bicyclo system</text>
<color>white</color>
</jmolAppletButton>
</jmol> was drawn and one of the -CH2-CH2- fragments were removed. Three C=C bonds were added and the previous one was taken away so a <jmol>
<jmolAppletButton>
<uploadedFileContents>Frozen_coorddh08.mol</uploadedFileContents>
<text>guess transition state</text>
<color>white</color>
</jmolAppletButton>
</jmol> was produced.

After this an optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while bonds between the termini of the fragments were frozen at 2.1Å. Following this the resulting file had two separate optimisation and frequency calculations run on it using the TS (Berny) with one using the semi-empirical/AM1 method and the other using the DFT/B3LYP/6-31G* level of theory. Both of these optimisations had the bonds unfrozen and the "derivative" option was chosen instead.

The DFT/B3LYP/6-31G* level of theory was used since it gave more accurate results in the calculations before and the energies it produces are consistent with the rest of the report (semi-empirical/AM1 method gives very different values).

The QST2 optimisation was not used since it required numbering the atoms, which will take a lot of time.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11: Comparison of two methods of optimsations for the transition state '''
| || '''semi-empirical/AM1 method''' || '''DFT/B3LYP/6-31G* method'''
|-
| Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>SE_AM1_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>DFT_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Energy (a. u) || 0.11165608 || -234.54389573
|-
| Bond lengths || [[Image:DistancesSEdh08.jpg|150x150px]]|| [[Image:DistancesDFTdh08.jpg|150x150px]]
|-
|Imaginary frequency (cm-1) || -957 || -525
|-
|Animation || [[Image:-957dh08.gif]] || [[Image:-525dh08.gif]]
|-
| Lowest real frequency (cm-1) || 148 || 136
|-
|Animation || [[Image: 148dh08.gif]] || [[Image: 136dh08.gif]]
|}

The C-C bond forming length between the termini carbons is 0.15 Å larger for the DFT/B3LYP/6-31G* level of theory and this is because this method gives carries out the calculation to a greater degree of accuracy. This larger bond length leads a lower potential energy gradient to the transition state, which gives a lower total energy for the transition state. Therefore the given values below will be the DFT/B3LYP/6-31G* and not the semi-empirical/AM1 method.

A typical sp3 C-C bond length is 1.54 Å and a typical sp2 C=C bond length is 1.34 Å<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref>. Within the optimised transition molecule above the C-C bond lengths are 1.40 Å and the C=C bond length is 1.38 Å. This suggests that the π-bonds are breaking in order for the C-C bonds to form and additionally to this a π-bond is forming in between the two sp3 carbons.

The Van der Waals radius of a C-atom is 1.72 Å<ref name="ref 3">M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer, D. G. Truhlar, 1995, 346, pp. 63-69.[http://pubs.acs.org/doi/abs/10.1021/jp8111556]</ref>. The partly formed C-C length is 2.27 Å, which is shorter than two times the the van der Waals radius of a carbon atom. Therefore the carbon atoms are within a close enough distance to each other to share electron density and yet the bond has not formed at this point.

From looking at the animations for the imaginary frequency above it can be seen that the formation of the two bonds between the termini of the reactants is synchronous. This is expected of the Diels-Alder reaction since the mechanism is concerted. However the lowest real frequency is asynchronous with respect to the two bonds forming between the termini.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12: Frontier molecular orbitals of the transition state '''
| '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| LUMO || [[Image:LUMO1TSdh08.jpg|100x100px]] || [[Image:LUMO2TSdh08.jpg|100x100px]]|| Symmetric || +0.023
|-
| HOMO || [[Image:HOMO1TSdh08.jpg|100x100px]] || [[Image:HOMO2TSdh08.jpg|100x100px]]|| Anti-symmetric || -0.323
|}

From examining the images from table 12 above it can be deduced that the LUMO of the transition state originates from the cis-butadiene's LUMO and the ethylene's HOMO, which represents the σ*-antibonding orbitals. However the HOMO of the transition state originates from the cis-butadiene's HOMO and the ethylene's LUMO, which represents the σ-bonding orbitals within the two single C-C bonds that are formed within the reaction.

==='''cis-butadiene and ethylene: activation energy'''===

The cis-butadiene and ethylene were re-optimised at the DFT/B3LYP/6-31G* level of theory so the activation energy of the reaction could be calculated.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 13: Calculation for the activation energy for the reaction'''
| '''molecule''' || '''DFT/B3LYP/6-31G* energy (a. u)''' || '''Calculations''' || '''Energy '''
|-
| cis-butadiene || -155.98595591 || transition state || -234.54389573 a. u
|-
| ethylene || -78.58745828 || ΔE, activation energy || 0.02951846 a. u
|-
| cis-butadiene and ethylene combined || -234.57341419 || ΔE, activation energy || 77.50 KJ mol-1
|}

The calculated activation energy is 77.50 KJ mol-1, which is of the right magnitude when compared to the literature value (115 ± 8 KJ mol-1 at 0 K<ref name="ref 5">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref>) but there is still an error of 37.50 KJ mol-1. This shows that Gaussview is not perfect but still gives an idea of the activation energy required for the reaction to take place.

==='''Cyclohexa-1,3-diene reaction with maleic anhydride'''===

The reaction scheme below shows the stereoselective Diels-Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene. The reaction is kinetically controlled with the major product being the endo product, which will be explained later on in the exercise.

[[Image:Reaction_schemedh081.jpg|center|500x500px]]

The <jmol>
<jmolAppletButton>
<uploadedFileContents>FrozenTS22dh08.mol</uploadedFileContents>
<text>exo</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_frozenTS22dh08.mol</uploadedFileContents>
<text>endo</text>
<color>white</color>
</jmolAppletButton>
</jmol> transition states were drawn on Gaussview and then the optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while the distance of the forming bonds were frozen at 2.1Å between the respective carbons. After this the structures changed slightly (<jmol>
<jmolAppletButton>
<uploadedFileContents>TS2_afteropt1_exo1dh08.mol</uploadedFileContents>
<text>exo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_TS2_afteropt1dh08.mol</uploadedFileContents>
<text>endo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol>) and the resulting files had an optimisation and frequency calculation run on it using the TA (Berny) with using the semi-empirical/AM1 level of theory while the unfrozen option as changed to "derivative". After this the endo and exo optimised transition state structures were re-optimised as they were before but using the DFT/B3LYP/6-31G* level of theory instead with the additional keyword "Opt=NoEigen".

{| class="wikitable" border="1" align="center"
|+ '''Table 14: Comparison of the exo and endo transition states '''
! rowspan="2"|-!! colspan="2"|''exo'' !! colspan="2"|''endo''
|-
| Semi-empirical/AM1 || DFT/B3LYP/6-31G* || Semi-empirical/AM1 || DFT/B3LYP/6-31G*
|-
| align="center"|Structure ||align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>After_opt2_exodh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>Endo_TS2_afteropt2dh081.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| align="center"| Energy (a. u) ||align="center"| -0.05041975 ||align="center"| -612.67931095 ||align="center"| -0.05150470 ||align="center"| -612.68339684
|-
| C-C bond forming length (Å) ||align="center"| 2.17 ||align="center"| 2.29 ||align="center"| 2.16 ||align="center"| 2.27
|-align="center"
| Imaginary frequency (cm-1) ||align="center"| -812 ||align="center"| -448 ||align="center"| -806 ||align="center"| -447
|-align="center"
| Animation ||colspan="2"|[[Image:-812_SEdh08.gif]] || colspan="2"|[[Image:-806_cm-1dh08.gif]]
|}

Again the C-C bond forming length between the termini carbons is larger (by 0.11 and 0.12 Å) for the DFT/B3LYP/6-31G* level of theory and this is due the same reasons talked about above. However the C-C bond forming length between the termini carbons are very similar in both the endo and exo transition states (0.02 Å difference).

From looking at the animations for the imaginary frequency above it can also be seen that the formation of the two bonds between the carbons in this case as well is synchronous. Again this is expected of the Diels-Alder reaction since it is concerted.

{| align=center
|+ '''Strain diagrams for the endo and exo transition states'''
| [[Image:Exostraindh08.jpg|thumb|200x200px|'''Exo TS''']]
| [[Image:Strainendodh08.jpg|thumb|200x200px|'''Endo TS''']]
|}

From studying the energies in table 14 it can be seen that the endo transition state is more stable than the exo by 4.08589 x 10-3 a. u (10.73 KJ mol-1). However by looking at the strain of both transition states one would expect the exo TS to be more stable. This is because the distance between the cyclohexa-1,3-diene ring and the -(C=O)-O-(C=O)- maleic anhydride fragment is 2.99 Å in the endo TS and 3.03 Å in the exo TS. Thus the endo form would have more strain due to steric hindrance.

However within the exo TS the vicinal hydrogens on the CH2-CH2 (carbon numbers 18 and 21) fragment on the cyclohexa-1,3-diene ring are found below the plane and next to the maleic anhydride. This leads to steric hindrance is this is where the strain originates from within the exo TS. Therefore this disfavours the exo form, which makes the endo more stable.

Another factor making the endo TS more favourable is secondary orbital overlap. These orbitals contribute to the bonds forming and they interact with each other through space. Within the endo TS below there are 2 more orbitals interacting (secondary orbitals) than within the exo TS. This therefore makes the endo TS more stable and therefore more favoured.

[[Image:Secondary_orbital_shizzledh08.jpg|500x500px|center]]

{| class="wikitable" border="1" align="center"
|+ '''Table 15: Frontier molecular orbitals of the the transition state for the cyclohexa-1,3-diene reaction '''
! rowspan="2"|Molecular Orbital !! colspan="4"|''exo'' !! colspan="4"|''endo''
|- align="center"
| '''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''||'''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''
|- align="center"
| LUMO || [[Image:LUMO1exoh081.jpg|150x150px]]|| [[Image:LUMO2exodh081.jpg|150x150px]]|| Anti-symmetric || -0.078 || [[Image:LUMO1ENDOdh08.jpg|150x150px]]||[[Image:LUMO2ENDOdh08.jpg|150x150px]] || Anti-symmetric || -0.067
|- align="center"
| HOMO || [[Image:HOMO1exodh081.jpg|150x150px]]||[[Image:HOMO2exodh081.jpg|150x150px]] || Anti-symmetric|| -0.242 || [[Image:HOMO1ENDOdh08.jpg|150x150px]]|| [[Image:HOMO2ENDOdh08.jpg|150x150px]]||Anti-symmetric || -0.242
|}

Digital repositories:

exo: http://hdl.handle.net/10042/to-5596

endo: http://hdl.handle.net/10042/to-5597

From examining table 15 it is clear to see that both the HOMO and LUMO are anti-symmetric for both the transition states. The LUMO of the maleic anhydride fragment is the π* anti-bonding orbital of the C=C bond. Within the reaction then this has electrons transferred to it and the C=C bond breaks, which leaves just a C-C bond. The LUMOs for both transition states are both anti-bonding with respect to the two fragments. The HOMOs of both transitions state represent the σ-bonding that will be formed between the termini carbons on the products and reactants. This is most likely from the overlap of the cyclohexa-1,3-diene's HOMO and the maleic anhydride's LUMO.

='''References'''=
<references/>

Rep:Mod:hampluscheese

2010-11-11T13:07:56Z

Dh08: /* '''Conclusion''' */

='''Module 3:Physical Computational Chemistry Lab'''=

Douglas Hunt

=='''Objectives'''==

The aim of this experiment is to is to use various levels of theory (HF/3-21G, DFT/B3LYP/6-31G (d) and AM1/semi-empirical) on Gaussian to calculate various energies, molecular orbitals, geometrical parameters, reaction coordinate pathways and low and imaginary frequencies for a range of molecules and transition states. This will include using different optimisation methods throughout and will lead to a range of findings in various areas of chemical interest.

=='''The Cope Rearrangement'''==

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangement] is a [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement |3,3|-sigmatropic rearrangement] that takes place through a single [http://en.wikipedia.org/wiki/Pericyclic pericyclic], concerted [http://en.wikipedia.org/wiki/Transition_state transition state]. Within this rearrangement a σ bond is formed in concert with another being broken within a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] π system.

The Cope rearrangement we will be studying will be that of 1,5-hexadiene, which is shown below.

[[Image:Cope_rearrangement_schemedh08.jpg|300x300px|center]]

The objective is to locate the low-energy minima and transition structures on the C6H10 potential energy surface and to determine the preferred reaction mechanism by using [http://en.wikipedia.org/wiki/GAUSSIAN Gaussian], with the [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)] methods.

==='''Optimisation of 1,5-hexadiene Conformers'''===

Rough <jmol>
<jmolAppletButton>
<uploadedFileContents>Anti_2_drawndh08.mol</uploadedFileContents>
<text>anti</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche3_drawndh08.mol</uploadedFileContents>
<text>gauche</text>
<color>white</color>
</jmolAppletButton>
</jmol> conformers of 1,5-hexadiene were drawn on Gaussview and their structures were tidied by the clean function. They were then further optimised by the HF/3-21G method and were identified which conformer they were from their point group and appendix 1 on the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3 physical lab page].

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 1: Optimisation and comparison of anti and gauche 1,5-hexadiene conformers using HF/3-21G methods
| ''' Conformer''' || '''Optimised Structure''' || '''Energy (Atomic Units)''' || '''[http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1 Energies (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (Anti2 - Gauche3)''' || '''Point group'''
|-
| '''Anti2''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Anti_2_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692535 || -231.69254 ||-6.08 x 105 || 1.2606 x 10-4 a. u|| Ci
|-
| '''Gauche3''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Gauche3_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692661 || -231.69266 || -6.08 x 105 || 3.31 x 10-1 KJ mol-1 || C1
|}

From studying table 1 it can be seen that the gauche conformer is 0.33 KJ mol-1 more stable than the anti conformation. One would assume that the anti2 conformer would be more stable due to [http://en.wikipedia.org/wiki/Steric steric] factors/repulsion within the gauche3 conformation. However within the gauche3 conformer there is a favourable donation of [http://en.wikipedia.org/wiki/Electron_density electron density] from the C=C π orbital to the C-H σ* orbital from the adjacent proton, which stabilises the conformer<ref name="ref 1">B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446.[http://www.informaworld.com/smpp/content~db=all?content=10.1080/00268970110081412]</ref>.

 

The anti2 conformer was then optimised using the DFT/B3LYP/6-31G* (6-31G (d)) method.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 2: Comparison of the anti2 conformer optimisations carried out at different calculation methods
| ''' Method''' || '''Structure and bond distances (Å)''' || '''Dihedral angle (C1 to C4)''' || '''Energy (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (HF/3-21G - DFT/B3LYP/6-31G*)'''
|-
| HF/3-21G || [[Image:Anti2_opt1dh08.jpg|300x300px]] || 114.6 o || -231.692535 || -6.08 x 105 || 2.919171 a. u
|-
| DFT/B3LYP/6-31G* || [[Image:Anti2_opt2DH08.jpg|300x300px]] || 118.8 o || -234.611716 || -6.16 x 105 || 7.66 x 103 KJ mol-1
|}

From examining table 2 at first it is possible to see that the bond lengths are very similar to each other and that of the literature bond lengths<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref> (C1=C2: 1.340 Å, C2-C3: 1.508 Å and C3-C4: 1.538 Å). It is also possible to see that the C=C bond length is more accurate for the DFT/B3LYP/6-31G* method by it being 0.01 Å closer to the literature value. Another main difference in terms of geometry is that the DFT/B3LYP/6-31G* method gives a larger [http://en.wikipedia.org/wiki/Dihedral_angle dihedral angle] (and most likely more accurate) of 118.8o compared to that of 114.6o.

In terms of energy the DFT/B3LYP/6-31G* method produces a conformer of a higher stability by it possessing a lower energy with a difference of 7.66 x 103 KJ mol-1.

==='''Frequency analysis of 1,5-hexadiene Conformers'''===

[[Image:Vibrations_anti2dh08.jpg|right|300x300px]]

For the frequency analysis the DFT/B3LYP/6-31G* method was used.

Frequency analysis is essential within this calculation due to it being the second derivative of the potential energy surface. This means that if all the frequencies are all positive the system is at a minimum, if one of them is negative the optimised system is at a transition state, and if more than one are negative then the calculation has failed to find a critical point, which means the optimisation has failed. This therefore gives us a method of seeing whether the calculation has worked.

From looking to the image on the right it can deduced that the optimisation has taken place successfully by there being no negative vibrations and therefore the conformation is at a minimum.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 3: Energies under the thermochemistry section at 298.15 K and 0 K within the frequency logfiles'''
| '''Energies''' || '''298.15 K (atomic units)''' ||| '''0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (KJ mol-1)''' || '''Discussion'''
|-
| Sum of electronic and zero-point energies || -234.469177 || -234.466673 || -2.504 x 10-3 || -6.57 || This is the potential energy at 0 K, which includes the zero-point vibrational energy (E = Eelec + ZPE).

|-
| Sum of electronic and thermal energies || -234.461835 || -234.461085 || -7.5 x 10-4|| -1.97 || This is the energy at 298.15 K (0 K for 0 K calculation) and 1 atm of pressure, which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans).

|-
| Sum of electronic and thermal enthalpies || -234.460891 || -234.460141 || -7.5 x 10-4 || -1.97 || This contains an additional correction for RT (H = E + RT), which is especially important when looking at dissociation reactions.
|-
| Sum of electronic and thermal free energies || -234.500739 || -234.495931 || 4.808 x 10-3 || -12.62 || This includes the entropic contribution to the free energy (G = H - TS).
|}

{|align=right
|+
| [[Image:Anti2_freq_IRdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 298.15 K''']]
| [[Image:Anti2_freq_0Kdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 0 K''']]
|}

From looking at table 3 above it can be deduced that the anti2 conformer is more stable at room temperature to that of 0 K with respect to all the types of energies. The spectra also seem to not change much between the two temperatures but the main change is the change in the sum of electronic and thermal free energies, which is heavily dependent of temperature by it including the entropic contribution to the free energy (G = H - '''T'''S)

 
 
 
 

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

Within the Cope Rearrangement there are two possible transition states, the chair (C2h) and the boat (C2v), which are shown within [https://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_2 appendix two ].

==='''Comparison of optimisation methods for the chair transition state'''===

An <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2hdh08.mol</uploadedFileContents>
<text>allyl fragment</text>
<color>white</color>
</jmolAppletButton>
</jmol>, CH2CHCH2 was drawn on Gaussview and was optimized through the HF/3-21G method. After this the original allyl fragment was copied and pasted into a new window and place exactly above the first fragment with a bond distance of 2.20 Å between the terminal end carbons. This therefore produced the <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2h_fulldh08.mol</uploadedFileContents>
<text>chair transition structure.</text>
<color>white</color>
</jmolAppletButton>
</jmol>

Gaussian was then set up for an optimisation for the transition state with the additional key words "Opt=NoEigen", the optimisation set to TS (Berny), the force constants set to "once" and using the HF/3-21G method.

After this the "chair" transition state was optimised an alternative way by freezing the coordinates of the two sets of terminal carbons (where the bonds form/break during the rearrangement) within the Redundant Coord Editor on Gauusview.
Following this a second calculation was run where the freeze coordinates option was changed to "Derivative" and the force constants were set to "never".

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 4: Comparison of chair transition state optimisation methods
| || '''Optimised through a TS (Berny)''' || '''Optimised through a frozen coordinate method'''
|-
| '''Optimised TS Structure''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>C2h_optfreq_TS_Bdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| '''Terminal C-C bond length (Å)''' || 2.02 || 2.02
|-
|'''Energy (atomic units)''' || -231.61932247 || -231.61932190
|-
| '''ΔE, TS (Berny) - frozen coordinate''' || 5.7 x 10-7 a.u || 1.50 x 10-3 KJ mol-1
|-
| '''Imaginary frequency (cm-1)''' || -818 || -818
|-
| '''Animation''' ||[[Image:818dh08TSB.gif|350x350px]] || [[Image:818cm-1Fdh08.gif]]
|}

The data in table 4 shows that both methods give the same results and they both give the same transition state. They both produced the same energy with a negligible energy difference between them and the bond lengths between the terminal carbons are both 2.02 Å. In addition to this the imaginary frequencies were both -818cm-1 and within the animations above it is possible to see that when one C-C bond is forming the other is breaking. This means it is a asynchronous bond formation, which is expected for the Cope Rearrangement since it is a pericyclic reaction.

The only difference between the two optimisations is the way in which the transition states were calculated. This is due to the TS (Berny) method calculating the force constant matrix within the first optimisation step and is then being recalculated as the optimisation carries on. When the frozen coordinate method freezes the reaction coordinate in order for the rest of the structure to be optimised around it, which does not require the whole hessian (force constant matrix) to be calculated but only that the reaction coordinate is differentiated (second step). The main advantage of this, is that you do not need to calculate the entire force constant matrix and just need to differentiate the reaction coordinate in order to obtain a good enough guess for the initial force constant matrix. The frozen coordinate method will therefore take less time to compute the transition state, which can be very expensive within certain systems. However to carry out this method one needs to have an idea of how far apart the terminal carbons are.

==='''Boat transition state'''===

===='''QST2 optimisation'''====

The optimisation for the boat transition state had a different method and this was the QST2 method. This calculation interpolates between two structures (product and reactant) and tries to find a transition state between them.

The reactants and products were numbered in the same way. Therefore both 1,5-hexadiene molecules had their numbering manually changed for the product molecule so it corresponded to the numbering of the reactant if it rearranged. This is illustrated in the reaction scheme below.

[[Image:Boat_rerrangmentdh08.jpg|300x300px|center]]

The chk file for the optimized anti2 conformer was opened and copied into a new window. A second window was opened within the new window and the same anti2 conformer was copied into the new window. The numbering was changed within the second molecule so the two looked like they do in the image below.

[[Image:TS_(QST2)_numbering_1dh08.jpg|400x400px|center]]

The job type was set as "Opt+Freq" and "TS (QST2)" was chosen to optimise the transition state. However this failed and produced
a structure a bit like a <jmol>
<jmolAppletButton>
<uploadedFileContents>Trialone_BPdh08.mol</uploadedFileContents>
<text>chair transition structure </text>
<color>white</color>
</jmolAppletButton>
</jmol> but more dissociated. This is because it interpolated between the two structures and translated the top allyl fragment and therefore did not consider the possibility of a rotation around the central bonds.

To get around this the C-C-C-C dihedral angle (C2-C3-C4-C5) was set to 0 o and the C-C-C angle (C2-C3-C4 and C3-C4-C5 for the product molecule) was set to 100 o for both molecules. This gave rise to the two structures below.

[[Image:TS_(QST2)_numbering_2dh08.jpg|400x400px|center]]

After this the QST2 calculation was run again and the results in table 5 were produced. The imaginary frequency at -839 cm-1 shows that while one bond is forming the other one is breaking, which is again an asynchronous bond formation. It is also possible to see that the boat transition state has a longer terminal C-C bond length by 0.12 Å to that of the chair.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5: Data for the boat transition state from the QST2 optimisation method
| Optimised TS Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.14
|-
|Energy (atomic units) || -231.60280151
|-
| Energy (KJ mol-1) || -6.08 x 105
|-
| Imaginary frequency (cm-1) || -839
|-
| Animation ||[[Image:-839dh08.gif]]
|}

===='''Intrinsic Reaction Coordinate'''====

It is not possible to predict which product the chair and boat transition state will lead to. To solve this problem within Gaussian one can use the IRC method, which allows one to follow the minimum energy path from a transition structure down to its local minimum on a potential energy surface. This creates a succession of points by taking small geometry steps in the direction to where the gradient of the energy surface is at its steepest until the minimum in energy is reached.

The chk file for the optimized chair transition structure was opened and the job type selected was "IRC", the reaction coordinate was computed in the forward direction (since molecule is symmetric) with 50 points along the IRC. This was done with two type of force constants being calculated, once and always.

Within the "once" method the output showed no bond formation process by the 26th step and therefore the calculation was incomplete, since a complete 1,5-hexadiene structure was supposed to be formed. From looking at the IRC pathway it can be seen that the minimum on the potential energy surface was not reached at that point, since the points on the pathway did not form an asymptote and the gradient did not go close enough to zero.

For the "always" method the computation of force constants were carried out at every step. This therefore lets the IRC calculation distinguish whether it is moving along the minimum energy pathway while going down the potential energy surface. While this is happening it can make changes to the pathway if necessary. Within the "always" case the bond forming arrives at 1,5-hexadiene and this is backed up by the IRC pathway reaching a minimum within its energy by the gradient going close to zero and the pathway reaches an asymptote.

The structure produced through the always method had an energy of -231.69199434 atomic units, which is very close to the energy of the gauche2 conformer in the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 appendix 1]. This suggests that the reaction takes place through the chair transition state, which later produces the gauche2 conformer of 1,5-hexadiene.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 6: Comparison of IRC calculations for the chair structure
| || '''Calculate force constants once''' || '''Calculate force constants always'''
|-
| '''Number of points along IRC''' || 26 || 47
|-
| '''Structure at the last point of IRC''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Once_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Always_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|'''Energy (atomic units)''' || -231.68906923 || -231.69199434
|-
| '''ΔE, once - always''' || 2.92511 x 10-3 a.u || 7.68 KJ mol-1
|-
| '''IRC pathway''' || [[Image:One_pathwaydh08.jpg|400x400px]] || [[Image:Always_pathwaydh08.jpg|400x400px]]
|-
| '''IRC gradient''' || [[Image:One_gradientdh08.jpg|400x400px]]|| [[Image:Always_gradientdh08.jpg|400x400px]]
|}

 
 
 

===='''Comparison of activation energies for the transition states'''====

The HF/3-21G optimised chair and boat transition structures were both reoptimised using the B3LYP/6-31G* (B3LYP/6-31G (d)) method. This gives a more accurate minimisation however this method takes longer to run. The values for the zero-point energies and thermal energies were found within the thermochemistry section of the log file.

The values for the HF/3-21G reactant (anti2) optimisation were calculated and the B3LYP/6-31G* values were taken from the "Frequency analysis of 1,5-hexadiene Conformers" earlier in this module.

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 7: Energy data for chair, boat and the anti2 conformation from the HF/3-21G and B3LYP/6-31G* methods of optimisation '''
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466677
| align="center" | -231.461320
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450936
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402342
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539524
| align="center" | -231.532562
| align="center" | -234.611716
| align="center" | -234.469177
| align="center" | -234.461835
|}
Digital repositories

Chair:
HF/3-21G: http://hdl.handle.net/10042/to-5569
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5568

Boat:
HF/3-21G: http://hdl.handle.net/10042/to-5570
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5571

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 8: Summary of activation energies for the chair and boat conformations (ΔE = chair TS or boat TS minus anti2)'''
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| align="center" | 1 a.u = 627.509 kcal mol-1
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE, chair (a.u)'''
| align="center" | 0.072847
| align="center" | 0.071242
| align="center" | 0.054248
| align="center" | 0.052827
| align="center" | -
|-
| '''ΔE, chair (kcal mol-1)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE, boat (a. u)'''
| align="center" | 0.088588
| align="center" | 0.087259
| align="center" | 0.066835
| align="center" | 0.065827
| align="center" | -
|-
| '''ΔE, boat (kcal mol-1)'''
| align="center" | 55.59
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.31
| align="center" | 44.7 ± 2.0
|}

By examining table 8 it is clear to see that the B3LYP/6-31G* method gives values which are much closer to the experimental ones. This is due to the B3LYP/6-31G* giving a higher level of optimisation than the HF/3-21G method. Table 8 also shows that the chair transition state has a lower activation energy to that of the boat transition state by 8.16 kcal mol-1. This suggests that the Cope rearrangement mechanism would most likely proceed through the chair transition state to that of the boat. Therefore the reaction would take place from the anti2 conformer to the chair transition state and then onto the gauche2 conformer within the Cope rearrangement.

It is also possible to see that the calcualted energies are very close to the literature values with the chair conformation being closer to its literature value to that of the boat.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9: Geometric parameters for the optimised chair and boat structures '''
| || ''' HF/3-21G chair TS''' || ''' B3LYP/6-31G* chair TS''' || '''HF/3-21G boat TS''' || '''B3LYP/6-31G* boat TS'''
|-
| Optimised Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Chair_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Boat_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.02 (C5-C10 or C3-C14)|| 1.97 (C5-C10 or C3-C14) || 2.14 (C3-C4 or C1-C6) || 2.21 (C3-C4 or C1-C6)
|-
| Fragment angle || 120 o (C3-C1-C5) || 120 o (C3-C1-C5) || 122 o (C4-C5-C6) || 122 o (C4-C5-C6)
|-
| Dihedral angle || 55 o (C1-C3-C14-C11)|| 54 o (C1-C3-C14-C11)|| 0 o (C2-C1-C6-C5) || 0 o (C2-C1-C6-C5)
|-
| Numbering of carbons || [[Image:Numbering_chairdh08.jpg|200x200px]]|| [[Image:Numbering_chairdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]]
|}

From studying table 9 it can be seen that the C-C bond breaking/forming length has changed between the two different types of optimisation methods. However the angles between the bonds remains unchanged.

=='''The Diels Alder Cycloaddition'''==

The Diels-Alder reaction is a cycloaddition between a conjugated diene and a dienophile (alkene system), which forms a cyclohexene system. It is often referred to as a [4s+2s] cycloaddition and proceeds through a single cyclic, concerted transition state where 2 σ bonds are formed in between the termini carbons of the 2 п systems. This is shown in the reaction scheme below where ethene is reacting with cis-butadiene.

[[Image:Reaction_scheme2dh08.jpg|300x300px|center]]

The π orbitals of the dienophile form new σ bonds with the π orbitals of the diene. The number of π electrons involved determine whether the reaction proceeds in a concerted stereospecific fashion (allowed) or not (forbidden). The HOMO/LUMO of one fragment interacts with the HOMO/LUMO of the other fragment to form two new bonding and anti-bonding MOs. The nodal properties allow one to predict how the reaction will occur.

Semi-Empirical/AM1 method and the B3LYP/6-31G* method of optimisation will be used on Gaussian to determine particular properites of the reactants to predict the way in which the reaction will proceed.

==='''cis-butadiene and ethylene: reagents'''===

<jmol>
<jmolAppletButton>
<uploadedFileContents>Cis_butadienedh08.mol</uploadedFileContents>
<text>Cis-butadiene</text>
<color>whi-te</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Ethenedh08.mol</uploadedFileContents>
<text>ethylene</text>
<color>white</color>
</jmolAppletButton>
</jmol> were optimised separately using the AM1 semi-empirical method on Gaussview.The cis-butadiene had an energy of 0.04879725 a. u (1.28 x 102 KJ mol-1) and ethylene had an energy of 0.02619027 a. u (68.76 KJ mol-1).

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10: Frontier molecular orbitals of cis-butadiene and ethylene '''
| '''Reagent''' || '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| cis-butadiene || LUMO || [[Image:LUMO1_cis_butadieneDH08.jpg|100x100px]] || [[Image:LUMO2_cis_butadienedh08.jpg|100x100px]]|| Symmetric || +0.017
|-
| cis-butadiene || HOMO || [[Image:HOMO1_cis_butadienedh08.jpg|100x100px]] || [[Image:HOMO2_cis_butadienedh08.jpg|100x100px]]|| Anti-symmetric || -0.343
|-
| ethylene || LUMO || [[Image:LUMO1_ethenedh08.jpg|100x100px]] || [[Image:LUMO2_ethenedh08.jpg|100x100px]]|| Anti-symmetric || +0.052
|-
| ethylene || HOMO || [[Image:HOMO1_ethenedh08.jpg|100x100px]] || [[Image:HOMO2_ethenedh08.jpg|100x100px]]|| Symmetric || -0.387
|}

 
The symmetries of the LUMO and HOMO orbitals for cis-butadiene are opposite of those for the ethylene. This therefore means that the only interactions that can take place are those between ethylene's LUMO and cis-butadiene's HOMO and in addition to this the interaction between cis-butadiene's LUMO and ethylene's HOMO. This is because orbital symmetry has to be conserved (the conservation of orbital symmetry states that the symmetry of an new orbital formed by the overlap of the 2 original orbitals must be the same as the symmetry of the initial orbitals).

==='''cis-butadiene and ethylene: transition state'''===

A <jmol>
<jmolAppletButton>
<uploadedFileContents>Bicyclic_startingdh08.mol</uploadedFileContents>
<text>bicyclo system</text>
<color>white</color>
</jmolAppletButton>
</jmol> was drawn and one of the -CH2-CH2- fragments were removed. Three C=C bonds were added and the previous one was taken away so a <jmol>
<jmolAppletButton>
<uploadedFileContents>Frozen_coorddh08.mol</uploadedFileContents>
<text>guess transition state</text>
<color>white</color>
</jmolAppletButton>
</jmol> was produced.

After this an optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while bonds between the termini of the fragments were frozen at 2.1Å. Following this the resulting file had two separate optimisation and frequency calculations run on it using the TS (Berny) with one using the semi-empirical/AM1 method and the other using the DFT/B3LYP/6-31G* level of theory. Both of these optimisations had the bonds unfrozen and the "derivative" option was chosen instead.

The DFT/B3LYP/6-31G* level of theory was used since it gave more accurate results in the calculations before and the energies it produces are consistent with the rest of the report (semi-empirical/AM1 method gives very different values).

The QST2 optimisation was not used since it required numbering the atoms, which will take a lot of time.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11: Comparison of two methods of optimsations for the transition state '''
| || '''semi-empirical/AM1 method''' || '''DFT/B3LYP/6-31G* method'''
|-
| Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>SE_AM1_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>DFT_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Energy (a. u) || 0.11165608 || -234.54389573
|-
| Bond lengths || [[Image:DistancesSEdh08.jpg|150x150px]]|| [[Image:DistancesDFTdh08.jpg|150x150px]]
|-
|Imaginary frequency (cm-1) || -957 || -525
|-
|Animation || [[Image:-957dh08.gif]] || [[Image:-525dh08.gif]]
|-
| Lowest real frequency (cm-1) || 148 || 136
|-
|Animation || [[Image: 148dh08.gif]] || [[Image: 136dh08.gif]]
|}

The C-C bond forming length between the termini carbons is 0.15 Å larger for the DFT/B3LYP/6-31G* level of theory and this is because this method gives carries out the calculation to a greater degree of accuracy. This larger bond length leads a lower potential energy gradient to the transition state, which gives a lower total energy for the transition state. Therefore the given values below will be the DFT/B3LYP/6-31G* and not the semi-empirical/AM1 method.

A typical sp3 C-C bond length is 1.54 Å and a typical sp2 C=C bond length is 1.34 Å<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref>. Within the optimised transition molecule above the C-C bond lengths are 1.40 Å and the C=C bond length is 1.38 Å. This suggests that the π-bonds are breaking in order for the C-C bonds to form and additionally to this a π-bond is forming in between the two sp3 carbons.

The Van der Waals radius of a C-atom is 1.72 Å<ref name="ref 3">M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer, D. G. Truhlar, 1995, 346, pp. 63-69.[http://pubs.acs.org/doi/abs/10.1021/jp8111556]</ref>. The partly formed C-C length is 2.27 Å, which is shorter than two times the the van der Waals radius of a carbon atom. Therefore the carbon atoms are within a close enough distance to each other to share electron density and yet the bond has not formed at this point.

From looking at the animations for the imaginary frequency above it can be seen that the formation of the two bonds between the termini of the reactants is synchronous. This is expected of the Diels-Alder reaction since the mechanism is concerted. However the lowest real frequency is asynchronous with respect to the two bonds forming between the termini.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12: Frontier molecular orbitals of the transition state '''
| '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| LUMO || [[Image:LUMO1TSdh08.jpg|100x100px]] || [[Image:LUMO2TSdh08.jpg|100x100px]]|| Symmetric || +0.023
|-
| HOMO || [[Image:HOMO1TSdh08.jpg|100x100px]] || [[Image:HOMO2TSdh08.jpg|100x100px]]|| Anti-symmetric || -0.323
|}

From examining the images from table 12 above it can be deduced that the LUMO of the transition state originates from the cis-butadiene's LUMO and the ethylene's HOMO, which represents the σ*-antibonding orbitals. However the HOMO of the transition state originates from the cis-butadiene's HOMO and the ethylene's LUMO, which represents the σ-bonding orbitals within the two single C-C bonds that are formed within the reaction.

==='''cis-butadiene and ethylene: activation energy'''===

The cis-butadiene and ethylene were re-optimised at the DFT/B3LYP/6-31G* level of theory so the activation energy of the reaction could be calculated.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 13: Calculation for the activation energy for the reaction'''
| '''molecule''' || '''DFT/B3LYP/6-31G* energy (a. u)''' || '''Calculations''' || '''Energy '''
|-
| cis-butadiene || -155.98595591 || transition state || -234.54389573 a. u
|-
| ethylene || -78.58745828 || ΔE, activation energy || 0.02951846 a. u
|-
| cis-butadiene and ethylene combined || -234.57341419 || ΔE, activation energy || 77.50 KJ mol-1
|}

The calculated activation energy is 77.50 KJ mol-1, which is of the right magnitude when compared to the literature value (115 ± 8 KJ mol-1 at 0 K) but there is still an error of 37.50 KJ mol-1. This shows that Gaussview is not perfect but still gives an idea of the activation energy required for the reaction to take place.

==='''Cyclohexa-1,3-diene reaction with maleic anhydride'''===

The reaction scheme below shows the stereoselective Diels-Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene. The reaction is kinetically controlled with the major product being the endo product, which will be explained later on in the exercise.

[[Image:Reaction_schemedh081.jpg|center|500x500px]]

The <jmol>
<jmolAppletButton>
<uploadedFileContents>FrozenTS22dh08.mol</uploadedFileContents>
<text>exo</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_frozenTS22dh08.mol</uploadedFileContents>
<text>endo</text>
<color>white</color>
</jmolAppletButton>
</jmol> transition states were drawn on Gaussview and then the optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while the distance of the forming bonds were frozen at 2.1Å between the respective carbons. After this the structures changed slightly (<jmol>
<jmolAppletButton>
<uploadedFileContents>TS2_afteropt1_exo1dh08.mol</uploadedFileContents>
<text>exo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_TS2_afteropt1dh08.mol</uploadedFileContents>
<text>endo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol>) and the resulting files had an optimisation and frequency calculation run on it using the TA (Berny) with using the semi-empirical/AM1 level of theory while the unfrozen option as changed to "derivative". After this the endo and exo optimised transition state structures were re-optimised as they were before but using the DFT/B3LYP/6-31G* level of theory instead with the additional keyword "Opt=NoEigen".

{| class="wikitable" border="1" align="center"
|+ '''Table 14: Comparison of the exo and endo transition states '''
! rowspan="2"|-!! colspan="2"|''exo'' !! colspan="2"|''endo''
|-
| Semi-empirical/AM1 || DFT/B3LYP/6-31G* || Semi-empirical/AM1 || DFT/B3LYP/6-31G*
|-
| align="center"|Structure ||align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>After_opt2_exodh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>Endo_TS2_afteropt2dh081.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| align="center"| Energy (a. u) ||align="center"| -0.05041975 ||align="center"| -612.67931095 ||align="center"| -0.05150470 ||align="center"| -612.68339684
|-
| C-C bond forming length (Å) ||align="center"| 2.17 ||align="center"| 2.29 ||align="center"| 2.16 ||align="center"| 2.27
|-align="center"
| Imaginary frequency (cm-1) ||align="center"| -812 ||align="center"| -448 ||align="center"| -806 ||align="center"| -447
|-align="center"
| Animation ||colspan="2"|[[Image:-812_SEdh08.gif]] || colspan="2"|[[Image:-806_cm-1dh08.gif]]
|}

Again the C-C bond forming length between the termini carbons is larger (by 0.11 and 0.12 Å) for the DFT/B3LYP/6-31G* level of theory and this is due the same reasons talked about above. However the C-C bond forming length between the termini carbons are very similar in both the endo and exo transition states (0.02 Å difference).

From looking at the animations for the imaginary frequency above it can also be seen that the formation of the two bonds between the carbons in this case as well is synchronous. Again this is expected of the Diels-Alder reaction since it is concerted.

{| align=center
|+ '''Strain diagrams for the endo and exo transition states'''
| [[Image:Exostraindh08.jpg|thumb|200x200px|'''Exo TS''']]
| [[Image:Strainendodh08.jpg|thumb|200x200px|'''Endo TS''']]
|}

From studying the energies in table 14 it can be seen that the endo transition state is more stable than the exo by 4.08589 x 10-3 a. u (10.73 KJ mol-1). However by looking at the strain of both transition states one would expect the exo TS to be more stable. This is because the distance between the cyclohexa-1,3-diene ring and the -(C=O)-O-(C=O)- maleic anhydride fragment is 2.99 Å in the endo TS and 3.03 Å in the exo TS. Thus the endo form would have more strain due to steric hindrance.

However within the exo TS the vicinal hydrogens on the CH2-CH2 (carbon numbers 18 and 21) fragment on the cyclohexa-1,3-diene ring are found below the plane and next to the maleic anhydride. This leads to steric hindrance is this is where the strain originates from within the exo TS. Therefore this disfavours the exo form, which makes the endo more stable.

Another factor making the endo TS more favourable is secondary orbital overlap. These orbitals contribute to the bonds forming and they interact with each other through space. Within the endo TS below there are 2 more orbitals interacting (secondary orbitals) than within the exo TS. This therefore makes the endo TS more stable and therefore more favoured.

[[Image:Secondary_orbital_shizzledh08.jpg|500x500px|center]]

{| class="wikitable" border="1" align="center"
|+ '''Table 15: Frontier molecular orbitals of the the transition state for the cyclohexa-1,3-diene reaction '''
! rowspan="2"|Molecular Orbital !! colspan="4"|''exo'' !! colspan="4"|''endo''
|- align="center"
| '''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''||'''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''
|- align="center"
| LUMO || [[Image:LUMO1exoh081.jpg|150x150px]]|| [[Image:LUMO2exodh081.jpg|150x150px]]|| Anti-symmetric || -0.078 || [[Image:LUMO1ENDOdh08.jpg|150x150px]]||[[Image:LUMO2ENDOdh08.jpg|150x150px]] || Anti-symmetric || -0.067
|- align="center"
| HOMO || [[Image:HOMO1exodh081.jpg|150x150px]]||[[Image:HOMO2exodh081.jpg|150x150px]] || Anti-symmetric|| -0.242 || [[Image:HOMO1ENDOdh08.jpg|150x150px]]|| [[Image:HOMO2ENDOdh08.jpg|150x150px]]||Anti-symmetric || -0.242
|}

Digital repositories:

exo: http://hdl.handle.net/10042/to-5596

endo: http://hdl.handle.net/10042/to-5597

From examining table 15 it is clear to see that both the HOMO and LUMO are anti-symmetric for both the transition states. The LUMO of the maleic anhydride fragment is the π* anti-bonding orbital of the C=C bond. Within the reaction then this has electrons transferred to it and the C=C bond breaks, which leaves just a C-C bond. The LUMOs for both transition states are both anti-bonding with respect to the two fragments. The HOMOs of both transitions state represent the σ-bonding that will be formed between the termini carbons on the products and reactants. This is most likely from the overlap of the cyclohexa-1,3-diene's HOMO and the maleic anhydride's LUMO.

='''References'''=
<references/>

Rep:Mod:hampluscheese

2010-11-11T13:02:27Z

Dh08: /* '''Optimisation of 1,5-hexadiene Conformers''' */

='''Module 3:Physical Computational Chemistry Lab'''=

Douglas Hunt

=='''Objectives'''==

The aim of this experiment is to is to use various levels of theory (HF/3-21G, DFT/B3LYP/6-31G (d) and AM1/semi-empirical) on Gaussian to calculate various energies, molecular orbitals, geometrical parameters, reaction coordinate pathways and low and imaginary frequencies for a range of molecules and transition states. This will include using different optimisation methods throughout and will lead to a range of findings in various areas of chemical interest.

=='''The Cope Rearrangement'''==

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangement] is a [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement |3,3|-sigmatropic rearrangement] that takes place through a single [http://en.wikipedia.org/wiki/Pericyclic pericyclic], concerted [http://en.wikipedia.org/wiki/Transition_state transition state]. Within this rearrangement a σ bond is formed in concert with another being broken within a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] π system.

The Cope rearrangement we will be studying will be that of 1,5-hexadiene, which is shown below.

[[Image:Cope_rearrangement_schemedh08.jpg|300x300px|center]]

The objective is to locate the low-energy minima and transition structures on the C6H10 potential energy surface and to determine the preferred reaction mechanism by using [http://en.wikipedia.org/wiki/GAUSSIAN Gaussian], with the [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)] methods.

==='''Optimisation of 1,5-hexadiene Conformers'''===

Rough <jmol>
<jmolAppletButton>
<uploadedFileContents>Anti_2_drawndh08.mol</uploadedFileContents>
<text>anti</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche3_drawndh08.mol</uploadedFileContents>
<text>gauche</text>
<color>white</color>
</jmolAppletButton>
</jmol> conformers of 1,5-hexadiene were drawn on Gaussview and their structures were tidied by the clean function. They were then further optimised by the HF/3-21G method and were identified which conformer they were from their point group and appendix 1 on the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3 physical lab page].

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 1: Optimisation and comparison of anti and gauche 1,5-hexadiene conformers using HF/3-21G methods
| ''' Conformer''' || '''Optimised Structure''' || '''Energy (Atomic Units)''' || '''[http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1 Energies (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (Anti2 - Gauche3)''' || '''Point group'''
|-
| '''Anti2''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Anti_2_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692535 || -231.69254 ||-6.08 x 105 || 1.2606 x 10-4 a. u|| Ci
|-
| '''Gauche3''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Gauche3_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692661 || -231.69266 || -6.08 x 105 || 3.31 x 10-1 KJ mol-1 || C1
|}

From studying table 1 it can be seen that the gauche conformer is 0.33 KJ mol-1 more stable than the anti conformation. One would assume that the anti2 conformer would be more stable due to [http://en.wikipedia.org/wiki/Steric steric] factors/repulsion within the gauche3 conformation. However within the gauche3 conformer there is a favourable donation of [http://en.wikipedia.org/wiki/Electron_density electron density] from the C=C π orbital to the C-H σ* orbital from the adjacent proton, which stabilises the conformer<ref name="ref 1">B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446.[http://www.informaworld.com/smpp/content~db=all?content=10.1080/00268970110081412]</ref>.

 

The anti2 conformer was then optimised using the DFT/B3LYP/6-31G* (6-31G (d)) method.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 2: Comparison of the anti2 conformer optimisations carried out at different calculation methods
| ''' Method''' || '''Structure and bond distances (Å)''' || '''Dihedral angle (C1 to C4)''' || '''Energy (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (HF/3-21G - DFT/B3LYP/6-31G*)'''
|-
| HF/3-21G || [[Image:Anti2_opt1dh08.jpg|300x300px]] || 114.6 o || -231.692535 || -6.08 x 105 || 2.919171 a. u
|-
| DFT/B3LYP/6-31G* || [[Image:Anti2_opt2DH08.jpg|300x300px]] || 118.8 o || -234.611716 || -6.16 x 105 || 7.66 x 103 KJ mol-1
|}

From examining table 2 at first it is possible to see that the bond lengths are very similar to each other and that of the literature bond lengths<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref> (C1=C2: 1.340 Å, C2-C3: 1.508 Å and C3-C4: 1.538 Å). It is also possible to see that the C=C bond length is more accurate for the DFT/B3LYP/6-31G* method by it being 0.01 Å closer to the literature value. Another main difference in terms of geometry is that the DFT/B3LYP/6-31G* method gives a larger [http://en.wikipedia.org/wiki/Dihedral_angle dihedral angle] (and most likely more accurate) of 118.8o compared to that of 114.6o.

In terms of energy the DFT/B3LYP/6-31G* method produces a conformer of a higher stability by it possessing a lower energy with a difference of 7.66 x 103 KJ mol-1.

==='''Frequency analysis of 1,5-hexadiene Conformers'''===

[[Image:Vibrations_anti2dh08.jpg|right|300x300px]]

For the frequency analysis the DFT/B3LYP/6-31G* method was used.

Frequency analysis is essential within this calculation due to it being the second derivative of the potential energy surface. This means that if all the frequencies are all positive the system is at a minimum, if one of them is negative the optimised system is at a transition state, and if more than one are negative then the calculation has failed to find a critical point, which means the optimisation has failed. This therefore gives us a method of seeing whether the calculation has worked.

From looking to the image on the right it can deduced that the optimisation has taken place successfully by there being no negative vibrations and therefore the conformation is at a minimum.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 3: Energies under the thermochemistry section at 298.15 K and 0 K within the frequency logfiles'''
| '''Energies''' || '''298.15 K (atomic units)''' ||| '''0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (KJ mol-1)''' || '''Discussion'''
|-
| Sum of electronic and zero-point energies || -234.469177 || -234.466673 || -2.504 x 10-3 || -6.57 || This is the potential energy at 0 K, which includes the zero-point vibrational energy (E = Eelec + ZPE).

|-
| Sum of electronic and thermal energies || -234.461835 || -234.461085 || -7.5 x 10-4|| -1.97 || This is the energy at 298.15 K (0 K for 0 K calculation) and 1 atm of pressure, which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans).

|-
| Sum of electronic and thermal enthalpies || -234.460891 || -234.460141 || -7.5 x 10-4 || -1.97 || This contains an additional correction for RT (H = E + RT), which is especially important when looking at dissociation reactions.
|-
| Sum of electronic and thermal free energies || -234.500739 || -234.495931 || 4.808 x 10-3 || -12.62 || This includes the entropic contribution to the free energy (G = H - TS).
|}

{|align=right
|+
| [[Image:Anti2_freq_IRdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 298.15 K''']]
| [[Image:Anti2_freq_0Kdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 0 K''']]
|}

From looking at table 3 above it can be deduced that the anti2 conformer is more stable at room temperature to that of 0 K with respect to all the types of energies. The spectra also seem to not change much between the two temperatures but the main change is the change in the sum of electronic and thermal free energies, which is heavily dependent of temperature by it including the entropic contribution to the free energy (G = H - '''T'''S)

 
 
 
 

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

Within the Cope Rearrangement there are two possible transition states, the chair (C2h) and the boat (C2v), which are shown within [https://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_2 appendix two ].

==='''Comparison of optimisation methods for the chair transition state'''===

An <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2hdh08.mol</uploadedFileContents>
<text>allyl fragment</text>
<color>white</color>
</jmolAppletButton>
</jmol>, CH2CHCH2 was drawn on Gaussview and was optimized through the HF/3-21G method. After this the original allyl fragment was copied and pasted into a new window and place exactly above the first fragment with a bond distance of 2.20 Å between the terminal end carbons. This therefore produced the <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2h_fulldh08.mol</uploadedFileContents>
<text>chair transition structure.</text>
<color>white</color>
</jmolAppletButton>
</jmol>

Gaussian was then set up for an optimisation for the transition state with the additional key words "Opt=NoEigen", the optimisation set to TS (Berny), the force constants set to "once" and using the HF/3-21G method.

After this the "chair" transition state was optimised an alternative way by freezing the coordinates of the two sets of terminal carbons (where the bonds form/break during the rearrangement) within the Redundant Coord Editor on Gauusview.
Following this a second calculation was run where the freeze coordinates option was changed to "Derivative" and the force constants were set to "never".

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 4: Comparison of chair transition state optimisation methods
| || '''Optimised through a TS (Berny)''' || '''Optimised through a frozen coordinate method'''
|-
| '''Optimised TS Structure''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>C2h_optfreq_TS_Bdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| '''Terminal C-C bond length (Å)''' || 2.02 || 2.02
|-
|'''Energy (atomic units)''' || -231.61932247 || -231.61932190
|-
| '''ΔE, TS (Berny) - frozen coordinate''' || 5.7 x 10-7 a.u || 1.50 x 10-3 KJ mol-1
|-
| '''Imaginary frequency (cm-1)''' || -818 || -818
|-
| '''Animation''' ||[[Image:818dh08TSB.gif|350x350px]] || [[Image:818cm-1Fdh08.gif]]
|}

The data in table 4 shows that both methods give the same results and they both give the same transition state. They both produced the same energy with a negligible energy difference between them and the bond lengths between the terminal carbons are both 2.02 Å. In addition to this the imaginary frequencies were both -818cm-1 and within the animations above it is possible to see that when one C-C bond is forming the other is breaking. This means it is a asynchronous bond formation, which is expected for the Cope Rearrangement since it is a pericyclic reaction.

The only difference between the two optimisations is the way in which the transition states were calculated. This is due to the TS (Berny) method calculating the force constant matrix within the first optimisation step and is then being recalculated as the optimisation carries on. When the frozen coordinate method freezes the reaction coordinate in order for the rest of the structure to be optimised around it, which does not require the whole hessian (force constant matrix) to be calculated but only that the reaction coordinate is differentiated (second step). The main advantage of this, is that you do not need to calculate the entire force constant matrix and just need to differentiate the reaction coordinate in order to obtain a good enough guess for the initial force constant matrix. The frozen coordinate method will therefore take less time to compute the transition state, which can be very expensive within certain systems. However to carry out this method one needs to have an idea of how far apart the terminal carbons are.

==='''Boat transition state'''===

===='''QST2 optimisation'''====

The optimisation for the boat transition state had a different method and this was the QST2 method. This calculation interpolates between two structures (product and reactant) and tries to find a transition state between them.

The reactants and products were numbered in the same way. Therefore both 1,5-hexadiene molecules had their numbering manually changed for the product molecule so it corresponded to the numbering of the reactant if it rearranged. This is illustrated in the reaction scheme below.

[[Image:Boat_rerrangmentdh08.jpg|300x300px|center]]

The chk file for the optimized anti2 conformer was opened and copied into a new window. A second window was opened within the new window and the same anti2 conformer was copied into the new window. The numbering was changed within the second molecule so the two looked like they do in the image below.

[[Image:TS_(QST2)_numbering_1dh08.jpg|400x400px|center]]

The job type was set as "Opt+Freq" and "TS (QST2)" was chosen to optimise the transition state. However this failed and produced
a structure a bit like a <jmol>
<jmolAppletButton>
<uploadedFileContents>Trialone_BPdh08.mol</uploadedFileContents>
<text>chair transition structure </text>
<color>white</color>
</jmolAppletButton>
</jmol> but more dissociated. This is because it interpolated between the two structures and translated the top allyl fragment and therefore did not consider the possibility of a rotation around the central bonds.

To get around this the C-C-C-C dihedral angle (C2-C3-C4-C5) was set to 0 o and the C-C-C angle (C2-C3-C4 and C3-C4-C5 for the product molecule) was set to 100 o for both molecules. This gave rise to the two structures below.

[[Image:TS_(QST2)_numbering_2dh08.jpg|400x400px|center]]

After this the QST2 calculation was run again and the results in table 5 were produced. The imaginary frequency at -839 cm-1 shows that while one bond is forming the other one is breaking, which is again an asynchronous bond formation. It is also possible to see that the boat transition state has a longer terminal C-C bond length by 0.12 Å to that of the chair.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5: Data for the boat transition state from the QST2 optimisation method
| Optimised TS Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.14
|-
|Energy (atomic units) || -231.60280151
|-
| Energy (KJ mol-1) || -6.08 x 105
|-
| Imaginary frequency (cm-1) || -839
|-
| Animation ||[[Image:-839dh08.gif]]
|}

===='''Intrinsic Reaction Coordinate'''====

It is not possible to predict which product the chair and boat transition state will lead to. To solve this problem within Gaussian one can use the IRC method, which allows one to follow the minimum energy path from a transition structure down to its local minimum on a potential energy surface. This creates a succession of points by taking small geometry steps in the direction to where the gradient of the energy surface is at its steepest until the minimum in energy is reached.

The chk file for the optimized chair transition structure was opened and the job type selected was "IRC", the reaction coordinate was computed in the forward direction (since molecule is symmetric) with 50 points along the IRC. This was done with two type of force constants being calculated, once and always.

Within the "once" method the output showed no bond formation process by the 26th step and therefore the calculation was incomplete, since a complete 1,5-hexadiene structure was supposed to be formed. From looking at the IRC pathway it can be seen that the minimum on the potential energy surface was not reached at that point, since the points on the pathway did not form an asymptote and the gradient did not go close enough to zero.

For the "always" method the computation of force constants were carried out at every step. This therefore lets the IRC calculation distinguish whether it is moving along the minimum energy pathway while going down the potential energy surface. While this is happening it can make changes to the pathway if necessary. Within the "always" case the bond forming arrives at 1,5-hexadiene and this is backed up by the IRC pathway reaching a minimum within its energy by the gradient going close to zero and the pathway reaches an asymptote.

The structure produced through the always method had an energy of -231.69199434 atomic units, which is very close to the energy of the gauche2 conformer in the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 appendix 1]. This suggests that the reaction takes place through the chair transition state, which later produces the gauche2 conformer of 1,5-hexadiene.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 6: Comparison of IRC calculations for the chair structure
| || '''Calculate force constants once''' || '''Calculate force constants always'''
|-
| '''Number of points along IRC''' || 26 || 47
|-
| '''Structure at the last point of IRC''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Once_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Always_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|'''Energy (atomic units)''' || -231.68906923 || -231.69199434
|-
| '''ΔE, once - always''' || 2.92511 x 10-3 a.u || 7.68 KJ mol-1
|-
| '''IRC pathway''' || [[Image:One_pathwaydh08.jpg|400x400px]] || [[Image:Always_pathwaydh08.jpg|400x400px]]
|-
| '''IRC gradient''' || [[Image:One_gradientdh08.jpg|400x400px]]|| [[Image:Always_gradientdh08.jpg|400x400px]]
|}

 
 
 

===='''Comparison of activation energies for the transition states'''====

The HF/3-21G optimised chair and boat transition structures were both reoptimised using the B3LYP/6-31G* (B3LYP/6-31G (d)) method. This gives a more accurate minimisation however this method takes longer to run. The values for the zero-point energies and thermal energies were found within the thermochemistry section of the log file.

The values for the HF/3-21G reactant (anti2) optimisation were calculated and the B3LYP/6-31G* values were taken from the "Frequency analysis of 1,5-hexadiene Conformers" earlier in this module.

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 7: Energy data for chair, boat and the anti2 conformation from the HF/3-21G and B3LYP/6-31G* methods of optimisation '''
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466677
| align="center" | -231.461320
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450936
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402342
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539524
| align="center" | -231.532562
| align="center" | -234.611716
| align="center" | -234.469177
| align="center" | -234.461835
|}
Digital repositories

Chair:
HF/3-21G: http://hdl.handle.net/10042/to-5569
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5568

Boat:
HF/3-21G: http://hdl.handle.net/10042/to-5570
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5571

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 8: Summary of activation energies for the chair and boat conformations (ΔE = chair TS or boat TS minus anti2)'''
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| align="center" | 1 a.u = 627.509 kcal mol-1
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE, chair (a.u)'''
| align="center" | 0.072847
| align="center" | 0.071242
| align="center" | 0.054248
| align="center" | 0.052827
| align="center" | -
|-
| '''ΔE, chair (kcal mol-1)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE, boat (a. u)'''
| align="center" | 0.088588
| align="center" | 0.087259
| align="center" | 0.066835
| align="center" | 0.065827
| align="center" | -
|-
| '''ΔE, boat (kcal mol-1)'''
| align="center" | 55.59
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.31
| align="center" | 44.7 ± 2.0
|}

By examining table 8 it is clear to see that the B3LYP/6-31G* method gives values which are much closer to the experimental ones. This is due to the B3LYP/6-31G* giving a higher level of optimisation than the HF/3-21G method. Table 8 also shows that the chair transition state has a lower activation energy to that of the boat transition state by 8.16 kcal mol-1. This suggests that the Cope rearrangement mechanism would most likely proceed through the chair transition state to that of the boat. Therefore the reaction would take place from the anti2 conformer to the chair transition state and then onto the gauche2 conformer within the Cope rearrangement.

It is also possible to see that the calcualted energies are very close to the literature values with the chair conformation being closer to its literature value to that of the boat.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9: Geometric parameters for the optimised chair and boat structures '''
| || ''' HF/3-21G chair TS''' || ''' B3LYP/6-31G* chair TS''' || '''HF/3-21G boat TS''' || '''B3LYP/6-31G* boat TS'''
|-
| Optimised Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Chair_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Boat_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.02 (C5-C10 or C3-C14)|| 1.97 (C5-C10 or C3-C14) || 2.14 (C3-C4 or C1-C6) || 2.21 (C3-C4 or C1-C6)
|-
| Fragment angle || 120 o (C3-C1-C5) || 120 o (C3-C1-C5) || 122 o (C4-C5-C6) || 122 o (C4-C5-C6)
|-
| Dihedral angle || 55 o (C1-C3-C14-C11)|| 54 o (C1-C3-C14-C11)|| 0 o (C2-C1-C6-C5) || 0 o (C2-C1-C6-C5)
|-
| Numbering of carbons || [[Image:Numbering_chairdh08.jpg|200x200px]]|| [[Image:Numbering_chairdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]]
|}

From studying table 9 it can be seen that the C-C bond breaking/forming length has changed between the two different types of optimisation methods. However the angles between the bonds remains unchanged.

=='''The Diels Alder Cycloaddition'''==

The Diels-Alder reaction is a cycloaddition between a conjugated diene and a dienophile (alkene system), which forms a cyclohexene system. It is often referred to as a [4s+2s] cycloaddition and proceeds through a single cyclic, concerted transition state where 2 σ bonds are formed in between the termini carbons of the 2 п systems. This is shown in the reaction scheme below where ethene is reacting with cis-butadiene.

[[Image:Reaction_scheme2dh08.jpg|300x300px|center]]

The π orbitals of the dienophile form new σ bonds with the π orbitals of the diene. The number of π electrons involved determine whether the reaction proceeds in a concerted stereospecific fashion (allowed) or not (forbidden). The HOMO/LUMO of one fragment interacts with the HOMO/LUMO of the other fragment to form two new bonding and anti-bonding MOs. The nodal properties allow one to predict how the reaction will occur.

Semi-Empirical/AM1 method and the B3LYP/6-31G* method of optimisation will be used on Gaussian to determine particular properites of the reactants to predict the way in which the reaction will proceed.

==='''cis-butadiene and ethylene: reagents'''===

<jmol>
<jmolAppletButton>
<uploadedFileContents>Cis_butadienedh08.mol</uploadedFileContents>
<text>Cis-butadiene</text>
<color>whi-te</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Ethenedh08.mol</uploadedFileContents>
<text>ethylene</text>
<color>white</color>
</jmolAppletButton>
</jmol> were optimised separately using the AM1 semi-empirical method on Gaussview.The cis-butadiene had an energy of 0.04879725 a. u (1.28 x 102 KJ mol-1) and ethylene had an energy of 0.02619027 a. u (68.76 KJ mol-1).

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10: Frontier molecular orbitals of cis-butadiene and ethylene '''
| '''Reagent''' || '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| cis-butadiene || LUMO || [[Image:LUMO1_cis_butadieneDH08.jpg|100x100px]] || [[Image:LUMO2_cis_butadienedh08.jpg|100x100px]]|| Symmetric || +0.017
|-
| cis-butadiene || HOMO || [[Image:HOMO1_cis_butadienedh08.jpg|100x100px]] || [[Image:HOMO2_cis_butadienedh08.jpg|100x100px]]|| Anti-symmetric || -0.343
|-
| ethylene || LUMO || [[Image:LUMO1_ethenedh08.jpg|100x100px]] || [[Image:LUMO2_ethenedh08.jpg|100x100px]]|| Anti-symmetric || +0.052
|-
| ethylene || HOMO || [[Image:HOMO1_ethenedh08.jpg|100x100px]] || [[Image:HOMO2_ethenedh08.jpg|100x100px]]|| Symmetric || -0.387
|}

 
The symmetries of the LUMO and HOMO orbitals for cis-butadiene are opposite of those for the ethylene. This therefore means that the only interactions that can take place are those between ethylene's LUMO and cis-butadiene's HOMO and in addition to this the interaction between cis-butadiene's LUMO and ethylene's HOMO. This is because orbital symmetry has to be conserved (the conservation of orbital symmetry states that the symmetry of an new orbital formed by the overlap of the 2 original orbitals must be the same as the symmetry of the initial orbitals).

==='''cis-butadiene and ethylene: transition state'''===

A <jmol>
<jmolAppletButton>
<uploadedFileContents>Bicyclic_startingdh08.mol</uploadedFileContents>
<text>bicyclo system</text>
<color>white</color>
</jmolAppletButton>
</jmol> was drawn and one of the -CH2-CH2- fragments were removed. Three C=C bonds were added and the previous one was taken away so a <jmol>
<jmolAppletButton>
<uploadedFileContents>Frozen_coorddh08.mol</uploadedFileContents>
<text>guess transition state</text>
<color>white</color>
</jmolAppletButton>
</jmol> was produced.

After this an optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while bonds between the termini of the fragments were frozen at 2.1Å. Following this the resulting file had two separate optimisation and frequency calculations run on it using the TS (Berny) with one using the semi-empirical/AM1 method and the other using the DFT/B3LYP/6-31G* level of theory. Both of these optimisations had the bonds unfrozen and the "derivative" option was chosen instead.

The DFT/B3LYP/6-31G* level of theory was used since it gave more accurate results in the calculations before and the energies it produces are consistent with the rest of the report (semi-empirical/AM1 method gives very different values).

The QST2 optimisation was not used since it required numbering the atoms, which will take a lot of time.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11: Comparison of two methods of optimsations for the transition state '''
| || '''semi-empirical/AM1 method''' || '''DFT/B3LYP/6-31G* method'''
|-
| Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>SE_AM1_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>DFT_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Energy (a. u) || 0.11165608 || -234.54389573
|-
| Bond lengths || [[Image:DistancesSEdh08.jpg|150x150px]]|| [[Image:DistancesDFTdh08.jpg|150x150px]]
|-
|Imaginary frequency (cm-1) || -957 || -525
|-
|Animation || [[Image:-957dh08.gif]] || [[Image:-525dh08.gif]]
|-
| Lowest real frequency (cm-1) || 148 || 136
|-
|Animation || [[Image: 148dh08.gif]] || [[Image: 136dh08.gif]]
|}

The C-C bond forming length between the termini carbons is 0.15 Å larger for the DFT/B3LYP/6-31G* level of theory and this is because this method gives carries out the calculation to a greater degree of accuracy. This larger bond length leads a lower potential energy gradient to the transition state, which gives a lower total energy for the transition state. Therefore the given values below will be the DFT/B3LYP/6-31G* and not the semi-empirical/AM1 method.

A typical sp3 C-C bond length is 1.54 Å and a typical sp2 C=C bond length is 1.34 Å<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref>. Within the optimised transition molecule above the C-C bond lengths are 1.40 Å and the C=C bond length is 1.38 Å. This suggests that the π-bonds are breaking in order for the C-C bonds to form and additionally to this a π-bond is forming in between the two sp3 carbons.

The Van der Waals radius of a C-atom is 1.72 Å<ref name="ref 3">M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer, D. G. Truhlar, 1995, 346, pp. 63-69.[http://pubs.acs.org/doi/abs/10.1021/jp8111556]</ref>. The partly formed C-C length is 2.27 Å, which is shorter than two times the the van der Waals radius of a carbon atom. Therefore the carbon atoms are within a close enough distance to each other to share electron density and yet the bond has not formed at this point.

From looking at the animations for the imaginary frequency above it can be seen that the formation of the two bonds between the termini of the reactants is synchronous. This is expected of the Diels-Alder reaction since the mechanism is concerted. However the lowest real frequency is asynchronous with respect to the two bonds forming between the termini.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12: Frontier molecular orbitals of the transition state '''
| '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| LUMO || [[Image:LUMO1TSdh08.jpg|100x100px]] || [[Image:LUMO2TSdh08.jpg|100x100px]]|| Symmetric || +0.023
|-
| HOMO || [[Image:HOMO1TSdh08.jpg|100x100px]] || [[Image:HOMO2TSdh08.jpg|100x100px]]|| Anti-symmetric || -0.323
|}

From examining the images from table 12 above it can be deduced that the LUMO of the transition state originates from the cis-butadiene's LUMO and the ethylene's HOMO, which represents the σ*-antibonding orbitals. However the HOMO of the transition state originates from the cis-butadiene's HOMO and the ethylene's LUMO, which represents the σ-bonding orbitals within the two single C-C bonds that are formed within the reaction.

==='''cis-butadiene and ethylene: activation energy'''===

The cis-butadiene and ethylene were re-optimised at the DFT/B3LYP/6-31G* level of theory so the activation energy of the reaction could be calculated.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 13: Calculation for the activation energy for the reaction'''
| '''molecule''' || '''DFT/B3LYP/6-31G* energy (a. u)''' || '''Calculations''' || '''Energy '''
|-
| cis-butadiene || -155.98595591 || transition state || -234.54389573 a. u
|-
| ethylene || -78.58745828 || ΔE, activation energy || 0.02951846 a. u
|-
| cis-butadiene and ethylene combined || -234.57341419 || ΔE, activation energy || 77.50 KJ mol-1
|}

The calculated activation energy is 77.50 KJ mol-1, which is of the right magnitude when compared to the literature value (115 ± 8 KJ mol-1 at 0 K) but there is still an error of 37.50 KJ mol-1. This shows that Gaussview is not perfect but still gives an idea of the activation energy required for the reaction to take place.

==='''Cyclohexa-1,3-diene reaction with maleic anhydride'''===

The reaction scheme below shows the stereoselective Diels-Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene. The reaction is kinetically controlled with the major product being the endo product, which will be explained later on in the exercise.

[[Image:Reaction_schemedh081.jpg|center|500x500px]]

The <jmol>
<jmolAppletButton>
<uploadedFileContents>FrozenTS22dh08.mol</uploadedFileContents>
<text>exo</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_frozenTS22dh08.mol</uploadedFileContents>
<text>endo</text>
<color>white</color>
</jmolAppletButton>
</jmol> transition states were drawn on Gaussview and then the optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while the distance of the forming bonds were frozen at 2.1Å between the respective carbons. After this the structures changed slightly (<jmol>
<jmolAppletButton>
<uploadedFileContents>TS2_afteropt1_exo1dh08.mol</uploadedFileContents>
<text>exo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_TS2_afteropt1dh08.mol</uploadedFileContents>
<text>endo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol>) and the resulting files had an optimisation and frequency calculation run on it using the TA (Berny) with using the semi-empirical/AM1 level of theory while the unfrozen option as changed to "derivative". After this the endo and exo optimised transition state structures were re-optimised as they were before but using the DFT/B3LYP/6-31G* level of theory instead with the additional keyword "Opt=NoEigen".

{| class="wikitable" border="1" align="center"
|+ '''Table 14: Comparison of the exo and endo transition states '''
! rowspan="2"|-!! colspan="2"|''exo'' !! colspan="2"|''endo''
|-
| Semi-empirical/AM1 || DFT/B3LYP/6-31G* || Semi-empirical/AM1 || DFT/B3LYP/6-31G*
|-
| align="center"|Structure ||align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>After_opt2_exodh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>Endo_TS2_afteropt2dh081.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| align="center"| Energy (a. u) ||align="center"| -0.05041975 ||align="center"| -612.67931095 ||align="center"| -0.05150470 ||align="center"| -612.68339684
|-
| C-C bond forming length (Å) ||align="center"| 2.17 ||align="center"| 2.29 ||align="center"| 2.16 ||align="center"| 2.27
|-align="center"
| Imaginary frequency (cm-1) ||align="center"| -812 ||align="center"| -448 ||align="center"| -806 ||align="center"| -447
|-align="center"
| Animation ||colspan="2"|[[Image:-812_SEdh08.gif]] || colspan="2"|[[Image:-806_cm-1dh08.gif]]
|}

Again the C-C bond forming length between the termini carbons is larger (by 0.11 and 0.12 Å) for the DFT/B3LYP/6-31G* level of theory and this is due the same reasons talked about above. However the C-C bond forming length between the termini carbons are very similar in both the endo and exo transition states (0.02 Å difference).

From looking at the animations for the imaginary frequency above it can also be seen that the formation of the two bonds between the carbons in this case as well is synchronous. Again this is expected of the Diels-Alder reaction since it is concerted.

{| align=center
|+ '''Strain diagrams for the endo and exo transition states'''
| [[Image:Exostraindh08.jpg|thumb|200x200px|'''Exo TS''']]
| [[Image:Strainendodh08.jpg|thumb|200x200px|'''Endo TS''']]
|}

From studying the energies in table 14 it can be seen that the endo transition state is more stable than the exo by 4.08589 x 10-3 a. u (10.73 KJ mol-1). However by looking at the strain of both transition states one would expect the exo TS to be more stable. This is because the distance between the cyclohexa-1,3-diene ring and the -(C=O)-O-(C=O)- maleic anhydride fragment is 2.99 Å in the endo TS and 3.03 Å in the exo TS. Thus the endo form would have more strain due to steric hindrance.

However within the exo TS the vicinal hydrogens on the CH2-CH2 (carbon numbers 18 and 21) fragment on the cyclohexa-1,3-diene ring are found below the plane and next to the maleic anhydride. This leads to steric hindrance is this is where the strain originates from within the exo TS. Therefore this disfavours the exo form, which makes the endo more stable.

Another factor making the endo TS more favourable is secondary orbital overlap. These orbitals contribute to the bonds forming and they interact with each other through space. Within the endo TS below there are 2 more orbitals interacting (secondary orbitals) than within the exo TS. This therefore makes the endo TS more stable and therefore more favoured.

[[Image:Secondary_orbital_shizzledh08.jpg|500x500px|center]]

{| class="wikitable" border="1" align="center"
|+ '''Table 15: Frontier molecular orbitals of the the transition state for the cyclohexa-1,3-diene reaction '''
! rowspan="2"|Molecular Orbital !! colspan="4"|''exo'' !! colspan="4"|''endo''
|- align="center"
| '''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''||'''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''
|- align="center"
| LUMO || [[Image:LUMO1exoh081.jpg|150x150px]]|| [[Image:LUMO2exodh081.jpg|150x150px]]|| Anti-symmetric || -0.078 || [[Image:LUMO1ENDOdh08.jpg|150x150px]]||[[Image:LUMO2ENDOdh08.jpg|150x150px]] || Anti-symmetric || -0.067
|- align="center"
| HOMO || [[Image:HOMO1exodh081.jpg|150x150px]]||[[Image:HOMO2exodh081.jpg|150x150px]] || Anti-symmetric|| -0.242 || [[Image:HOMO1ENDOdh08.jpg|150x150px]]|| [[Image:HOMO2ENDOdh08.jpg|150x150px]]||Anti-symmetric || -0.242
|}

Digital repositories:

exo: http://hdl.handle.net/10042/to-5596

endo: http://hdl.handle.net/10042/to-5597

From examining table 15 it is clear to see that both the HOMO and LUMO are anti-symmetric for both the transition states. The LUMO of the maleic anhydride fragment is the π* anti-bonding orbital of the C=C bond. Within the reaction then this has electrons transferred to it and the C=C bond breaks, which leaves just a C-C bond. The LUMOs for both transition states are both anti-bonding with respect to the two fragments. The HOMOs of both transitions state represent the σ-bonding that will be formed between the termini carbons on the products and reactants. This is most likely from the overlap of the cyclohexa-1,3-diene's HOMO and the maleic anhydride's LUMO.

=='''Conclusion'''==

='''References'''=
<references/>

Rep:Mod:hampluscheese

2010-11-11T13:01:43Z

Dh08: /* '''cis-butadiene and ethylene: transition state''' */

='''Module 3:Physical Computational Chemistry Lab'''=

Douglas Hunt

=='''Objectives'''==

The aim of this experiment is to is to use various levels of theory (HF/3-21G, DFT/B3LYP/6-31G (d) and AM1/semi-empirical) on Gaussian to calculate various energies, molecular orbitals, geometrical parameters, reaction coordinate pathways and low and imaginary frequencies for a range of molecules and transition states. This will include using different optimisation methods throughout and will lead to a range of findings in various areas of chemical interest.

=='''The Cope Rearrangement'''==

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangement] is a [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement |3,3|-sigmatropic rearrangement] that takes place through a single [http://en.wikipedia.org/wiki/Pericyclic pericyclic], concerted [http://en.wikipedia.org/wiki/Transition_state transition state]. Within this rearrangement a σ bond is formed in concert with another being broken within a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] π system.

The Cope rearrangement we will be studying will be that of 1,5-hexadiene, which is shown below.

[[Image:Cope_rearrangement_schemedh08.jpg|300x300px|center]]

The objective is to locate the low-energy minima and transition structures on the C6H10 potential energy surface and to determine the preferred reaction mechanism by using [http://en.wikipedia.org/wiki/GAUSSIAN Gaussian], with the [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)] methods.

==='''Optimisation of 1,5-hexadiene Conformers'''===

Rough <jmol>
<jmolAppletButton>
<uploadedFileContents>Anti_2_drawndh08.mol</uploadedFileContents>
<text>anti</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche3_drawndh08.mol</uploadedFileContents>
<text>gauche</text>
<color>white</color>
</jmolAppletButton>
</jmol> conformers of 1,5-hexadiene were drawn on Gaussview and their structures were tidied by the clean function. They were then further optimised by the HF/3-21G method and were identified which conformer they were from their point group and appendix 1 on the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3 physical lab page].

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 1: Optimisation and comparison of anti and gauche 1,5-hexadiene conformers using HF/3-21G methods
| ''' Conformer''' || '''Optimised Structure''' || '''Energy (Atomic Units)''' || '''[http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1 Energies (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (Anti2 - Gauche3)''' || '''Point group'''
|-
| '''Anti2''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Anti_2_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692535 || -231.69254 ||-6.08 x 105 || 1.2606 x 10-4 a. u|| Ci
|-
| '''Gauche3''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Gauche3_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692661 || -231.69266 || -6.08 x 105 || 3.31 x 10-1 KJ mol-1 || C1
|}

From studying table 1 it can be seen that the gauche conformer is 0.33 KJ mol-1 more stable than the anti conformation. One would assume that the anti2 conformer would be more stable due to [http://en.wikipedia.org/wiki/Steric steric] factors/repulsion within the gauche3 conformation. However within the gauche3 conformer there is a favourable donation of [http://en.wikipedia.org/wiki/Electron_density electron density] from the C=C π orbital to the C-H σ* orbital from the adjacent proton, which stabilises the conformer<ref name="ref 1">B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446.[http://www.informaworld.com/smpp/content~db=all?content=10.1080/00268970110081412]</ref>.

 

The anti2 conformer was then optimised using the B3LYP 6-31G* (6-31G (d)) method.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 2: Comparison of the anti2 conformer optimisations carried out at different calculation methods
| ''' Method''' || '''Structure and bond distances (Å)''' || '''Dihedral angle (C1 to C4)''' || '''Energy (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (HF/3-21G - DFT/B3LYP/6-31G)'''
|-
| HF/3-21G || [[Image:Anti2_opt1dh08.jpg|300x300px]] || 114.6 o || -231.692535 || -6.08 x 105 || 2.919171 a. u
|-
| DFT/B3LYP/6-31G* || [[Image:Anti2_opt2DH08.jpg|300x300px]] || 118.8 o || -234.611716 || -6.16 x 105 || 7.66 x 103 KJ mol-1
|}

From examining table 2 at first it is possible to see that the bond lengths are very similar to each other and that of the literature bond lengths<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref> (C1=C2: 1.340 Å, C2-C3: 1.508 Å and C3-C4: 1.538 Å). It is also possible to see that the C=C bond length is more accurate for the DFT/B3LYP/6-31G* method by it being 0.01 Å closer to the literature value. Another main difference in terms of geometry is that the DFT/B3LYP/6-31G* method gives a larger [http://en.wikipedia.org/wiki/Dihedral_angle dihedral angle] (and most likely more accurate) of 118.8o compared to that of 114.6o.

In terms of energy the DFT/B3LYP/6-31G* method produces a conformer of a higher stability by it possessing a lower energy with a difference of 7.66 x 103 KJ mol-1.

==='''Frequency analysis of 1,5-hexadiene Conformers'''===

[[Image:Vibrations_anti2dh08.jpg|right|300x300px]]

For the frequency analysis the DFT/B3LYP/6-31G* method was used.

Frequency analysis is essential within this calculation due to it being the second derivative of the potential energy surface. This means that if all the frequencies are all positive the system is at a minimum, if one of them is negative the optimised system is at a transition state, and if more than one are negative then the calculation has failed to find a critical point, which means the optimisation has failed. This therefore gives us a method of seeing whether the calculation has worked.

From looking to the image on the right it can deduced that the optimisation has taken place successfully by there being no negative vibrations and therefore the conformation is at a minimum.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 3: Energies under the thermochemistry section at 298.15 K and 0 K within the frequency logfiles'''
| '''Energies''' || '''298.15 K (atomic units)''' ||| '''0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (KJ mol-1)''' || '''Discussion'''
|-
| Sum of electronic and zero-point energies || -234.469177 || -234.466673 || -2.504 x 10-3 || -6.57 || This is the potential energy at 0 K, which includes the zero-point vibrational energy (E = Eelec + ZPE).

|-
| Sum of electronic and thermal energies || -234.461835 || -234.461085 || -7.5 x 10-4|| -1.97 || This is the energy at 298.15 K (0 K for 0 K calculation) and 1 atm of pressure, which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans).

|-
| Sum of electronic and thermal enthalpies || -234.460891 || -234.460141 || -7.5 x 10-4 || -1.97 || This contains an additional correction for RT (H = E + RT), which is especially important when looking at dissociation reactions.
|-
| Sum of electronic and thermal free energies || -234.500739 || -234.495931 || 4.808 x 10-3 || -12.62 || This includes the entropic contribution to the free energy (G = H - TS).
|}

{|align=right
|+
| [[Image:Anti2_freq_IRdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 298.15 K''']]
| [[Image:Anti2_freq_0Kdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 0 K''']]
|}

From looking at table 3 above it can be deduced that the anti2 conformer is more stable at room temperature to that of 0 K with respect to all the types of energies. The spectra also seem to not change much between the two temperatures but the main change is the change in the sum of electronic and thermal free energies, which is heavily dependent of temperature by it including the entropic contribution to the free energy (G = H - '''T'''S)

 
 
 
 

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

Within the Cope Rearrangement there are two possible transition states, the chair (C2h) and the boat (C2v), which are shown within [https://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_2 appendix two ].

==='''Comparison of optimisation methods for the chair transition state'''===

An <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2hdh08.mol</uploadedFileContents>
<text>allyl fragment</text>
<color>white</color>
</jmolAppletButton>
</jmol>, CH2CHCH2 was drawn on Gaussview and was optimized through the HF/3-21G method. After this the original allyl fragment was copied and pasted into a new window and place exactly above the first fragment with a bond distance of 2.20 Å between the terminal end carbons. This therefore produced the <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2h_fulldh08.mol</uploadedFileContents>
<text>chair transition structure.</text>
<color>white</color>
</jmolAppletButton>
</jmol>

Gaussian was then set up for an optimisation for the transition state with the additional key words "Opt=NoEigen", the optimisation set to TS (Berny), the force constants set to "once" and using the HF/3-21G method.

After this the "chair" transition state was optimised an alternative way by freezing the coordinates of the two sets of terminal carbons (where the bonds form/break during the rearrangement) within the Redundant Coord Editor on Gauusview.
Following this a second calculation was run where the freeze coordinates option was changed to "Derivative" and the force constants were set to "never".

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 4: Comparison of chair transition state optimisation methods
| || '''Optimised through a TS (Berny)''' || '''Optimised through a frozen coordinate method'''
|-
| '''Optimised TS Structure''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>C2h_optfreq_TS_Bdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| '''Terminal C-C bond length (Å)''' || 2.02 || 2.02
|-
|'''Energy (atomic units)''' || -231.61932247 || -231.61932190
|-
| '''ΔE, TS (Berny) - frozen coordinate''' || 5.7 x 10-7 a.u || 1.50 x 10-3 KJ mol-1
|-
| '''Imaginary frequency (cm-1)''' || -818 || -818
|-
| '''Animation''' ||[[Image:818dh08TSB.gif|350x350px]] || [[Image:818cm-1Fdh08.gif]]
|}

The data in table 4 shows that both methods give the same results and they both give the same transition state. They both produced the same energy with a negligible energy difference between them and the bond lengths between the terminal carbons are both 2.02 Å. In addition to this the imaginary frequencies were both -818cm-1 and within the animations above it is possible to see that when one C-C bond is forming the other is breaking. This means it is a asynchronous bond formation, which is expected for the Cope Rearrangement since it is a pericyclic reaction.

The only difference between the two optimisations is the way in which the transition states were calculated. This is due to the TS (Berny) method calculating the force constant matrix within the first optimisation step and is then being recalculated as the optimisation carries on. When the frozen coordinate method freezes the reaction coordinate in order for the rest of the structure to be optimised around it, which does not require the whole hessian (force constant matrix) to be calculated but only that the reaction coordinate is differentiated (second step). The main advantage of this, is that you do not need to calculate the entire force constant matrix and just need to differentiate the reaction coordinate in order to obtain a good enough guess for the initial force constant matrix. The frozen coordinate method will therefore take less time to compute the transition state, which can be very expensive within certain systems. However to carry out this method one needs to have an idea of how far apart the terminal carbons are.

==='''Boat transition state'''===

===='''QST2 optimisation'''====

The optimisation for the boat transition state had a different method and this was the QST2 method. This calculation interpolates between two structures (product and reactant) and tries to find a transition state between them.

The reactants and products were numbered in the same way. Therefore both 1,5-hexadiene molecules had their numbering manually changed for the product molecule so it corresponded to the numbering of the reactant if it rearranged. This is illustrated in the reaction scheme below.

[[Image:Boat_rerrangmentdh08.jpg|300x300px|center]]

The chk file for the optimized anti2 conformer was opened and copied into a new window. A second window was opened within the new window and the same anti2 conformer was copied into the new window. The numbering was changed within the second molecule so the two looked like they do in the image below.

[[Image:TS_(QST2)_numbering_1dh08.jpg|400x400px|center]]

The job type was set as "Opt+Freq" and "TS (QST2)" was chosen to optimise the transition state. However this failed and produced
a structure a bit like a <jmol>
<jmolAppletButton>
<uploadedFileContents>Trialone_BPdh08.mol</uploadedFileContents>
<text>chair transition structure </text>
<color>white</color>
</jmolAppletButton>
</jmol> but more dissociated. This is because it interpolated between the two structures and translated the top allyl fragment and therefore did not consider the possibility of a rotation around the central bonds.

To get around this the C-C-C-C dihedral angle (C2-C3-C4-C5) was set to 0 o and the C-C-C angle (C2-C3-C4 and C3-C4-C5 for the product molecule) was set to 100 o for both molecules. This gave rise to the two structures below.

[[Image:TS_(QST2)_numbering_2dh08.jpg|400x400px|center]]

After this the QST2 calculation was run again and the results in table 5 were produced. The imaginary frequency at -839 cm-1 shows that while one bond is forming the other one is breaking, which is again an asynchronous bond formation. It is also possible to see that the boat transition state has a longer terminal C-C bond length by 0.12 Å to that of the chair.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5: Data for the boat transition state from the QST2 optimisation method
| Optimised TS Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.14
|-
|Energy (atomic units) || -231.60280151
|-
| Energy (KJ mol-1) || -6.08 x 105
|-
| Imaginary frequency (cm-1) || -839
|-
| Animation ||[[Image:-839dh08.gif]]
|}

===='''Intrinsic Reaction Coordinate'''====

It is not possible to predict which product the chair and boat transition state will lead to. To solve this problem within Gaussian one can use the IRC method, which allows one to follow the minimum energy path from a transition structure down to its local minimum on a potential energy surface. This creates a succession of points by taking small geometry steps in the direction to where the gradient of the energy surface is at its steepest until the minimum in energy is reached.

The chk file for the optimized chair transition structure was opened and the job type selected was "IRC", the reaction coordinate was computed in the forward direction (since molecule is symmetric) with 50 points along the IRC. This was done with two type of force constants being calculated, once and always.

Within the "once" method the output showed no bond formation process by the 26th step and therefore the calculation was incomplete, since a complete 1,5-hexadiene structure was supposed to be formed. From looking at the IRC pathway it can be seen that the minimum on the potential energy surface was not reached at that point, since the points on the pathway did not form an asymptote and the gradient did not go close enough to zero.

For the "always" method the computation of force constants were carried out at every step. This therefore lets the IRC calculation distinguish whether it is moving along the minimum energy pathway while going down the potential energy surface. While this is happening it can make changes to the pathway if necessary. Within the "always" case the bond forming arrives at 1,5-hexadiene and this is backed up by the IRC pathway reaching a minimum within its energy by the gradient going close to zero and the pathway reaches an asymptote.

The structure produced through the always method had an energy of -231.69199434 atomic units, which is very close to the energy of the gauche2 conformer in the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 appendix 1]. This suggests that the reaction takes place through the chair transition state, which later produces the gauche2 conformer of 1,5-hexadiene.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 6: Comparison of IRC calculations for the chair structure
| || '''Calculate force constants once''' || '''Calculate force constants always'''
|-
| '''Number of points along IRC''' || 26 || 47
|-
| '''Structure at the last point of IRC''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Once_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Always_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|'''Energy (atomic units)''' || -231.68906923 || -231.69199434
|-
| '''ΔE, once - always''' || 2.92511 x 10-3 a.u || 7.68 KJ mol-1
|-
| '''IRC pathway''' || [[Image:One_pathwaydh08.jpg|400x400px]] || [[Image:Always_pathwaydh08.jpg|400x400px]]
|-
| '''IRC gradient''' || [[Image:One_gradientdh08.jpg|400x400px]]|| [[Image:Always_gradientdh08.jpg|400x400px]]
|}

 
 
 

===='''Comparison of activation energies for the transition states'''====

The HF/3-21G optimised chair and boat transition structures were both reoptimised using the B3LYP/6-31G* (B3LYP/6-31G (d)) method. This gives a more accurate minimisation however this method takes longer to run. The values for the zero-point energies and thermal energies were found within the thermochemistry section of the log file.

The values for the HF/3-21G reactant (anti2) optimisation were calculated and the B3LYP/6-31G* values were taken from the "Frequency analysis of 1,5-hexadiene Conformers" earlier in this module.

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 7: Energy data for chair, boat and the anti2 conformation from the HF/3-21G and B3LYP/6-31G* methods of optimisation '''
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466677
| align="center" | -231.461320
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450936
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402342
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539524
| align="center" | -231.532562
| align="center" | -234.611716
| align="center" | -234.469177
| align="center" | -234.461835
|}
Digital repositories

Chair:
HF/3-21G: http://hdl.handle.net/10042/to-5569
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5568

Boat:
HF/3-21G: http://hdl.handle.net/10042/to-5570
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5571

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 8: Summary of activation energies for the chair and boat conformations (ΔE = chair TS or boat TS minus anti2)'''
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| align="center" | 1 a.u = 627.509 kcal mol-1
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE, chair (a.u)'''
| align="center" | 0.072847
| align="center" | 0.071242
| align="center" | 0.054248
| align="center" | 0.052827
| align="center" | -
|-
| '''ΔE, chair (kcal mol-1)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE, boat (a. u)'''
| align="center" | 0.088588
| align="center" | 0.087259
| align="center" | 0.066835
| align="center" | 0.065827
| align="center" | -
|-
| '''ΔE, boat (kcal mol-1)'''
| align="center" | 55.59
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.31
| align="center" | 44.7 ± 2.0
|}

By examining table 8 it is clear to see that the B3LYP/6-31G* method gives values which are much closer to the experimental ones. This is due to the B3LYP/6-31G* giving a higher level of optimisation than the HF/3-21G method. Table 8 also shows that the chair transition state has a lower activation energy to that of the boat transition state by 8.16 kcal mol-1. This suggests that the Cope rearrangement mechanism would most likely proceed through the chair transition state to that of the boat. Therefore the reaction would take place from the anti2 conformer to the chair transition state and then onto the gauche2 conformer within the Cope rearrangement.

It is also possible to see that the calcualted energies are very close to the literature values with the chair conformation being closer to its literature value to that of the boat.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9: Geometric parameters for the optimised chair and boat structures '''
| || ''' HF/3-21G chair TS''' || ''' B3LYP/6-31G* chair TS''' || '''HF/3-21G boat TS''' || '''B3LYP/6-31G* boat TS'''
|-
| Optimised Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Chair_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Boat_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.02 (C5-C10 or C3-C14)|| 1.97 (C5-C10 or C3-C14) || 2.14 (C3-C4 or C1-C6) || 2.21 (C3-C4 or C1-C6)
|-
| Fragment angle || 120 o (C3-C1-C5) || 120 o (C3-C1-C5) || 122 o (C4-C5-C6) || 122 o (C4-C5-C6)
|-
| Dihedral angle || 55 o (C1-C3-C14-C11)|| 54 o (C1-C3-C14-C11)|| 0 o (C2-C1-C6-C5) || 0 o (C2-C1-C6-C5)
|-
| Numbering of carbons || [[Image:Numbering_chairdh08.jpg|200x200px]]|| [[Image:Numbering_chairdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]]
|}

From studying table 9 it can be seen that the C-C bond breaking/forming length has changed between the two different types of optimisation methods. However the angles between the bonds remains unchanged.

=='''The Diels Alder Cycloaddition'''==

The Diels-Alder reaction is a cycloaddition between a conjugated diene and a dienophile (alkene system), which forms a cyclohexene system. It is often referred to as a [4s+2s] cycloaddition and proceeds through a single cyclic, concerted transition state where 2 σ bonds are formed in between the termini carbons of the 2 п systems. This is shown in the reaction scheme below where ethene is reacting with cis-butadiene.

[[Image:Reaction_scheme2dh08.jpg|300x300px|center]]

The π orbitals of the dienophile form new σ bonds with the π orbitals of the diene. The number of π electrons involved determine whether the reaction proceeds in a concerted stereospecific fashion (allowed) or not (forbidden). The HOMO/LUMO of one fragment interacts with the HOMO/LUMO of the other fragment to form two new bonding and anti-bonding MOs. The nodal properties allow one to predict how the reaction will occur.

Semi-Empirical/AM1 method and the B3LYP/6-31G* method of optimisation will be used on Gaussian to determine particular properites of the reactants to predict the way in which the reaction will proceed.

==='''cis-butadiene and ethylene: reagents'''===

<jmol>
<jmolAppletButton>
<uploadedFileContents>Cis_butadienedh08.mol</uploadedFileContents>
<text>Cis-butadiene</text>
<color>whi-te</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Ethenedh08.mol</uploadedFileContents>
<text>ethylene</text>
<color>white</color>
</jmolAppletButton>
</jmol> were optimised separately using the AM1 semi-empirical method on Gaussview.The cis-butadiene had an energy of 0.04879725 a. u (1.28 x 102 KJ mol-1) and ethylene had an energy of 0.02619027 a. u (68.76 KJ mol-1).

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10: Frontier molecular orbitals of cis-butadiene and ethylene '''
| '''Reagent''' || '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| cis-butadiene || LUMO || [[Image:LUMO1_cis_butadieneDH08.jpg|100x100px]] || [[Image:LUMO2_cis_butadienedh08.jpg|100x100px]]|| Symmetric || +0.017
|-
| cis-butadiene || HOMO || [[Image:HOMO1_cis_butadienedh08.jpg|100x100px]] || [[Image:HOMO2_cis_butadienedh08.jpg|100x100px]]|| Anti-symmetric || -0.343
|-
| ethylene || LUMO || [[Image:LUMO1_ethenedh08.jpg|100x100px]] || [[Image:LUMO2_ethenedh08.jpg|100x100px]]|| Anti-symmetric || +0.052
|-
| ethylene || HOMO || [[Image:HOMO1_ethenedh08.jpg|100x100px]] || [[Image:HOMO2_ethenedh08.jpg|100x100px]]|| Symmetric || -0.387
|}

 
The symmetries of the LUMO and HOMO orbitals for cis-butadiene are opposite of those for the ethylene. This therefore means that the only interactions that can take place are those between ethylene's LUMO and cis-butadiene's HOMO and in addition to this the interaction between cis-butadiene's LUMO and ethylene's HOMO. This is because orbital symmetry has to be conserved (the conservation of orbital symmetry states that the symmetry of an new orbital formed by the overlap of the 2 original orbitals must be the same as the symmetry of the initial orbitals).

==='''cis-butadiene and ethylene: transition state'''===

A <jmol>
<jmolAppletButton>
<uploadedFileContents>Bicyclic_startingdh08.mol</uploadedFileContents>
<text>bicyclo system</text>
<color>white</color>
</jmolAppletButton>
</jmol> was drawn and one of the -CH2-CH2- fragments were removed. Three C=C bonds were added and the previous one was taken away so a <jmol>
<jmolAppletButton>
<uploadedFileContents>Frozen_coorddh08.mol</uploadedFileContents>
<text>guess transition state</text>
<color>white</color>
</jmolAppletButton>
</jmol> was produced.

After this an optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while bonds between the termini of the fragments were frozen at 2.1Å. Following this the resulting file had two separate optimisation and frequency calculations run on it using the TS (Berny) with one using the semi-empirical/AM1 method and the other using the DFT/B3LYP/6-31G* level of theory. Both of these optimisations had the bonds unfrozen and the "derivative" option was chosen instead.

The DFT/B3LYP/6-31G* level of theory was used since it gave more accurate results in the calculations before and the energies it produces are consistent with the rest of the report (semi-empirical/AM1 method gives very different values).

The QST2 optimisation was not used since it required numbering the atoms, which will take a lot of time.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11: Comparison of two methods of optimsations for the transition state '''
| || '''semi-empirical/AM1 method''' || '''DFT/B3LYP/6-31G* method'''
|-
| Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>SE_AM1_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>DFT_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Energy (a. u) || 0.11165608 || -234.54389573
|-
| Bond lengths || [[Image:DistancesSEdh08.jpg|150x150px]]|| [[Image:DistancesDFTdh08.jpg|150x150px]]
|-
|Imaginary frequency (cm-1) || -957 || -525
|-
|Animation || [[Image:-957dh08.gif]] || [[Image:-525dh08.gif]]
|-
| Lowest real frequency (cm-1) || 148 || 136
|-
|Animation || [[Image: 148dh08.gif]] || [[Image: 136dh08.gif]]
|}

The C-C bond forming length between the termini carbons is 0.15 Å larger for the DFT/B3LYP/6-31G* level of theory and this is because this method gives carries out the calculation to a greater degree of accuracy. This larger bond length leads a lower potential energy gradient to the transition state, which gives a lower total energy for the transition state. Therefore the given values below will be the DFT/B3LYP/6-31G* and not the semi-empirical/AM1 method.

A typical sp3 C-C bond length is 1.54 Å and a typical sp2 C=C bond length is 1.34 Å<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref>. Within the optimised transition molecule above the C-C bond lengths are 1.40 Å and the C=C bond length is 1.38 Å. This suggests that the π-bonds are breaking in order for the C-C bonds to form and additionally to this a π-bond is forming in between the two sp3 carbons.

The Van der Waals radius of a C-atom is 1.72 Å<ref name="ref 3">M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer, D. G. Truhlar, 1995, 346, pp. 63-69.[http://pubs.acs.org/doi/abs/10.1021/jp8111556]</ref>. The partly formed C-C length is 2.27 Å, which is shorter than two times the the van der Waals radius of a carbon atom. Therefore the carbon atoms are within a close enough distance to each other to share electron density and yet the bond has not formed at this point.

From looking at the animations for the imaginary frequency above it can be seen that the formation of the two bonds between the termini of the reactants is synchronous. This is expected of the Diels-Alder reaction since the mechanism is concerted. However the lowest real frequency is asynchronous with respect to the two bonds forming between the termini.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12: Frontier molecular orbitals of the transition state '''
| '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| LUMO || [[Image:LUMO1TSdh08.jpg|100x100px]] || [[Image:LUMO2TSdh08.jpg|100x100px]]|| Symmetric || +0.023
|-
| HOMO || [[Image:HOMO1TSdh08.jpg|100x100px]] || [[Image:HOMO2TSdh08.jpg|100x100px]]|| Anti-symmetric || -0.323
|}

From examining the images from table 12 above it can be deduced that the LUMO of the transition state originates from the cis-butadiene's LUMO and the ethylene's HOMO, which represents the σ*-antibonding orbitals. However the HOMO of the transition state originates from the cis-butadiene's HOMO and the ethylene's LUMO, which represents the σ-bonding orbitals within the two single C-C bonds that are formed within the reaction.

==='''cis-butadiene and ethylene: activation energy'''===

The cis-butadiene and ethylene were re-optimised at the DFT/B3LYP/6-31G* level of theory so the activation energy of the reaction could be calculated.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 13: Calculation for the activation energy for the reaction'''
| '''molecule''' || '''DFT/B3LYP/6-31G* energy (a. u)''' || '''Calculations''' || '''Energy '''
|-
| cis-butadiene || -155.98595591 || transition state || -234.54389573 a. u
|-
| ethylene || -78.58745828 || ΔE, activation energy || 0.02951846 a. u
|-
| cis-butadiene and ethylene combined || -234.57341419 || ΔE, activation energy || 77.50 KJ mol-1
|}

The calculated activation energy is 77.50 KJ mol-1, which is of the right magnitude when compared to the literature value (115 ± 8 KJ mol-1 at 0 K) but there is still an error of 37.50 KJ mol-1. This shows that Gaussview is not perfect but still gives an idea of the activation energy required for the reaction to take place.

==='''Cyclohexa-1,3-diene reaction with maleic anhydride'''===

The reaction scheme below shows the stereoselective Diels-Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene. The reaction is kinetically controlled with the major product being the endo product, which will be explained later on in the exercise.

[[Image:Reaction_schemedh081.jpg|center|500x500px]]

The <jmol>
<jmolAppletButton>
<uploadedFileContents>FrozenTS22dh08.mol</uploadedFileContents>
<text>exo</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_frozenTS22dh08.mol</uploadedFileContents>
<text>endo</text>
<color>white</color>
</jmolAppletButton>
</jmol> transition states were drawn on Gaussview and then the optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while the distance of the forming bonds were frozen at 2.1Å between the respective carbons. After this the structures changed slightly (<jmol>
<jmolAppletButton>
<uploadedFileContents>TS2_afteropt1_exo1dh08.mol</uploadedFileContents>
<text>exo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_TS2_afteropt1dh08.mol</uploadedFileContents>
<text>endo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol>) and the resulting files had an optimisation and frequency calculation run on it using the TA (Berny) with using the semi-empirical/AM1 level of theory while the unfrozen option as changed to "derivative". After this the endo and exo optimised transition state structures were re-optimised as they were before but using the DFT/B3LYP/6-31G* level of theory instead with the additional keyword "Opt=NoEigen".

{| class="wikitable" border="1" align="center"
|+ '''Table 14: Comparison of the exo and endo transition states '''
! rowspan="2"|-!! colspan="2"|''exo'' !! colspan="2"|''endo''
|-
| Semi-empirical/AM1 || DFT/B3LYP/6-31G* || Semi-empirical/AM1 || DFT/B3LYP/6-31G*
|-
| align="center"|Structure ||align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>After_opt2_exodh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>Endo_TS2_afteropt2dh081.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| align="center"| Energy (a. u) ||align="center"| -0.05041975 ||align="center"| -612.67931095 ||align="center"| -0.05150470 ||align="center"| -612.68339684
|-
| C-C bond forming length (Å) ||align="center"| 2.17 ||align="center"| 2.29 ||align="center"| 2.16 ||align="center"| 2.27
|-align="center"
| Imaginary frequency (cm-1) ||align="center"| -812 ||align="center"| -448 ||align="center"| -806 ||align="center"| -447
|-align="center"
| Animation ||colspan="2"|[[Image:-812_SEdh08.gif]] || colspan="2"|[[Image:-806_cm-1dh08.gif]]
|}

Again the C-C bond forming length between the termini carbons is larger (by 0.11 and 0.12 Å) for the DFT/B3LYP/6-31G* level of theory and this is due the same reasons talked about above. However the C-C bond forming length between the termini carbons are very similar in both the endo and exo transition states (0.02 Å difference).

From looking at the animations for the imaginary frequency above it can also be seen that the formation of the two bonds between the carbons in this case as well is synchronous. Again this is expected of the Diels-Alder reaction since it is concerted.

{| align=center
|+ '''Strain diagrams for the endo and exo transition states'''
| [[Image:Exostraindh08.jpg|thumb|200x200px|'''Exo TS''']]
| [[Image:Strainendodh08.jpg|thumb|200x200px|'''Endo TS''']]
|}

From studying the energies in table 14 it can be seen that the endo transition state is more stable than the exo by 4.08589 x 10-3 a. u (10.73 KJ mol-1). However by looking at the strain of both transition states one would expect the exo TS to be more stable. This is because the distance between the cyclohexa-1,3-diene ring and the -(C=O)-O-(C=O)- maleic anhydride fragment is 2.99 Å in the endo TS and 3.03 Å in the exo TS. Thus the endo form would have more strain due to steric hindrance.

However within the exo TS the vicinal hydrogens on the CH2-CH2 (carbon numbers 18 and 21) fragment on the cyclohexa-1,3-diene ring are found below the plane and next to the maleic anhydride. This leads to steric hindrance is this is where the strain originates from within the exo TS. Therefore this disfavours the exo form, which makes the endo more stable.

Another factor making the endo TS more favourable is secondary orbital overlap. These orbitals contribute to the bonds forming and they interact with each other through space. Within the endo TS below there are 2 more orbitals interacting (secondary orbitals) than within the exo TS. This therefore makes the endo TS more stable and therefore more favoured.

[[Image:Secondary_orbital_shizzledh08.jpg|500x500px|center]]

{| class="wikitable" border="1" align="center"
|+ '''Table 15: Frontier molecular orbitals of the the transition state for the cyclohexa-1,3-diene reaction '''
! rowspan="2"|Molecular Orbital !! colspan="4"|''exo'' !! colspan="4"|''endo''
|- align="center"
| '''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''||'''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''
|- align="center"
| LUMO || [[Image:LUMO1exoh081.jpg|150x150px]]|| [[Image:LUMO2exodh081.jpg|150x150px]]|| Anti-symmetric || -0.078 || [[Image:LUMO1ENDOdh08.jpg|150x150px]]||[[Image:LUMO2ENDOdh08.jpg|150x150px]] || Anti-symmetric || -0.067
|- align="center"
| HOMO || [[Image:HOMO1exodh081.jpg|150x150px]]||[[Image:HOMO2exodh081.jpg|150x150px]] || Anti-symmetric|| -0.242 || [[Image:HOMO1ENDOdh08.jpg|150x150px]]|| [[Image:HOMO2ENDOdh08.jpg|150x150px]]||Anti-symmetric || -0.242
|}

Digital repositories:

exo: http://hdl.handle.net/10042/to-5596

endo: http://hdl.handle.net/10042/to-5597

From examining table 15 it is clear to see that both the HOMO and LUMO are anti-symmetric for both the transition states. The LUMO of the maleic anhydride fragment is the π* anti-bonding orbital of the C=C bond. Within the reaction then this has electrons transferred to it and the C=C bond breaks, which leaves just a C-C bond. The LUMOs for both transition states are both anti-bonding with respect to the two fragments. The HOMOs of both transitions state represent the σ-bonding that will be formed between the termini carbons on the products and reactants. This is most likely from the overlap of the cyclohexa-1,3-diene's HOMO and the maleic anhydride's LUMO.

=='''Conclusion'''==

='''References'''=
<references/>

Rep:Mod:hampluscheese

2010-11-11T13:01:21Z

Dh08: /* '''cis-butadiene and ethylene: transition state''' */

='''Module 3:Physical Computational Chemistry Lab'''=

Douglas Hunt

=='''Objectives'''==

The aim of this experiment is to is to use various levels of theory (HF/3-21G, DFT/B3LYP/6-31G (d) and AM1/semi-empirical) on Gaussian to calculate various energies, molecular orbitals, geometrical parameters, reaction coordinate pathways and low and imaginary frequencies for a range of molecules and transition states. This will include using different optimisation methods throughout and will lead to a range of findings in various areas of chemical interest.

=='''The Cope Rearrangement'''==

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangement] is a [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement |3,3|-sigmatropic rearrangement] that takes place through a single [http://en.wikipedia.org/wiki/Pericyclic pericyclic], concerted [http://en.wikipedia.org/wiki/Transition_state transition state]. Within this rearrangement a σ bond is formed in concert with another being broken within a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] π system.

The Cope rearrangement we will be studying will be that of 1,5-hexadiene, which is shown below.

[[Image:Cope_rearrangement_schemedh08.jpg|300x300px|center]]

The objective is to locate the low-energy minima and transition structures on the C6H10 potential energy surface and to determine the preferred reaction mechanism by using [http://en.wikipedia.org/wiki/GAUSSIAN Gaussian], with the [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)] methods.

==='''Optimisation of 1,5-hexadiene Conformers'''===

Rough <jmol>
<jmolAppletButton>
<uploadedFileContents>Anti_2_drawndh08.mol</uploadedFileContents>
<text>anti</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche3_drawndh08.mol</uploadedFileContents>
<text>gauche</text>
<color>white</color>
</jmolAppletButton>
</jmol> conformers of 1,5-hexadiene were drawn on Gaussview and their structures were tidied by the clean function. They were then further optimised by the HF/3-21G method and were identified which conformer they were from their point group and appendix 1 on the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3 physical lab page].

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 1: Optimisation and comparison of anti and gauche 1,5-hexadiene conformers using HF/3-21G methods
| ''' Conformer''' || '''Optimised Structure''' || '''Energy (Atomic Units)''' || '''[http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1 Energies (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (Anti2 - Gauche3)''' || '''Point group'''
|-
| '''Anti2''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Anti_2_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692535 || -231.69254 ||-6.08 x 105 || 1.2606 x 10-4 a. u|| Ci
|-
| '''Gauche3''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Gauche3_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692661 || -231.69266 || -6.08 x 105 || 3.31 x 10-1 KJ mol-1 || C1
|}

From studying table 1 it can be seen that the gauche conformer is 0.33 KJ mol-1 more stable than the anti conformation. One would assume that the anti2 conformer would be more stable due to [http://en.wikipedia.org/wiki/Steric steric] factors/repulsion within the gauche3 conformation. However within the gauche3 conformer there is a favourable donation of [http://en.wikipedia.org/wiki/Electron_density electron density] from the C=C π orbital to the C-H σ* orbital from the adjacent proton, which stabilises the conformer<ref name="ref 1">B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446.[http://www.informaworld.com/smpp/content~db=all?content=10.1080/00268970110081412]</ref>.

 

The anti2 conformer was then optimised using the B3LYP 6-31G* (6-31G (d)) method.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 2: Comparison of the anti2 conformer optimisations carried out at different calculation methods
| ''' Method''' || '''Structure and bond distances (Å)''' || '''Dihedral angle (C1 to C4)''' || '''Energy (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (HF/3-21G - DFT/B3LYP/6-31G)'''
|-
| HF/3-21G || [[Image:Anti2_opt1dh08.jpg|300x300px]] || 114.6 o || -231.692535 || -6.08 x 105 || 2.919171 a. u
|-
| DFT/B3LYP/6-31G* || [[Image:Anti2_opt2DH08.jpg|300x300px]] || 118.8 o || -234.611716 || -6.16 x 105 || 7.66 x 103 KJ mol-1
|}

From examining table 2 at first it is possible to see that the bond lengths are very similar to each other and that of the literature bond lengths<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref> (C1=C2: 1.340 Å, C2-C3: 1.508 Å and C3-C4: 1.538 Å). It is also possible to see that the C=C bond length is more accurate for the DFT/B3LYP/6-31G* method by it being 0.01 Å closer to the literature value. Another main difference in terms of geometry is that the DFT/B3LYP/6-31G* method gives a larger [http://en.wikipedia.org/wiki/Dihedral_angle dihedral angle] (and most likely more accurate) of 118.8o compared to that of 114.6o.

In terms of energy the DFT/B3LYP/6-31G* method produces a conformer of a higher stability by it possessing a lower energy with a difference of 7.66 x 103 KJ mol-1.

==='''Frequency analysis of 1,5-hexadiene Conformers'''===

[[Image:Vibrations_anti2dh08.jpg|right|300x300px]]

For the frequency analysis the DFT/B3LYP/6-31G* method was used.

Frequency analysis is essential within this calculation due to it being the second derivative of the potential energy surface. This means that if all the frequencies are all positive the system is at a minimum, if one of them is negative the optimised system is at a transition state, and if more than one are negative then the calculation has failed to find a critical point, which means the optimisation has failed. This therefore gives us a method of seeing whether the calculation has worked.

From looking to the image on the right it can deduced that the optimisation has taken place successfully by there being no negative vibrations and therefore the conformation is at a minimum.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 3: Energies under the thermochemistry section at 298.15 K and 0 K within the frequency logfiles'''
| '''Energies''' || '''298.15 K (atomic units)''' ||| '''0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (KJ mol-1)''' || '''Discussion'''
|-
| Sum of electronic and zero-point energies || -234.469177 || -234.466673 || -2.504 x 10-3 || -6.57 || This is the potential energy at 0 K, which includes the zero-point vibrational energy (E = Eelec + ZPE).

|-
| Sum of electronic and thermal energies || -234.461835 || -234.461085 || -7.5 x 10-4|| -1.97 || This is the energy at 298.15 K (0 K for 0 K calculation) and 1 atm of pressure, which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans).

|-
| Sum of electronic and thermal enthalpies || -234.460891 || -234.460141 || -7.5 x 10-4 || -1.97 || This contains an additional correction for RT (H = E + RT), which is especially important when looking at dissociation reactions.
|-
| Sum of electronic and thermal free energies || -234.500739 || -234.495931 || 4.808 x 10-3 || -12.62 || This includes the entropic contribution to the free energy (G = H - TS).
|}

{|align=right
|+
| [[Image:Anti2_freq_IRdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 298.15 K''']]
| [[Image:Anti2_freq_0Kdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 0 K''']]
|}

From looking at table 3 above it can be deduced that the anti2 conformer is more stable at room temperature to that of 0 K with respect to all the types of energies. The spectra also seem to not change much between the two temperatures but the main change is the change in the sum of electronic and thermal free energies, which is heavily dependent of temperature by it including the entropic contribution to the free energy (G = H - '''T'''S)

 
 
 
 

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

Within the Cope Rearrangement there are two possible transition states, the chair (C2h) and the boat (C2v), which are shown within [https://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_2 appendix two ].

==='''Comparison of optimisation methods for the chair transition state'''===

An <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2hdh08.mol</uploadedFileContents>
<text>allyl fragment</text>
<color>white</color>
</jmolAppletButton>
</jmol>, CH2CHCH2 was drawn on Gaussview and was optimized through the HF/3-21G method. After this the original allyl fragment was copied and pasted into a new window and place exactly above the first fragment with a bond distance of 2.20 Å between the terminal end carbons. This therefore produced the <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2h_fulldh08.mol</uploadedFileContents>
<text>chair transition structure.</text>
<color>white</color>
</jmolAppletButton>
</jmol>

Gaussian was then set up for an optimisation for the transition state with the additional key words "Opt=NoEigen", the optimisation set to TS (Berny), the force constants set to "once" and using the HF/3-21G method.

After this the "chair" transition state was optimised an alternative way by freezing the coordinates of the two sets of terminal carbons (where the bonds form/break during the rearrangement) within the Redundant Coord Editor on Gauusview.
Following this a second calculation was run where the freeze coordinates option was changed to "Derivative" and the force constants were set to "never".

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 4: Comparison of chair transition state optimisation methods
| || '''Optimised through a TS (Berny)''' || '''Optimised through a frozen coordinate method'''
|-
| '''Optimised TS Structure''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>C2h_optfreq_TS_Bdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| '''Terminal C-C bond length (Å)''' || 2.02 || 2.02
|-
|'''Energy (atomic units)''' || -231.61932247 || -231.61932190
|-
| '''ΔE, TS (Berny) - frozen coordinate''' || 5.7 x 10-7 a.u || 1.50 x 10-3 KJ mol-1
|-
| '''Imaginary frequency (cm-1)''' || -818 || -818
|-
| '''Animation''' ||[[Image:818dh08TSB.gif|350x350px]] || [[Image:818cm-1Fdh08.gif]]
|}

The data in table 4 shows that both methods give the same results and they both give the same transition state. They both produced the same energy with a negligible energy difference between them and the bond lengths between the terminal carbons are both 2.02 Å. In addition to this the imaginary frequencies were both -818cm-1 and within the animations above it is possible to see that when one C-C bond is forming the other is breaking. This means it is a asynchronous bond formation, which is expected for the Cope Rearrangement since it is a pericyclic reaction.

The only difference between the two optimisations is the way in which the transition states were calculated. This is due to the TS (Berny) method calculating the force constant matrix within the first optimisation step and is then being recalculated as the optimisation carries on. When the frozen coordinate method freezes the reaction coordinate in order for the rest of the structure to be optimised around it, which does not require the whole hessian (force constant matrix) to be calculated but only that the reaction coordinate is differentiated (second step). The main advantage of this, is that you do not need to calculate the entire force constant matrix and just need to differentiate the reaction coordinate in order to obtain a good enough guess for the initial force constant matrix. The frozen coordinate method will therefore take less time to compute the transition state, which can be very expensive within certain systems. However to carry out this method one needs to have an idea of how far apart the terminal carbons are.

==='''Boat transition state'''===

===='''QST2 optimisation'''====

The optimisation for the boat transition state had a different method and this was the QST2 method. This calculation interpolates between two structures (product and reactant) and tries to find a transition state between them.

The reactants and products were numbered in the same way. Therefore both 1,5-hexadiene molecules had their numbering manually changed for the product molecule so it corresponded to the numbering of the reactant if it rearranged. This is illustrated in the reaction scheme below.

[[Image:Boat_rerrangmentdh08.jpg|300x300px|center]]

The chk file for the optimized anti2 conformer was opened and copied into a new window. A second window was opened within the new window and the same anti2 conformer was copied into the new window. The numbering was changed within the second molecule so the two looked like they do in the image below.

[[Image:TS_(QST2)_numbering_1dh08.jpg|400x400px|center]]

The job type was set as "Opt+Freq" and "TS (QST2)" was chosen to optimise the transition state. However this failed and produced
a structure a bit like a <jmol>
<jmolAppletButton>
<uploadedFileContents>Trialone_BPdh08.mol</uploadedFileContents>
<text>chair transition structure </text>
<color>white</color>
</jmolAppletButton>
</jmol> but more dissociated. This is because it interpolated between the two structures and translated the top allyl fragment and therefore did not consider the possibility of a rotation around the central bonds.

To get around this the C-C-C-C dihedral angle (C2-C3-C4-C5) was set to 0 o and the C-C-C angle (C2-C3-C4 and C3-C4-C5 for the product molecule) was set to 100 o for both molecules. This gave rise to the two structures below.

[[Image:TS_(QST2)_numbering_2dh08.jpg|400x400px|center]]

After this the QST2 calculation was run again and the results in table 5 were produced. The imaginary frequency at -839 cm-1 shows that while one bond is forming the other one is breaking, which is again an asynchronous bond formation. It is also possible to see that the boat transition state has a longer terminal C-C bond length by 0.12 Å to that of the chair.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5: Data for the boat transition state from the QST2 optimisation method
| Optimised TS Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.14
|-
|Energy (atomic units) || -231.60280151
|-
| Energy (KJ mol-1) || -6.08 x 105
|-
| Imaginary frequency (cm-1) || -839
|-
| Animation ||[[Image:-839dh08.gif]]
|}

===='''Intrinsic Reaction Coordinate'''====

It is not possible to predict which product the chair and boat transition state will lead to. To solve this problem within Gaussian one can use the IRC method, which allows one to follow the minimum energy path from a transition structure down to its local minimum on a potential energy surface. This creates a succession of points by taking small geometry steps in the direction to where the gradient of the energy surface is at its steepest until the minimum in energy is reached.

The chk file for the optimized chair transition structure was opened and the job type selected was "IRC", the reaction coordinate was computed in the forward direction (since molecule is symmetric) with 50 points along the IRC. This was done with two type of force constants being calculated, once and always.

Within the "once" method the output showed no bond formation process by the 26th step and therefore the calculation was incomplete, since a complete 1,5-hexadiene structure was supposed to be formed. From looking at the IRC pathway it can be seen that the minimum on the potential energy surface was not reached at that point, since the points on the pathway did not form an asymptote and the gradient did not go close enough to zero.

For the "always" method the computation of force constants were carried out at every step. This therefore lets the IRC calculation distinguish whether it is moving along the minimum energy pathway while going down the potential energy surface. While this is happening it can make changes to the pathway if necessary. Within the "always" case the bond forming arrives at 1,5-hexadiene and this is backed up by the IRC pathway reaching a minimum within its energy by the gradient going close to zero and the pathway reaches an asymptote.

The structure produced through the always method had an energy of -231.69199434 atomic units, which is very close to the energy of the gauche2 conformer in the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 appendix 1]. This suggests that the reaction takes place through the chair transition state, which later produces the gauche2 conformer of 1,5-hexadiene.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 6: Comparison of IRC calculations for the chair structure
| || '''Calculate force constants once''' || '''Calculate force constants always'''
|-
| '''Number of points along IRC''' || 26 || 47
|-
| '''Structure at the last point of IRC''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Once_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Always_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|'''Energy (atomic units)''' || -231.68906923 || -231.69199434
|-
| '''ΔE, once - always''' || 2.92511 x 10-3 a.u || 7.68 KJ mol-1
|-
| '''IRC pathway''' || [[Image:One_pathwaydh08.jpg|400x400px]] || [[Image:Always_pathwaydh08.jpg|400x400px]]
|-
| '''IRC gradient''' || [[Image:One_gradientdh08.jpg|400x400px]]|| [[Image:Always_gradientdh08.jpg|400x400px]]
|}

 
 
 

===='''Comparison of activation energies for the transition states'''====

The HF/3-21G optimised chair and boat transition structures were both reoptimised using the B3LYP/6-31G* (B3LYP/6-31G (d)) method. This gives a more accurate minimisation however this method takes longer to run. The values for the zero-point energies and thermal energies were found within the thermochemistry section of the log file.

The values for the HF/3-21G reactant (anti2) optimisation were calculated and the B3LYP/6-31G* values were taken from the "Frequency analysis of 1,5-hexadiene Conformers" earlier in this module.

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 7: Energy data for chair, boat and the anti2 conformation from the HF/3-21G and B3LYP/6-31G* methods of optimisation '''
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466677
| align="center" | -231.461320
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450936
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402342
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539524
| align="center" | -231.532562
| align="center" | -234.611716
| align="center" | -234.469177
| align="center" | -234.461835
|}
Digital repositories

Chair:
HF/3-21G: http://hdl.handle.net/10042/to-5569
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5568

Boat:
HF/3-21G: http://hdl.handle.net/10042/to-5570
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5571

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 8: Summary of activation energies for the chair and boat conformations (ΔE = chair TS or boat TS minus anti2)'''
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| align="center" | 1 a.u = 627.509 kcal mol-1
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE, chair (a.u)'''
| align="center" | 0.072847
| align="center" | 0.071242
| align="center" | 0.054248
| align="center" | 0.052827
| align="center" | -
|-
| '''ΔE, chair (kcal mol-1)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE, boat (a. u)'''
| align="center" | 0.088588
| align="center" | 0.087259
| align="center" | 0.066835
| align="center" | 0.065827
| align="center" | -
|-
| '''ΔE, boat (kcal mol-1)'''
| align="center" | 55.59
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.31
| align="center" | 44.7 ± 2.0
|}

By examining table 8 it is clear to see that the B3LYP/6-31G* method gives values which are much closer to the experimental ones. This is due to the B3LYP/6-31G* giving a higher level of optimisation than the HF/3-21G method. Table 8 also shows that the chair transition state has a lower activation energy to that of the boat transition state by 8.16 kcal mol-1. This suggests that the Cope rearrangement mechanism would most likely proceed through the chair transition state to that of the boat. Therefore the reaction would take place from the anti2 conformer to the chair transition state and then onto the gauche2 conformer within the Cope rearrangement.

It is also possible to see that the calcualted energies are very close to the literature values with the chair conformation being closer to its literature value to that of the boat.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9: Geometric parameters for the optimised chair and boat structures '''
| || ''' HF/3-21G chair TS''' || ''' B3LYP/6-31G* chair TS''' || '''HF/3-21G boat TS''' || '''B3LYP/6-31G* boat TS'''
|-
| Optimised Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Chair_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Boat_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.02 (C5-C10 or C3-C14)|| 1.97 (C5-C10 or C3-C14) || 2.14 (C3-C4 or C1-C6) || 2.21 (C3-C4 or C1-C6)
|-
| Fragment angle || 120 o (C3-C1-C5) || 120 o (C3-C1-C5) || 122 o (C4-C5-C6) || 122 o (C4-C5-C6)
|-
| Dihedral angle || 55 o (C1-C3-C14-C11)|| 54 o (C1-C3-C14-C11)|| 0 o (C2-C1-C6-C5) || 0 o (C2-C1-C6-C5)
|-
| Numbering of carbons || [[Image:Numbering_chairdh08.jpg|200x200px]]|| [[Image:Numbering_chairdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]]
|}

From studying table 9 it can be seen that the C-C bond breaking/forming length has changed between the two different types of optimisation methods. However the angles between the bonds remains unchanged.

=='''The Diels Alder Cycloaddition'''==

The Diels-Alder reaction is a cycloaddition between a conjugated diene and a dienophile (alkene system), which forms a cyclohexene system. It is often referred to as a [4s+2s] cycloaddition and proceeds through a single cyclic, concerted transition state where 2 σ bonds are formed in between the termini carbons of the 2 п systems. This is shown in the reaction scheme below where ethene is reacting with cis-butadiene.

[[Image:Reaction_scheme2dh08.jpg|300x300px|center]]

The π orbitals of the dienophile form new σ bonds with the π orbitals of the diene. The number of π electrons involved determine whether the reaction proceeds in a concerted stereospecific fashion (allowed) or not (forbidden). The HOMO/LUMO of one fragment interacts with the HOMO/LUMO of the other fragment to form two new bonding and anti-bonding MOs. The nodal properties allow one to predict how the reaction will occur.

Semi-Empirical/AM1 method and the B3LYP/6-31G* method of optimisation will be used on Gaussian to determine particular properites of the reactants to predict the way in which the reaction will proceed.

==='''cis-butadiene and ethylene: reagents'''===

<jmol>
<jmolAppletButton>
<uploadedFileContents>Cis_butadienedh08.mol</uploadedFileContents>
<text>Cis-butadiene</text>
<color>whi-te</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Ethenedh08.mol</uploadedFileContents>
<text>ethylene</text>
<color>white</color>
</jmolAppletButton>
</jmol> were optimised separately using the AM1 semi-empirical method on Gaussview.The cis-butadiene had an energy of 0.04879725 a. u (1.28 x 102 KJ mol-1) and ethylene had an energy of 0.02619027 a. u (68.76 KJ mol-1).

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10: Frontier molecular orbitals of cis-butadiene and ethylene '''
| '''Reagent''' || '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| cis-butadiene || LUMO || [[Image:LUMO1_cis_butadieneDH08.jpg|100x100px]] || [[Image:LUMO2_cis_butadienedh08.jpg|100x100px]]|| Symmetric || +0.017
|-
| cis-butadiene || HOMO || [[Image:HOMO1_cis_butadienedh08.jpg|100x100px]] || [[Image:HOMO2_cis_butadienedh08.jpg|100x100px]]|| Anti-symmetric || -0.343
|-
| ethylene || LUMO || [[Image:LUMO1_ethenedh08.jpg|100x100px]] || [[Image:LUMO2_ethenedh08.jpg|100x100px]]|| Anti-symmetric || +0.052
|-
| ethylene || HOMO || [[Image:HOMO1_ethenedh08.jpg|100x100px]] || [[Image:HOMO2_ethenedh08.jpg|100x100px]]|| Symmetric || -0.387
|}

 
The symmetries of the LUMO and HOMO orbitals for cis-butadiene are opposite of those for the ethylene. This therefore means that the only interactions that can take place are those between ethylene's LUMO and cis-butadiene's HOMO and in addition to this the interaction between cis-butadiene's LUMO and ethylene's HOMO. This is because orbital symmetry has to be conserved (the conservation of orbital symmetry states that the symmetry of an new orbital formed by the overlap of the 2 original orbitals must be the same as the symmetry of the initial orbitals).

==='''cis-butadiene and ethylene: transition state'''===

A <jmol>
<jmolAppletButton>
<uploadedFileContents>Bicyclic_startingdh08.mol</uploadedFileContents>
<text>bicyclo system</text>
<color>white</color>
</jmolAppletButton>
</jmol> was drawn and one of the -CH2-CH2- fragments were removed. Three C=C bonds were added and the previous one was taken away so a <jmol>
<jmolAppletButton>
<uploadedFileContents>Frozen_coorddh08.mol</uploadedFileContents>
<text>guess transition state</text>
<color>white</color>
</jmolAppletButton>
</jmol> was produced.

After this an optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while bonds between the termini of the fragments were frozen at 2.1Å. Following this the resulting file had two separate optimisation and frequency calculations run on it using the TS (Berny) with one using the semi-empirical/AM1 method and the other using the DFT/B3LYP/6-31G* level of theory. Both of these optimisations had the bonds unfrozen and the "derivative" option was chosen instead.

The DFT/B3LYP/6-31G* level of theory was used since it gave more accurate results in the calculations before and the energies it produces are consistent with the rest of the report (semi-empirical/AM1 method gives very different values).

The QST2 optimisation was not used since it required numbering the atoms, which took a lot of time to do.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11: Comparison of two methods of optimsations for the transition state '''
| || '''semi-empirical/AM1 method''' || '''DFT/B3LYP/6-31G* method'''
|-
| Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>SE_AM1_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>DFT_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Energy (a. u) || 0.11165608 || -234.54389573
|-
| Bond lengths || [[Image:DistancesSEdh08.jpg|150x150px]]|| [[Image:DistancesDFTdh08.jpg|150x150px]]
|-
|Imaginary frequency (cm-1) || -957 || -525
|-
|Animation || [[Image:-957dh08.gif]] || [[Image:-525dh08.gif]]
|-
| Lowest real frequency (cm-1) || 148 || 136
|-
|Animation || [[Image: 148dh08.gif]] || [[Image: 136dh08.gif]]
|}

The C-C bond forming length between the termini carbons is 0.15 Å larger for the DFT/B3LYP/6-31G* level of theory and this is because this method gives carries out the calculation to a greater degree of accuracy. This larger bond length leads a lower potential energy gradient to the transition state, which gives a lower total energy for the transition state. Therefore the given values below will be the DFT/B3LYP/6-31G* and not the semi-empirical/AM1 method.

A typical sp3 C-C bond length is 1.54 Å and a typical sp2 C=C bond length is 1.34 Å<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref>. Within the optimised transition molecule above the C-C bond lengths are 1.40 Å and the C=C bond length is 1.38 Å. This suggests that the π-bonds are breaking in order for the C-C bonds to form and additionally to this a π-bond is forming in between the two sp3 carbons.

The Van der Waals radius of a C-atom is 1.72 Å<ref name="ref 3">M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer, D. G. Truhlar, 1995, 346, pp. 63-69.[http://pubs.acs.org/doi/abs/10.1021/jp8111556]</ref>. The partly formed C-C length is 2.27 Å, which is shorter than two times the the van der Waals radius of a carbon atom. Therefore the carbon atoms are within a close enough distance to each other to share electron density and yet the bond has not formed at this point.

From looking at the animations for the imaginary frequency above it can be seen that the formation of the two bonds between the termini of the reactants is synchronous. This is expected of the Diels-Alder reaction since the mechanism is concerted. However the lowest real frequency is asynchronous with respect to the two bonds forming between the termini.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12: Frontier molecular orbitals of the transition state '''
| '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| LUMO || [[Image:LUMO1TSdh08.jpg|100x100px]] || [[Image:LUMO2TSdh08.jpg|100x100px]]|| Symmetric || +0.023
|-
| HOMO || [[Image:HOMO1TSdh08.jpg|100x100px]] || [[Image:HOMO2TSdh08.jpg|100x100px]]|| Anti-symmetric || -0.323
|}

From examining the images from table 12 above it can be deduced that the LUMO of the transition state originates from the cis-butadiene's LUMO and the ethylene's HOMO, which represents the σ*-antibonding orbitals. However the HOMO of the transition state originates from the cis-butadiene's HOMO and the ethylene's LUMO, which represents the σ-bonding orbitals within the two single C-C bonds that are formed within the reaction.

==='''cis-butadiene and ethylene: activation energy'''===

The cis-butadiene and ethylene were re-optimised at the DFT/B3LYP/6-31G* level of theory so the activation energy of the reaction could be calculated.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 13: Calculation for the activation energy for the reaction'''
| '''molecule''' || '''DFT/B3LYP/6-31G* energy (a. u)''' || '''Calculations''' || '''Energy '''
|-
| cis-butadiene || -155.98595591 || transition state || -234.54389573 a. u
|-
| ethylene || -78.58745828 || ΔE, activation energy || 0.02951846 a. u
|-
| cis-butadiene and ethylene combined || -234.57341419 || ΔE, activation energy || 77.50 KJ mol-1
|}

The calculated activation energy is 77.50 KJ mol-1, which is of the right magnitude when compared to the literature value (115 ± 8 KJ mol-1 at 0 K) but there is still an error of 37.50 KJ mol-1. This shows that Gaussview is not perfect but still gives an idea of the activation energy required for the reaction to take place.

==='''Cyclohexa-1,3-diene reaction with maleic anhydride'''===

The reaction scheme below shows the stereoselective Diels-Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene. The reaction is kinetically controlled with the major product being the endo product, which will be explained later on in the exercise.

[[Image:Reaction_schemedh081.jpg|center|500x500px]]

The <jmol>
<jmolAppletButton>
<uploadedFileContents>FrozenTS22dh08.mol</uploadedFileContents>
<text>exo</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_frozenTS22dh08.mol</uploadedFileContents>
<text>endo</text>
<color>white</color>
</jmolAppletButton>
</jmol> transition states were drawn on Gaussview and then the optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while the distance of the forming bonds were frozen at 2.1Å between the respective carbons. After this the structures changed slightly (<jmol>
<jmolAppletButton>
<uploadedFileContents>TS2_afteropt1_exo1dh08.mol</uploadedFileContents>
<text>exo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_TS2_afteropt1dh08.mol</uploadedFileContents>
<text>endo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol>) and the resulting files had an optimisation and frequency calculation run on it using the TA (Berny) with using the semi-empirical/AM1 level of theory while the unfrozen option as changed to "derivative". After this the endo and exo optimised transition state structures were re-optimised as they were before but using the DFT/B3LYP/6-31G* level of theory instead with the additional keyword "Opt=NoEigen".

{| class="wikitable" border="1" align="center"
|+ '''Table 14: Comparison of the exo and endo transition states '''
! rowspan="2"|-!! colspan="2"|''exo'' !! colspan="2"|''endo''
|-
| Semi-empirical/AM1 || DFT/B3LYP/6-31G* || Semi-empirical/AM1 || DFT/B3LYP/6-31G*
|-
| align="center"|Structure ||align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>After_opt2_exodh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>Endo_TS2_afteropt2dh081.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| align="center"| Energy (a. u) ||align="center"| -0.05041975 ||align="center"| -612.67931095 ||align="center"| -0.05150470 ||align="center"| -612.68339684
|-
| C-C bond forming length (Å) ||align="center"| 2.17 ||align="center"| 2.29 ||align="center"| 2.16 ||align="center"| 2.27
|-align="center"
| Imaginary frequency (cm-1) ||align="center"| -812 ||align="center"| -448 ||align="center"| -806 ||align="center"| -447
|-align="center"
| Animation ||colspan="2"|[[Image:-812_SEdh08.gif]] || colspan="2"|[[Image:-806_cm-1dh08.gif]]
|}

Again the C-C bond forming length between the termini carbons is larger (by 0.11 and 0.12 Å) for the DFT/B3LYP/6-31G* level of theory and this is due the same reasons talked about above. However the C-C bond forming length between the termini carbons are very similar in both the endo and exo transition states (0.02 Å difference).

From looking at the animations for the imaginary frequency above it can also be seen that the formation of the two bonds between the carbons in this case as well is synchronous. Again this is expected of the Diels-Alder reaction since it is concerted.

{| align=center
|+ '''Strain diagrams for the endo and exo transition states'''
| [[Image:Exostraindh08.jpg|thumb|200x200px|'''Exo TS''']]
| [[Image:Strainendodh08.jpg|thumb|200x200px|'''Endo TS''']]
|}

From studying the energies in table 14 it can be seen that the endo transition state is more stable than the exo by 4.08589 x 10-3 a. u (10.73 KJ mol-1). However by looking at the strain of both transition states one would expect the exo TS to be more stable. This is because the distance between the cyclohexa-1,3-diene ring and the -(C=O)-O-(C=O)- maleic anhydride fragment is 2.99 Å in the endo TS and 3.03 Å in the exo TS. Thus the endo form would have more strain due to steric hindrance.

However within the exo TS the vicinal hydrogens on the CH2-CH2 (carbon numbers 18 and 21) fragment on the cyclohexa-1,3-diene ring are found below the plane and next to the maleic anhydride. This leads to steric hindrance is this is where the strain originates from within the exo TS. Therefore this disfavours the exo form, which makes the endo more stable.

Another factor making the endo TS more favourable is secondary orbital overlap. These orbitals contribute to the bonds forming and they interact with each other through space. Within the endo TS below there are 2 more orbitals interacting (secondary orbitals) than within the exo TS. This therefore makes the endo TS more stable and therefore more favoured.

[[Image:Secondary_orbital_shizzledh08.jpg|500x500px|center]]

{| class="wikitable" border="1" align="center"
|+ '''Table 15: Frontier molecular orbitals of the the transition state for the cyclohexa-1,3-diene reaction '''
! rowspan="2"|Molecular Orbital !! colspan="4"|''exo'' !! colspan="4"|''endo''
|- align="center"
| '''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''||'''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''
|- align="center"
| LUMO || [[Image:LUMO1exoh081.jpg|150x150px]]|| [[Image:LUMO2exodh081.jpg|150x150px]]|| Anti-symmetric || -0.078 || [[Image:LUMO1ENDOdh08.jpg|150x150px]]||[[Image:LUMO2ENDOdh08.jpg|150x150px]] || Anti-symmetric || -0.067
|- align="center"
| HOMO || [[Image:HOMO1exodh081.jpg|150x150px]]||[[Image:HOMO2exodh081.jpg|150x150px]] || Anti-symmetric|| -0.242 || [[Image:HOMO1ENDOdh08.jpg|150x150px]]|| [[Image:HOMO2ENDOdh08.jpg|150x150px]]||Anti-symmetric || -0.242
|}

Digital repositories:

exo: http://hdl.handle.net/10042/to-5596

endo: http://hdl.handle.net/10042/to-5597

From examining table 15 it is clear to see that both the HOMO and LUMO are anti-symmetric for both the transition states. The LUMO of the maleic anhydride fragment is the π* anti-bonding orbital of the C=C bond. Within the reaction then this has electrons transferred to it and the C=C bond breaks, which leaves just a C-C bond. The LUMOs for both transition states are both anti-bonding with respect to the two fragments. The HOMOs of both transitions state represent the σ-bonding that will be formed between the termini carbons on the products and reactants. This is most likely from the overlap of the cyclohexa-1,3-diene's HOMO and the maleic anhydride's LUMO.

=='''Conclusion'''==

='''References'''=
<references/>

Rep:Mod:hampluscheese

2010-11-11T13:00:28Z

Dh08: /* '''Objectives''' */

='''Module 3:Physical Computational Chemistry Lab'''=

Douglas Hunt

=='''Objectives'''==

The aim of this experiment is to is to use various levels of theory (HF/3-21G, DFT/B3LYP/6-31G (d) and AM1/semi-empirical) on Gaussian to calculate various energies, molecular orbitals, geometrical parameters, reaction coordinate pathways and low and imaginary frequencies for a range of molecules and transition states. This will include using different optimisation methods throughout and will lead to a range of findings in various areas of chemical interest.

=='''The Cope Rearrangement'''==

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangement] is a [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement |3,3|-sigmatropic rearrangement] that takes place through a single [http://en.wikipedia.org/wiki/Pericyclic pericyclic], concerted [http://en.wikipedia.org/wiki/Transition_state transition state]. Within this rearrangement a σ bond is formed in concert with another being broken within a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] π system.

The Cope rearrangement we will be studying will be that of 1,5-hexadiene, which is shown below.

[[Image:Cope_rearrangement_schemedh08.jpg|300x300px|center]]

The objective is to locate the low-energy minima and transition structures on the C6H10 potential energy surface and to determine the preferred reaction mechanism by using [http://en.wikipedia.org/wiki/GAUSSIAN Gaussian], with the [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)] methods.

==='''Optimisation of 1,5-hexadiene Conformers'''===

Rough <jmol>
<jmolAppletButton>
<uploadedFileContents>Anti_2_drawndh08.mol</uploadedFileContents>
<text>anti</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche3_drawndh08.mol</uploadedFileContents>
<text>gauche</text>
<color>white</color>
</jmolAppletButton>
</jmol> conformers of 1,5-hexadiene were drawn on Gaussview and their structures were tidied by the clean function. They were then further optimised by the HF/3-21G method and were identified which conformer they were from their point group and appendix 1 on the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3 physical lab page].

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 1: Optimisation and comparison of anti and gauche 1,5-hexadiene conformers using HF/3-21G methods
| ''' Conformer''' || '''Optimised Structure''' || '''Energy (Atomic Units)''' || '''[http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1 Energies (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (Anti2 - Gauche3)''' || '''Point group'''
|-
| '''Anti2''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Anti_2_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692535 || -231.69254 ||-6.08 x 105 || 1.2606 x 10-4 a. u|| Ci
|-
| '''Gauche3''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Gauche3_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692661 || -231.69266 || -6.08 x 105 || 3.31 x 10-1 KJ mol-1 || C1
|}

From studying table 1 it can be seen that the gauche conformer is 0.33 KJ mol-1 more stable than the anti conformation. One would assume that the anti2 conformer would be more stable due to [http://en.wikipedia.org/wiki/Steric steric] factors/repulsion within the gauche3 conformation. However within the gauche3 conformer there is a favourable donation of [http://en.wikipedia.org/wiki/Electron_density electron density] from the C=C π orbital to the C-H σ* orbital from the adjacent proton, which stabilises the conformer<ref name="ref 1">B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446.[http://www.informaworld.com/smpp/content~db=all?content=10.1080/00268970110081412]</ref>.

 

The anti2 conformer was then optimised using the B3LYP 6-31G* (6-31G (d)) method.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 2: Comparison of the anti2 conformer optimisations carried out at different calculation methods
| ''' Method''' || '''Structure and bond distances (Å)''' || '''Dihedral angle (C1 to C4)''' || '''Energy (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (HF/3-21G - DFT/B3LYP/6-31G)'''
|-
| HF/3-21G || [[Image:Anti2_opt1dh08.jpg|300x300px]] || 114.6 o || -231.692535 || -6.08 x 105 || 2.919171 a. u
|-
| DFT/B3LYP/6-31G* || [[Image:Anti2_opt2DH08.jpg|300x300px]] || 118.8 o || -234.611716 || -6.16 x 105 || 7.66 x 103 KJ mol-1
|}

From examining table 2 at first it is possible to see that the bond lengths are very similar to each other and that of the literature bond lengths<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref> (C1=C2: 1.340 Å, C2-C3: 1.508 Å and C3-C4: 1.538 Å). It is also possible to see that the C=C bond length is more accurate for the DFT/B3LYP/6-31G* method by it being 0.01 Å closer to the literature value. Another main difference in terms of geometry is that the DFT/B3LYP/6-31G* method gives a larger [http://en.wikipedia.org/wiki/Dihedral_angle dihedral angle] (and most likely more accurate) of 118.8o compared to that of 114.6o.

In terms of energy the DFT/B3LYP/6-31G* method produces a conformer of a higher stability by it possessing a lower energy with a difference of 7.66 x 103 KJ mol-1.

==='''Frequency analysis of 1,5-hexadiene Conformers'''===

[[Image:Vibrations_anti2dh08.jpg|right|300x300px]]

For the frequency analysis the DFT/B3LYP/6-31G* method was used.

Frequency analysis is essential within this calculation due to it being the second derivative of the potential energy surface. This means that if all the frequencies are all positive the system is at a minimum, if one of them is negative the optimised system is at a transition state, and if more than one are negative then the calculation has failed to find a critical point, which means the optimisation has failed. This therefore gives us a method of seeing whether the calculation has worked.

From looking to the image on the right it can deduced that the optimisation has taken place successfully by there being no negative vibrations and therefore the conformation is at a minimum.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 3: Energies under the thermochemistry section at 298.15 K and 0 K within the frequency logfiles'''
| '''Energies''' || '''298.15 K (atomic units)''' ||| '''0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (KJ mol-1)''' || '''Discussion'''
|-
| Sum of electronic and zero-point energies || -234.469177 || -234.466673 || -2.504 x 10-3 || -6.57 || This is the potential energy at 0 K, which includes the zero-point vibrational energy (E = Eelec + ZPE).

|-
| Sum of electronic and thermal energies || -234.461835 || -234.461085 || -7.5 x 10-4|| -1.97 || This is the energy at 298.15 K (0 K for 0 K calculation) and 1 atm of pressure, which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans).

|-
| Sum of electronic and thermal enthalpies || -234.460891 || -234.460141 || -7.5 x 10-4 || -1.97 || This contains an additional correction for RT (H = E + RT), which is especially important when looking at dissociation reactions.
|-
| Sum of electronic and thermal free energies || -234.500739 || -234.495931 || 4.808 x 10-3 || -12.62 || This includes the entropic contribution to the free energy (G = H - TS).
|}

{|align=right
|+
| [[Image:Anti2_freq_IRdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 298.15 K''']]
| [[Image:Anti2_freq_0Kdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 0 K''']]
|}

From looking at table 3 above it can be deduced that the anti2 conformer is more stable at room temperature to that of 0 K with respect to all the types of energies. The spectra also seem to not change much between the two temperatures but the main change is the change in the sum of electronic and thermal free energies, which is heavily dependent of temperature by it including the entropic contribution to the free energy (G = H - '''T'''S)

 
 
 
 

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

Within the Cope Rearrangement there are two possible transition states, the chair (C2h) and the boat (C2v), which are shown within [https://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_2 appendix two ].

==='''Comparison of optimisation methods for the chair transition state'''===

An <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2hdh08.mol</uploadedFileContents>
<text>allyl fragment</text>
<color>white</color>
</jmolAppletButton>
</jmol>, CH2CHCH2 was drawn on Gaussview and was optimized through the HF/3-21G method. After this the original allyl fragment was copied and pasted into a new window and place exactly above the first fragment with a bond distance of 2.20 Å between the terminal end carbons. This therefore produced the <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2h_fulldh08.mol</uploadedFileContents>
<text>chair transition structure.</text>
<color>white</color>
</jmolAppletButton>
</jmol>

Gaussian was then set up for an optimisation for the transition state with the additional key words "Opt=NoEigen", the optimisation set to TS (Berny), the force constants set to "once" and using the HF/3-21G method.

After this the "chair" transition state was optimised an alternative way by freezing the coordinates of the two sets of terminal carbons (where the bonds form/break during the rearrangement) within the Redundant Coord Editor on Gauusview.
Following this a second calculation was run where the freeze coordinates option was changed to "Derivative" and the force constants were set to "never".

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 4: Comparison of chair transition state optimisation methods
| || '''Optimised through a TS (Berny)''' || '''Optimised through a frozen coordinate method'''
|-
| '''Optimised TS Structure''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>C2h_optfreq_TS_Bdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| '''Terminal C-C bond length (Å)''' || 2.02 || 2.02
|-
|'''Energy (atomic units)''' || -231.61932247 || -231.61932190
|-
| '''ΔE, TS (Berny) - frozen coordinate''' || 5.7 x 10-7 a.u || 1.50 x 10-3 KJ mol-1
|-
| '''Imaginary frequency (cm-1)''' || -818 || -818
|-
| '''Animation''' ||[[Image:818dh08TSB.gif|350x350px]] || [[Image:818cm-1Fdh08.gif]]
|}

The data in table 4 shows that both methods give the same results and they both give the same transition state. They both produced the same energy with a negligible energy difference between them and the bond lengths between the terminal carbons are both 2.02 Å. In addition to this the imaginary frequencies were both -818cm-1 and within the animations above it is possible to see that when one C-C bond is forming the other is breaking. This means it is a asynchronous bond formation, which is expected for the Cope Rearrangement since it is a pericyclic reaction.

The only difference between the two optimisations is the way in which the transition states were calculated. This is due to the TS (Berny) method calculating the force constant matrix within the first optimisation step and is then being recalculated as the optimisation carries on. When the frozen coordinate method freezes the reaction coordinate in order for the rest of the structure to be optimised around it, which does not require the whole hessian (force constant matrix) to be calculated but only that the reaction coordinate is differentiated (second step). The main advantage of this, is that you do not need to calculate the entire force constant matrix and just need to differentiate the reaction coordinate in order to obtain a good enough guess for the initial force constant matrix. The frozen coordinate method will therefore take less time to compute the transition state, which can be very expensive within certain systems. However to carry out this method one needs to have an idea of how far apart the terminal carbons are.

==='''Boat transition state'''===

===='''QST2 optimisation'''====

The optimisation for the boat transition state had a different method and this was the QST2 method. This calculation interpolates between two structures (product and reactant) and tries to find a transition state between them.

The reactants and products were numbered in the same way. Therefore both 1,5-hexadiene molecules had their numbering manually changed for the product molecule so it corresponded to the numbering of the reactant if it rearranged. This is illustrated in the reaction scheme below.

[[Image:Boat_rerrangmentdh08.jpg|300x300px|center]]

The chk file for the optimized anti2 conformer was opened and copied into a new window. A second window was opened within the new window and the same anti2 conformer was copied into the new window. The numbering was changed within the second molecule so the two looked like they do in the image below.

[[Image:TS_(QST2)_numbering_1dh08.jpg|400x400px|center]]

The job type was set as "Opt+Freq" and "TS (QST2)" was chosen to optimise the transition state. However this failed and produced
a structure a bit like a <jmol>
<jmolAppletButton>
<uploadedFileContents>Trialone_BPdh08.mol</uploadedFileContents>
<text>chair transition structure </text>
<color>white</color>
</jmolAppletButton>
</jmol> but more dissociated. This is because it interpolated between the two structures and translated the top allyl fragment and therefore did not consider the possibility of a rotation around the central bonds.

To get around this the C-C-C-C dihedral angle (C2-C3-C4-C5) was set to 0 o and the C-C-C angle (C2-C3-C4 and C3-C4-C5 for the product molecule) was set to 100 o for both molecules. This gave rise to the two structures below.

[[Image:TS_(QST2)_numbering_2dh08.jpg|400x400px|center]]

After this the QST2 calculation was run again and the results in table 5 were produced. The imaginary frequency at -839 cm-1 shows that while one bond is forming the other one is breaking, which is again an asynchronous bond formation. It is also possible to see that the boat transition state has a longer terminal C-C bond length by 0.12 Å to that of the chair.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5: Data for the boat transition state from the QST2 optimisation method
| Optimised TS Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.14
|-
|Energy (atomic units) || -231.60280151
|-
| Energy (KJ mol-1) || -6.08 x 105
|-
| Imaginary frequency (cm-1) || -839
|-
| Animation ||[[Image:-839dh08.gif]]
|}

===='''Intrinsic Reaction Coordinate'''====

It is not possible to predict which product the chair and boat transition state will lead to. To solve this problem within Gaussian one can use the IRC method, which allows one to follow the minimum energy path from a transition structure down to its local minimum on a potential energy surface. This creates a succession of points by taking small geometry steps in the direction to where the gradient of the energy surface is at its steepest until the minimum in energy is reached.

The chk file for the optimized chair transition structure was opened and the job type selected was "IRC", the reaction coordinate was computed in the forward direction (since molecule is symmetric) with 50 points along the IRC. This was done with two type of force constants being calculated, once and always.

Within the "once" method the output showed no bond formation process by the 26th step and therefore the calculation was incomplete, since a complete 1,5-hexadiene structure was supposed to be formed. From looking at the IRC pathway it can be seen that the minimum on the potential energy surface was not reached at that point, since the points on the pathway did not form an asymptote and the gradient did not go close enough to zero.

For the "always" method the computation of force constants were carried out at every step. This therefore lets the IRC calculation distinguish whether it is moving along the minimum energy pathway while going down the potential energy surface. While this is happening it can make changes to the pathway if necessary. Within the "always" case the bond forming arrives at 1,5-hexadiene and this is backed up by the IRC pathway reaching a minimum within its energy by the gradient going close to zero and the pathway reaches an asymptote.

The structure produced through the always method had an energy of -231.69199434 atomic units, which is very close to the energy of the gauche2 conformer in the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 appendix 1]. This suggests that the reaction takes place through the chair transition state, which later produces the gauche2 conformer of 1,5-hexadiene.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 6: Comparison of IRC calculations for the chair structure
| || '''Calculate force constants once''' || '''Calculate force constants always'''
|-
| '''Number of points along IRC''' || 26 || 47
|-
| '''Structure at the last point of IRC''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Once_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Always_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|'''Energy (atomic units)''' || -231.68906923 || -231.69199434
|-
| '''ΔE, once - always''' || 2.92511 x 10-3 a.u || 7.68 KJ mol-1
|-
| '''IRC pathway''' || [[Image:One_pathwaydh08.jpg|400x400px]] || [[Image:Always_pathwaydh08.jpg|400x400px]]
|-
| '''IRC gradient''' || [[Image:One_gradientdh08.jpg|400x400px]]|| [[Image:Always_gradientdh08.jpg|400x400px]]
|}

 
 
 

===='''Comparison of activation energies for the transition states'''====

The HF/3-21G optimised chair and boat transition structures were both reoptimised using the B3LYP/6-31G* (B3LYP/6-31G (d)) method. This gives a more accurate minimisation however this method takes longer to run. The values for the zero-point energies and thermal energies were found within the thermochemistry section of the log file.

The values for the HF/3-21G reactant (anti2) optimisation were calculated and the B3LYP/6-31G* values were taken from the "Frequency analysis of 1,5-hexadiene Conformers" earlier in this module.

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 7: Energy data for chair, boat and the anti2 conformation from the HF/3-21G and B3LYP/6-31G* methods of optimisation '''
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466677
| align="center" | -231.461320
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450936
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402342
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539524
| align="center" | -231.532562
| align="center" | -234.611716
| align="center" | -234.469177
| align="center" | -234.461835
|}
Digital repositories

Chair:
HF/3-21G: http://hdl.handle.net/10042/to-5569
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5568

Boat:
HF/3-21G: http://hdl.handle.net/10042/to-5570
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5571

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 8: Summary of activation energies for the chair and boat conformations (ΔE = chair TS or boat TS minus anti2)'''
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| align="center" | 1 a.u = 627.509 kcal mol-1
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE, chair (a.u)'''
| align="center" | 0.072847
| align="center" | 0.071242
| align="center" | 0.054248
| align="center" | 0.052827
| align="center" | -
|-
| '''ΔE, chair (kcal mol-1)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE, boat (a. u)'''
| align="center" | 0.088588
| align="center" | 0.087259
| align="center" | 0.066835
| align="center" | 0.065827
| align="center" | -
|-
| '''ΔE, boat (kcal mol-1)'''
| align="center" | 55.59
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.31
| align="center" | 44.7 ± 2.0
|}

By examining table 8 it is clear to see that the B3LYP/6-31G* method gives values which are much closer to the experimental ones. This is due to the B3LYP/6-31G* giving a higher level of optimisation than the HF/3-21G method. Table 8 also shows that the chair transition state has a lower activation energy to that of the boat transition state by 8.16 kcal mol-1. This suggests that the Cope rearrangement mechanism would most likely proceed through the chair transition state to that of the boat. Therefore the reaction would take place from the anti2 conformer to the chair transition state and then onto the gauche2 conformer within the Cope rearrangement.

It is also possible to see that the calcualted energies are very close to the literature values with the chair conformation being closer to its literature value to that of the boat.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9: Geometric parameters for the optimised chair and boat structures '''
| || ''' HF/3-21G chair TS''' || ''' B3LYP/6-31G* chair TS''' || '''HF/3-21G boat TS''' || '''B3LYP/6-31G* boat TS'''
|-
| Optimised Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Chair_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Boat_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.02 (C5-C10 or C3-C14)|| 1.97 (C5-C10 or C3-C14) || 2.14 (C3-C4 or C1-C6) || 2.21 (C3-C4 or C1-C6)
|-
| Fragment angle || 120 o (C3-C1-C5) || 120 o (C3-C1-C5) || 122 o (C4-C5-C6) || 122 o (C4-C5-C6)
|-
| Dihedral angle || 55 o (C1-C3-C14-C11)|| 54 o (C1-C3-C14-C11)|| 0 o (C2-C1-C6-C5) || 0 o (C2-C1-C6-C5)
|-
| Numbering of carbons || [[Image:Numbering_chairdh08.jpg|200x200px]]|| [[Image:Numbering_chairdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]]
|}

From studying table 9 it can be seen that the C-C bond breaking/forming length has changed between the two different types of optimisation methods. However the angles between the bonds remains unchanged.

=='''The Diels Alder Cycloaddition'''==

The Diels-Alder reaction is a cycloaddition between a conjugated diene and a dienophile (alkene system), which forms a cyclohexene system. It is often referred to as a [4s+2s] cycloaddition and proceeds through a single cyclic, concerted transition state where 2 σ bonds are formed in between the termini carbons of the 2 п systems. This is shown in the reaction scheme below where ethene is reacting with cis-butadiene.

[[Image:Reaction_scheme2dh08.jpg|300x300px|center]]

The π orbitals of the dienophile form new σ bonds with the π orbitals of the diene. The number of π electrons involved determine whether the reaction proceeds in a concerted stereospecific fashion (allowed) or not (forbidden). The HOMO/LUMO of one fragment interacts with the HOMO/LUMO of the other fragment to form two new bonding and anti-bonding MOs. The nodal properties allow one to predict how the reaction will occur.

Semi-Empirical/AM1 method and the B3LYP/6-31G* method of optimisation will be used on Gaussian to determine particular properites of the reactants to predict the way in which the reaction will proceed.

==='''cis-butadiene and ethylene: reagents'''===

<jmol>
<jmolAppletButton>
<uploadedFileContents>Cis_butadienedh08.mol</uploadedFileContents>
<text>Cis-butadiene</text>
<color>whi-te</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Ethenedh08.mol</uploadedFileContents>
<text>ethylene</text>
<color>white</color>
</jmolAppletButton>
</jmol> were optimised separately using the AM1 semi-empirical method on Gaussview.The cis-butadiene had an energy of 0.04879725 a. u (1.28 x 102 KJ mol-1) and ethylene had an energy of 0.02619027 a. u (68.76 KJ mol-1).

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10: Frontier molecular orbitals of cis-butadiene and ethylene '''
| '''Reagent''' || '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| cis-butadiene || LUMO || [[Image:LUMO1_cis_butadieneDH08.jpg|100x100px]] || [[Image:LUMO2_cis_butadienedh08.jpg|100x100px]]|| Symmetric || +0.017
|-
| cis-butadiene || HOMO || [[Image:HOMO1_cis_butadienedh08.jpg|100x100px]] || [[Image:HOMO2_cis_butadienedh08.jpg|100x100px]]|| Anti-symmetric || -0.343
|-
| ethylene || LUMO || [[Image:LUMO1_ethenedh08.jpg|100x100px]] || [[Image:LUMO2_ethenedh08.jpg|100x100px]]|| Anti-symmetric || +0.052
|-
| ethylene || HOMO || [[Image:HOMO1_ethenedh08.jpg|100x100px]] || [[Image:HOMO2_ethenedh08.jpg|100x100px]]|| Symmetric || -0.387
|}

 
The symmetries of the LUMO and HOMO orbitals for cis-butadiene are opposite of those for the ethylene. This therefore means that the only interactions that can take place are those between ethylene's LUMO and cis-butadiene's HOMO and in addition to this the interaction between cis-butadiene's LUMO and ethylene's HOMO. This is because orbital symmetry has to be conserved (the conservation of orbital symmetry states that the symmetry of an new orbital formed by the overlap of the 2 original orbitals must be the same as the symmetry of the initial orbitals).

==='''cis-butadiene and ethylene: transition state'''===

A <jmol>
<jmolAppletButton>
<uploadedFileContents>Bicyclic_startingdh08.mol</uploadedFileContents>
<text>bicyclo system</text>
<color>white</color>
</jmolAppletButton>
</jmol> was drawn and one of the -CH2-CH2- fragments were removed. Three C=C bonds were added and the previous one was taken away so a <jmol>
<jmolAppletButton>
<uploadedFileContents>Frozen_coorddh08.mol</uploadedFileContents>
<text>guess transition state</text>
<color>white</color>
</jmolAppletButton>
</jmol> was produced.

After this an optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while bonds between the termini of the fragments were frozen at 2.1Å. Following this the resulting file had two separate optimisation and frequency calculations run on it using the TS (Berny) with one using the semi-empirical/AM1 method and the other using the DFT/B3LYP/6-31G* level of theory. Both of these optimisations had the bonds unfrozen and the "derivative" option was chosen instead.

The DFT/B3LYP/6-31G* level of theory was used since it gave more accurate results in the calculations before and the energies it produces are consistent with the rest of the report (semi-empirical/AM1 method gives very different values).

The QST2 optimisation was not used since it required numbering the atoms in both the reactants and products, which took a lot of time to do.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11: Comparison of two methods of optimsations for the transition state '''
| || '''semi-empirical/AM1 method''' || '''DFT/B3LYP/6-31G* method'''
|-
| Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>SE_AM1_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>DFT_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Energy (a. u) || 0.11165608 || -234.54389573
|-
| Bond lengths || [[Image:DistancesSEdh08.jpg|150x150px]]|| [[Image:DistancesDFTdh08.jpg|150x150px]]
|-
|Imaginary frequency (cm-1) || -957 || -525
|-
|Animation || [[Image:-957dh08.gif]] || [[Image:-525dh08.gif]]
|-
| Lowest real frequency (cm-1) || 148 || 136
|-
|Animation || [[Image: 148dh08.gif]] || [[Image: 136dh08.gif]]
|}

The C-C bond forming length between the termini carbons is 0.15 Å larger for the DFT/B3LYP/6-31G* level of theory and this is because this method gives carries out the calculation to a greater degree of accuracy. This larger bond length leads a lower potential energy gradient to the transition state, which gives a lower total energy for the transition state. Therefore the given values below will be the DFT/B3LYP/6-31G* and not the semi-empirical/AM1 method.

A typical sp3 C-C bond length is 1.54 Å and a typical sp2 C=C bond length is 1.34 Å<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref>. Within the optimised transition molecule above the C-C bond lengths are 1.40 Å and the C=C bond length is 1.38 Å. This suggests that the π-bonds are breaking in order for the C-C bonds to form and additionally to this a π-bond is forming in between the two sp3 carbons.

The Van der Waals radius of a C-atom is 1.72 Å<ref name="ref 3">M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer, D. G. Truhlar, 1995, 346, pp. 63-69.[http://pubs.acs.org/doi/abs/10.1021/jp8111556]</ref>. The partly formed C-C length is 2.27 Å, which is shorter than two times the the van der Waals radius of a carbon atom. Therefore the carbon atoms are within a close enough distance to each other to share electron density and yet the bond has not formed at this point.

From looking at the animations for the imaginary frequency above it can be seen that the formation of the two bonds between the termini of the reactants is synchronous. This is expected of the Diels-Alder reaction since the mechanism is concerted. However the lowest real frequency is asynchronous with respect to the two bonds forming between the termini.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12: Frontier molecular orbitals of the transition state '''
| '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| LUMO || [[Image:LUMO1TSdh08.jpg|100x100px]] || [[Image:LUMO2TSdh08.jpg|100x100px]]|| Symmetric || +0.023
|-
| HOMO || [[Image:HOMO1TSdh08.jpg|100x100px]] || [[Image:HOMO2TSdh08.jpg|100x100px]]|| Anti-symmetric || -0.323
|}

From examining the images from table 12 above it can be deduced that the LUMO of the transition state originates from the cis-butadiene's LUMO and the ethylene's HOMO, which represents the σ*-antibonding orbitals. However the HOMO of the transition state originates from the cis-butadiene's HOMO and the ethylene's LUMO, which represents the σ-bonding orbitals within the two single C-C bonds that are formed within the reaction.

==='''cis-butadiene and ethylene: activation energy'''===

The cis-butadiene and ethylene were re-optimised at the DFT/B3LYP/6-31G* level of theory so the activation energy of the reaction could be calculated.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 13: Calculation for the activation energy for the reaction'''
| '''molecule''' || '''DFT/B3LYP/6-31G* energy (a. u)''' || '''Calculations''' || '''Energy '''
|-
| cis-butadiene || -155.98595591 || transition state || -234.54389573 a. u
|-
| ethylene || -78.58745828 || ΔE, activation energy || 0.02951846 a. u
|-
| cis-butadiene and ethylene combined || -234.57341419 || ΔE, activation energy || 77.50 KJ mol-1
|}

The calculated activation energy is 77.50 KJ mol-1, which is of the right magnitude when compared to the literature value (115 ± 8 KJ mol-1 at 0 K) but there is still an error of 37.50 KJ mol-1. This shows that Gaussview is not perfect but still gives an idea of the activation energy required for the reaction to take place.

==='''Cyclohexa-1,3-diene reaction with maleic anhydride'''===

The reaction scheme below shows the stereoselective Diels-Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene. The reaction is kinetically controlled with the major product being the endo product, which will be explained later on in the exercise.

[[Image:Reaction_schemedh081.jpg|center|500x500px]]

The <jmol>
<jmolAppletButton>
<uploadedFileContents>FrozenTS22dh08.mol</uploadedFileContents>
<text>exo</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_frozenTS22dh08.mol</uploadedFileContents>
<text>endo</text>
<color>white</color>
</jmolAppletButton>
</jmol> transition states were drawn on Gaussview and then the optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while the distance of the forming bonds were frozen at 2.1Å between the respective carbons. After this the structures changed slightly (<jmol>
<jmolAppletButton>
<uploadedFileContents>TS2_afteropt1_exo1dh08.mol</uploadedFileContents>
<text>exo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_TS2_afteropt1dh08.mol</uploadedFileContents>
<text>endo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol>) and the resulting files had an optimisation and frequency calculation run on it using the TA (Berny) with using the semi-empirical/AM1 level of theory while the unfrozen option as changed to "derivative". After this the endo and exo optimised transition state structures were re-optimised as they were before but using the DFT/B3LYP/6-31G* level of theory instead with the additional keyword "Opt=NoEigen".

{| class="wikitable" border="1" align="center"
|+ '''Table 14: Comparison of the exo and endo transition states '''
! rowspan="2"|-!! colspan="2"|''exo'' !! colspan="2"|''endo''
|-
| Semi-empirical/AM1 || DFT/B3LYP/6-31G* || Semi-empirical/AM1 || DFT/B3LYP/6-31G*
|-
| align="center"|Structure ||align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>After_opt2_exodh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>Endo_TS2_afteropt2dh081.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| align="center"| Energy (a. u) ||align="center"| -0.05041975 ||align="center"| -612.67931095 ||align="center"| -0.05150470 ||align="center"| -612.68339684
|-
| C-C bond forming length (Å) ||align="center"| 2.17 ||align="center"| 2.29 ||align="center"| 2.16 ||align="center"| 2.27
|-align="center"
| Imaginary frequency (cm-1) ||align="center"| -812 ||align="center"| -448 ||align="center"| -806 ||align="center"| -447
|-align="center"
| Animation ||colspan="2"|[[Image:-812_SEdh08.gif]] || colspan="2"|[[Image:-806_cm-1dh08.gif]]
|}

Again the C-C bond forming length between the termini carbons is larger (by 0.11 and 0.12 Å) for the DFT/B3LYP/6-31G* level of theory and this is due the same reasons talked about above. However the C-C bond forming length between the termini carbons are very similar in both the endo and exo transition states (0.02 Å difference).

From looking at the animations for the imaginary frequency above it can also be seen that the formation of the two bonds between the carbons in this case as well is synchronous. Again this is expected of the Diels-Alder reaction since it is concerted.

{| align=center
|+ '''Strain diagrams for the endo and exo transition states'''
| [[Image:Exostraindh08.jpg|thumb|200x200px|'''Exo TS''']]
| [[Image:Strainendodh08.jpg|thumb|200x200px|'''Endo TS''']]
|}

From studying the energies in table 14 it can be seen that the endo transition state is more stable than the exo by 4.08589 x 10-3 a. u (10.73 KJ mol-1). However by looking at the strain of both transition states one would expect the exo TS to be more stable. This is because the distance between the cyclohexa-1,3-diene ring and the -(C=O)-O-(C=O)- maleic anhydride fragment is 2.99 Å in the endo TS and 3.03 Å in the exo TS. Thus the endo form would have more strain due to steric hindrance.

However within the exo TS the vicinal hydrogens on the CH2-CH2 (carbon numbers 18 and 21) fragment on the cyclohexa-1,3-diene ring are found below the plane and next to the maleic anhydride. This leads to steric hindrance is this is where the strain originates from within the exo TS. Therefore this disfavours the exo form, which makes the endo more stable.

Another factor making the endo TS more favourable is secondary orbital overlap. These orbitals contribute to the bonds forming and they interact with each other through space. Within the endo TS below there are 2 more orbitals interacting (secondary orbitals) than within the exo TS. This therefore makes the endo TS more stable and therefore more favoured.

[[Image:Secondary_orbital_shizzledh08.jpg|500x500px|center]]

{| class="wikitable" border="1" align="center"
|+ '''Table 15: Frontier molecular orbitals of the the transition state for the cyclohexa-1,3-diene reaction '''
! rowspan="2"|Molecular Orbital !! colspan="4"|''exo'' !! colspan="4"|''endo''
|- align="center"
| '''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''||'''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''
|- align="center"
| LUMO || [[Image:LUMO1exoh081.jpg|150x150px]]|| [[Image:LUMO2exodh081.jpg|150x150px]]|| Anti-symmetric || -0.078 || [[Image:LUMO1ENDOdh08.jpg|150x150px]]||[[Image:LUMO2ENDOdh08.jpg|150x150px]] || Anti-symmetric || -0.067
|- align="center"
| HOMO || [[Image:HOMO1exodh081.jpg|150x150px]]||[[Image:HOMO2exodh081.jpg|150x150px]] || Anti-symmetric|| -0.242 || [[Image:HOMO1ENDOdh08.jpg|150x150px]]|| [[Image:HOMO2ENDOdh08.jpg|150x150px]]||Anti-symmetric || -0.242
|}

Digital repositories:

exo: http://hdl.handle.net/10042/to-5596

endo: http://hdl.handle.net/10042/to-5597

From examining table 15 it is clear to see that both the HOMO and LUMO are anti-symmetric for both the transition states. The LUMO of the maleic anhydride fragment is the π* anti-bonding orbital of the C=C bond. Within the reaction then this has electrons transferred to it and the C=C bond breaks, which leaves just a C-C bond. The LUMOs for both transition states are both anti-bonding with respect to the two fragments. The HOMOs of both transitions state represent the σ-bonding that will be formed between the termini carbons on the products and reactants. This is most likely from the overlap of the cyclohexa-1,3-diene's HOMO and the maleic anhydride's LUMO.

=='''Conclusion'''==

='''References'''=
<references/>

Rep:Mod:hampluscheese

2010-11-11T12:45:11Z

Dh08: /* '''The Diels Alder Cycloaddition''' */

='''Module 3:Physical Computational Chemistry Lab'''=

Douglas Hunt

=='''Objectives'''==

The aim of this experiment is to

=='''The Cope Rearrangement'''==

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangement] is a [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement |3,3|-sigmatropic rearrangement] that takes place through a single [http://en.wikipedia.org/wiki/Pericyclic pericyclic], concerted [http://en.wikipedia.org/wiki/Transition_state transition state]. Within this rearrangement a σ bond is formed in concert with another being broken within a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] π system.

The Cope rearrangement we will be studying will be that of 1,5-hexadiene, which is shown below.

[[Image:Cope_rearrangement_schemedh08.jpg|300x300px|center]]

The objective is to locate the low-energy minima and transition structures on the C6H10 potential energy surface and to determine the preferred reaction mechanism by using [http://en.wikipedia.org/wiki/GAUSSIAN Gaussian], with the [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)] methods.

==='''Optimisation of 1,5-hexadiene Conformers'''===

Rough <jmol>
<jmolAppletButton>
<uploadedFileContents>Anti_2_drawndh08.mol</uploadedFileContents>
<text>anti</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche3_drawndh08.mol</uploadedFileContents>
<text>gauche</text>
<color>white</color>
</jmolAppletButton>
</jmol> conformers of 1,5-hexadiene were drawn on Gaussview and their structures were tidied by the clean function. They were then further optimised by the HF/3-21G method and were identified which conformer they were from their point group and appendix 1 on the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3 physical lab page].

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 1: Optimisation and comparison of anti and gauche 1,5-hexadiene conformers using HF/3-21G methods
| ''' Conformer''' || '''Optimised Structure''' || '''Energy (Atomic Units)''' || '''[http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1 Energies (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (Anti2 - Gauche3)''' || '''Point group'''
|-
| '''Anti2''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Anti_2_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692535 || -231.69254 ||-6.08 x 105 || 1.2606 x 10-4 a. u|| Ci
|-
| '''Gauche3''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Gauche3_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692661 || -231.69266 || -6.08 x 105 || 3.31 x 10-1 KJ mol-1 || C1
|}

From studying table 1 it can be seen that the gauche conformer is 0.33 KJ mol-1 more stable than the anti conformation. One would assume that the anti2 conformer would be more stable due to [http://en.wikipedia.org/wiki/Steric steric] factors/repulsion within the gauche3 conformation. However within the gauche3 conformer there is a favourable donation of [http://en.wikipedia.org/wiki/Electron_density electron density] from the C=C π orbital to the C-H σ* orbital from the adjacent proton, which stabilises the conformer<ref name="ref 1">B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446.[http://www.informaworld.com/smpp/content~db=all?content=10.1080/00268970110081412]</ref>.

 

The anti2 conformer was then optimised using the B3LYP 6-31G* (6-31G (d)) method.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 2: Comparison of the anti2 conformer optimisations carried out at different calculation methods
| ''' Method''' || '''Structure and bond distances (Å)''' || '''Dihedral angle (C1 to C4)''' || '''Energy (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (HF/3-21G - DFT/B3LYP/6-31G)'''
|-
| HF/3-21G || [[Image:Anti2_opt1dh08.jpg|300x300px]] || 114.6 o || -231.692535 || -6.08 x 105 || 2.919171 a. u
|-
| DFT/B3LYP/6-31G* || [[Image:Anti2_opt2DH08.jpg|300x300px]] || 118.8 o || -234.611716 || -6.16 x 105 || 7.66 x 103 KJ mol-1
|}

From examining table 2 at first it is possible to see that the bond lengths are very similar to each other and that of the literature bond lengths<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref> (C1=C2: 1.340 Å, C2-C3: 1.508 Å and C3-C4: 1.538 Å). It is also possible to see that the C=C bond length is more accurate for the DFT/B3LYP/6-31G* method by it being 0.01 Å closer to the literature value. Another main difference in terms of geometry is that the DFT/B3LYP/6-31G* method gives a larger [http://en.wikipedia.org/wiki/Dihedral_angle dihedral angle] (and most likely more accurate) of 118.8o compared to that of 114.6o.

In terms of energy the DFT/B3LYP/6-31G* method produces a conformer of a higher stability by it possessing a lower energy with a difference of 7.66 x 103 KJ mol-1.

==='''Frequency analysis of 1,5-hexadiene Conformers'''===

[[Image:Vibrations_anti2dh08.jpg|right|300x300px]]

For the frequency analysis the DFT/B3LYP/6-31G* method was used.

Frequency analysis is essential within this calculation due to it being the second derivative of the potential energy surface. This means that if all the frequencies are all positive the system is at a minimum, if one of them is negative the optimised system is at a transition state, and if more than one are negative then the calculation has failed to find a critical point, which means the optimisation has failed. This therefore gives us a method of seeing whether the calculation has worked.

From looking to the image on the right it can deduced that the optimisation has taken place successfully by there being no negative vibrations and therefore the conformation is at a minimum.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 3: Energies under the thermochemistry section at 298.15 K and 0 K within the frequency logfiles'''
| '''Energies''' || '''298.15 K (atomic units)''' ||| '''0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (KJ mol-1)''' || '''Discussion'''
|-
| Sum of electronic and zero-point energies || -234.469177 || -234.466673 || -2.504 x 10-3 || -6.57 || This is the potential energy at 0 K, which includes the zero-point vibrational energy (E = Eelec + ZPE).

|-
| Sum of electronic and thermal energies || -234.461835 || -234.461085 || -7.5 x 10-4|| -1.97 || This is the energy at 298.15 K (0 K for 0 K calculation) and 1 atm of pressure, which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans).

|-
| Sum of electronic and thermal enthalpies || -234.460891 || -234.460141 || -7.5 x 10-4 || -1.97 || This contains an additional correction for RT (H = E + RT), which is especially important when looking at dissociation reactions.
|-
| Sum of electronic and thermal free energies || -234.500739 || -234.495931 || 4.808 x 10-3 || -12.62 || This includes the entropic contribution to the free energy (G = H - TS).
|}

{|align=right
|+
| [[Image:Anti2_freq_IRdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 298.15 K''']]
| [[Image:Anti2_freq_0Kdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 0 K''']]
|}

From looking at table 3 above it can be deduced that the anti2 conformer is more stable at room temperature to that of 0 K with respect to all the types of energies. The spectra also seem to not change much between the two temperatures but the main change is the change in the sum of electronic and thermal free energies, which is heavily dependent of temperature by it including the entropic contribution to the free energy (G = H - '''T'''S)

 
 
 
 

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

Within the Cope Rearrangement there are two possible transition states, the chair (C2h) and the boat (C2v), which are shown within [https://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_2 appendix two ].

==='''Comparison of optimisation methods for the chair transition state'''===

An <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2hdh08.mol</uploadedFileContents>
<text>allyl fragment</text>
<color>white</color>
</jmolAppletButton>
</jmol>, CH2CHCH2 was drawn on Gaussview and was optimized through the HF/3-21G method. After this the original allyl fragment was copied and pasted into a new window and place exactly above the first fragment with a bond distance of 2.20 Å between the terminal end carbons. This therefore produced the <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2h_fulldh08.mol</uploadedFileContents>
<text>chair transition structure.</text>
<color>white</color>
</jmolAppletButton>
</jmol>

Gaussian was then set up for an optimisation for the transition state with the additional key words "Opt=NoEigen", the optimisation set to TS (Berny), the force constants set to "once" and using the HF/3-21G method.

After this the "chair" transition state was optimised an alternative way by freezing the coordinates of the two sets of terminal carbons (where the bonds form/break during the rearrangement) within the Redundant Coord Editor on Gauusview.
Following this a second calculation was run where the freeze coordinates option was changed to "Derivative" and the force constants were set to "never".

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 4: Comparison of chair transition state optimisation methods
| || '''Optimised through a TS (Berny)''' || '''Optimised through a frozen coordinate method'''
|-
| '''Optimised TS Structure''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>C2h_optfreq_TS_Bdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| '''Terminal C-C bond length (Å)''' || 2.02 || 2.02
|-
|'''Energy (atomic units)''' || -231.61932247 || -231.61932190
|-
| '''ΔE, TS (Berny) - frozen coordinate''' || 5.7 x 10-7 a.u || 1.50 x 10-3 KJ mol-1
|-
| '''Imaginary frequency (cm-1)''' || -818 || -818
|-
| '''Animation''' ||[[Image:818dh08TSB.gif|350x350px]] || [[Image:818cm-1Fdh08.gif]]
|}

The data in table 4 shows that both methods give the same results and they both give the same transition state. They both produced the same energy with a negligible energy difference between them and the bond lengths between the terminal carbons are both 2.02 Å. In addition to this the imaginary frequencies were both -818cm-1 and within the animations above it is possible to see that when one C-C bond is forming the other is breaking. This means it is a asynchronous bond formation, which is expected for the Cope Rearrangement since it is a pericyclic reaction.

The only difference between the two optimisations is the way in which the transition states were calculated. This is due to the TS (Berny) method calculating the force constant matrix within the first optimisation step and is then being recalculated as the optimisation carries on. When the frozen coordinate method freezes the reaction coordinate in order for the rest of the structure to be optimised around it, which does not require the whole hessian (force constant matrix) to be calculated but only that the reaction coordinate is differentiated (second step). The main advantage of this, is that you do not need to calculate the entire force constant matrix and just need to differentiate the reaction coordinate in order to obtain a good enough guess for the initial force constant matrix. The frozen coordinate method will therefore take less time to compute the transition state, which can be very expensive within certain systems. However to carry out this method one needs to have an idea of how far apart the terminal carbons are.

==='''Boat transition state'''===

===='''QST2 optimisation'''====

The optimisation for the boat transition state had a different method and this was the QST2 method. This calculation interpolates between two structures (product and reactant) and tries to find a transition state between them.

The reactants and products were numbered in the same way. Therefore both 1,5-hexadiene molecules had their numbering manually changed for the product molecule so it corresponded to the numbering of the reactant if it rearranged. This is illustrated in the reaction scheme below.

[[Image:Boat_rerrangmentdh08.jpg|300x300px|center]]

The chk file for the optimized anti2 conformer was opened and copied into a new window. A second window was opened within the new window and the same anti2 conformer was copied into the new window. The numbering was changed within the second molecule so the two looked like they do in the image below.

[[Image:TS_(QST2)_numbering_1dh08.jpg|400x400px|center]]

The job type was set as "Opt+Freq" and "TS (QST2)" was chosen to optimise the transition state. However this failed and produced
a structure a bit like a <jmol>
<jmolAppletButton>
<uploadedFileContents>Trialone_BPdh08.mol</uploadedFileContents>
<text>chair transition structure </text>
<color>white</color>
</jmolAppletButton>
</jmol> but more dissociated. This is because it interpolated between the two structures and translated the top allyl fragment and therefore did not consider the possibility of a rotation around the central bonds.

To get around this the C-C-C-C dihedral angle (C2-C3-C4-C5) was set to 0 o and the C-C-C angle (C2-C3-C4 and C3-C4-C5 for the product molecule) was set to 100 o for both molecules. This gave rise to the two structures below.

[[Image:TS_(QST2)_numbering_2dh08.jpg|400x400px|center]]

After this the QST2 calculation was run again and the results in table 5 were produced. The imaginary frequency at -839 cm-1 shows that while one bond is forming the other one is breaking, which is again an asynchronous bond formation. It is also possible to see that the boat transition state has a longer terminal C-C bond length by 0.12 Å to that of the chair.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5: Data for the boat transition state from the QST2 optimisation method
| Optimised TS Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.14
|-
|Energy (atomic units) || -231.60280151
|-
| Energy (KJ mol-1) || -6.08 x 105
|-
| Imaginary frequency (cm-1) || -839
|-
| Animation ||[[Image:-839dh08.gif]]
|}

===='''Intrinsic Reaction Coordinate'''====

It is not possible to predict which product the chair and boat transition state will lead to. To solve this problem within Gaussian one can use the IRC method, which allows one to follow the minimum energy path from a transition structure down to its local minimum on a potential energy surface. This creates a succession of points by taking small geometry steps in the direction to where the gradient of the energy surface is at its steepest until the minimum in energy is reached.

The chk file for the optimized chair transition structure was opened and the job type selected was "IRC", the reaction coordinate was computed in the forward direction (since molecule is symmetric) with 50 points along the IRC. This was done with two type of force constants being calculated, once and always.

Within the "once" method the output showed no bond formation process by the 26th step and therefore the calculation was incomplete, since a complete 1,5-hexadiene structure was supposed to be formed. From looking at the IRC pathway it can be seen that the minimum on the potential energy surface was not reached at that point, since the points on the pathway did not form an asymptote and the gradient did not go close enough to zero.

For the "always" method the computation of force constants were carried out at every step. This therefore lets the IRC calculation distinguish whether it is moving along the minimum energy pathway while going down the potential energy surface. While this is happening it can make changes to the pathway if necessary. Within the "always" case the bond forming arrives at 1,5-hexadiene and this is backed up by the IRC pathway reaching a minimum within its energy by the gradient going close to zero and the pathway reaches an asymptote.

The structure produced through the always method had an energy of -231.69199434 atomic units, which is very close to the energy of the gauche2 conformer in the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 appendix 1]. This suggests that the reaction takes place through the chair transition state, which later produces the gauche2 conformer of 1,5-hexadiene.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 6: Comparison of IRC calculations for the chair structure
| || '''Calculate force constants once''' || '''Calculate force constants always'''
|-
| '''Number of points along IRC''' || 26 || 47
|-
| '''Structure at the last point of IRC''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Once_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Always_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|'''Energy (atomic units)''' || -231.68906923 || -231.69199434
|-
| '''ΔE, once - always''' || 2.92511 x 10-3 a.u || 7.68 KJ mol-1
|-
| '''IRC pathway''' || [[Image:One_pathwaydh08.jpg|400x400px]] || [[Image:Always_pathwaydh08.jpg|400x400px]]
|-
| '''IRC gradient''' || [[Image:One_gradientdh08.jpg|400x400px]]|| [[Image:Always_gradientdh08.jpg|400x400px]]
|}

 
 
 

===='''Comparison of activation energies for the transition states'''====

The HF/3-21G optimised chair and boat transition structures were both reoptimised using the B3LYP/6-31G* (B3LYP/6-31G (d)) method. This gives a more accurate minimisation however this method takes longer to run. The values for the zero-point energies and thermal energies were found within the thermochemistry section of the log file.

The values for the HF/3-21G reactant (anti2) optimisation were calculated and the B3LYP/6-31G* values were taken from the "Frequency analysis of 1,5-hexadiene Conformers" earlier in this module.

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 7: Energy data for chair, boat and the anti2 conformation from the HF/3-21G and B3LYP/6-31G* methods of optimisation '''
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466677
| align="center" | -231.461320
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450936
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402342
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539524
| align="center" | -231.532562
| align="center" | -234.611716
| align="center" | -234.469177
| align="center" | -234.461835
|}
Digital repositories

Chair:
HF/3-21G: http://hdl.handle.net/10042/to-5569
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5568

Boat:
HF/3-21G: http://hdl.handle.net/10042/to-5570
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5571

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 8: Summary of activation energies for the chair and boat conformations (ΔE = chair TS or boat TS minus anti2)'''
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| align="center" | 1 a.u = 627.509 kcal mol-1
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE, chair (a.u)'''
| align="center" | 0.072847
| align="center" | 0.071242
| align="center" | 0.054248
| align="center" | 0.052827
| align="center" | -
|-
| '''ΔE, chair (kcal mol-1)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE, boat (a. u)'''
| align="center" | 0.088588
| align="center" | 0.087259
| align="center" | 0.066835
| align="center" | 0.065827
| align="center" | -
|-
| '''ΔE, boat (kcal mol-1)'''
| align="center" | 55.59
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.31
| align="center" | 44.7 ± 2.0
|}

By examining table 8 it is clear to see that the B3LYP/6-31G* method gives values which are much closer to the experimental ones. This is due to the B3LYP/6-31G* giving a higher level of optimisation than the HF/3-21G method. Table 8 also shows that the chair transition state has a lower activation energy to that of the boat transition state by 8.16 kcal mol-1. This suggests that the Cope rearrangement mechanism would most likely proceed through the chair transition state to that of the boat. Therefore the reaction would take place from the anti2 conformer to the chair transition state and then onto the gauche2 conformer within the Cope rearrangement.

It is also possible to see that the calcualted energies are very close to the literature values with the chair conformation being closer to its literature value to that of the boat.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9: Geometric parameters for the optimised chair and boat structures '''
| || ''' HF/3-21G chair TS''' || ''' B3LYP/6-31G* chair TS''' || '''HF/3-21G boat TS''' || '''B3LYP/6-31G* boat TS'''
|-
| Optimised Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Chair_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Boat_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.02 (C5-C10 or C3-C14)|| 1.97 (C5-C10 or C3-C14) || 2.14 (C3-C4 or C1-C6) || 2.21 (C3-C4 or C1-C6)
|-
| Fragment angle || 120 o (C3-C1-C5) || 120 o (C3-C1-C5) || 122 o (C4-C5-C6) || 122 o (C4-C5-C6)
|-
| Dihedral angle || 55 o (C1-C3-C14-C11)|| 54 o (C1-C3-C14-C11)|| 0 o (C2-C1-C6-C5) || 0 o (C2-C1-C6-C5)
|-
| Numbering of carbons || [[Image:Numbering_chairdh08.jpg|200x200px]]|| [[Image:Numbering_chairdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]]
|}

From studying table 9 it can be seen that the C-C bond breaking/forming length has changed between the two different types of optimisation methods. However the angles between the bonds remains unchanged.

=='''The Diels Alder Cycloaddition'''==

The Diels-Alder reaction is a cycloaddition between a conjugated diene and a dienophile (alkene system), which forms a cyclohexene system. It is often referred to as a [4s+2s] cycloaddition and proceeds through a single cyclic, concerted transition state where 2 σ bonds are formed in between the termini carbons of the 2 п systems. This is shown in the reaction scheme below where ethene is reacting with cis-butadiene.

[[Image:Reaction_scheme2dh08.jpg|300x300px|center]]

The π orbitals of the dienophile form new σ bonds with the π orbitals of the diene. The number of π electrons involved determine whether the reaction proceeds in a concerted stereospecific fashion (allowed) or not (forbidden). The HOMO/LUMO of one fragment interacts with the HOMO/LUMO of the other fragment to form two new bonding and anti-bonding MOs. The nodal properties allow one to predict how the reaction will occur.

Semi-Empirical/AM1 method and the B3LYP/6-31G* method of optimisation will be used on Gaussian to determine particular properites of the reactants to predict the way in which the reaction will proceed.

==='''cis-butadiene and ethylene: reagents'''===

<jmol>
<jmolAppletButton>
<uploadedFileContents>Cis_butadienedh08.mol</uploadedFileContents>
<text>Cis-butadiene</text>
<color>whi-te</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Ethenedh08.mol</uploadedFileContents>
<text>ethylene</text>
<color>white</color>
</jmolAppletButton>
</jmol> were optimised separately using the AM1 semi-empirical method on Gaussview.The cis-butadiene had an energy of 0.04879725 a. u (1.28 x 102 KJ mol-1) and ethylene had an energy of 0.02619027 a. u (68.76 KJ mol-1).

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10: Frontier molecular orbitals of cis-butadiene and ethylene '''
| '''Reagent''' || '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| cis-butadiene || LUMO || [[Image:LUMO1_cis_butadieneDH08.jpg|100x100px]] || [[Image:LUMO2_cis_butadienedh08.jpg|100x100px]]|| Symmetric || +0.017
|-
| cis-butadiene || HOMO || [[Image:HOMO1_cis_butadienedh08.jpg|100x100px]] || [[Image:HOMO2_cis_butadienedh08.jpg|100x100px]]|| Anti-symmetric || -0.343
|-
| ethylene || LUMO || [[Image:LUMO1_ethenedh08.jpg|100x100px]] || [[Image:LUMO2_ethenedh08.jpg|100x100px]]|| Anti-symmetric || +0.052
|-
| ethylene || HOMO || [[Image:HOMO1_ethenedh08.jpg|100x100px]] || [[Image:HOMO2_ethenedh08.jpg|100x100px]]|| Symmetric || -0.387
|}

 
The symmetries of the LUMO and HOMO orbitals for cis-butadiene are opposite of those for the ethylene. This therefore means that the only interactions that can take place are those between ethylene's LUMO and cis-butadiene's HOMO and in addition to this the interaction between cis-butadiene's LUMO and ethylene's HOMO. This is because orbital symmetry has to be conserved (the conservation of orbital symmetry states that the symmetry of an new orbital formed by the overlap of the 2 original orbitals must be the same as the symmetry of the initial orbitals).

==='''cis-butadiene and ethylene: transition state'''===

A <jmol>
<jmolAppletButton>
<uploadedFileContents>Bicyclic_startingdh08.mol</uploadedFileContents>
<text>bicyclo system</text>
<color>white</color>
</jmolAppletButton>
</jmol> was drawn and one of the -CH2-CH2- fragments were removed. Three C=C bonds were added and the previous one was taken away so a <jmol>
<jmolAppletButton>
<uploadedFileContents>Frozen_coorddh08.mol</uploadedFileContents>
<text>guess transition state</text>
<color>white</color>
</jmolAppletButton>
</jmol> was produced.

After this an optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while bonds between the termini of the fragments were frozen at 2.1Å. Following this the resulting file had two separate optimisation and frequency calculations run on it using the TS (Berny) with one using the semi-empirical/AM1 method and the other using the DFT/B3LYP/6-31G* level of theory. Both of these optimisations had the bonds unfrozen and the "derivative" option was chosen instead.

The DFT/B3LYP/6-31G* level of theory was used since it gave more accurate results in the calculations before and the energies it produces are consistent with the rest of the report (semi-empirical/AM1 method gives very different values).

The QST2 optimisation was not used since it required numbering the atoms in both the reactants and products, which took a lot of time to do.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11: Comparison of two methods of optimsations for the transition state '''
| || '''semi-empirical/AM1 method''' || '''DFT/B3LYP/6-31G* method'''
|-
| Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>SE_AM1_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>DFT_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Energy (a. u) || 0.11165608 || -234.54389573
|-
| Bond lengths || [[Image:DistancesSEdh08.jpg|150x150px]]|| [[Image:DistancesDFTdh08.jpg|150x150px]]
|-
|Imaginary frequency (cm-1) || -957 || -525
|-
|Animation || [[Image:-957dh08.gif]] || [[Image:-525dh08.gif]]
|-
| Lowest real frequency (cm-1) || 148 || 136
|-
|Animation || [[Image: 148dh08.gif]] || [[Image: 136dh08.gif]]
|}

The C-C bond forming length between the termini carbons is 0.15 Å larger for the DFT/B3LYP/6-31G* level of theory and this is because this method gives carries out the calculation to a greater degree of accuracy. This larger bond length leads a lower potential energy gradient to the transition state, which gives a lower total energy for the transition state. Therefore the given values below will be the DFT/B3LYP/6-31G* and not the semi-empirical/AM1 method.

A typical sp3 C-C bond length is 1.54 Å and a typical sp2 C=C bond length is 1.34 Å<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref>. Within the optimised transition molecule above the C-C bond lengths are 1.40 Å and the C=C bond length is 1.38 Å. This suggests that the π-bonds are breaking in order for the C-C bonds to form and additionally to this a π-bond is forming in between the two sp3 carbons.

The Van der Waals radius of a C-atom is 1.72 Å<ref name="ref 3">M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer, D. G. Truhlar, 1995, 346, pp. 63-69.[http://pubs.acs.org/doi/abs/10.1021/jp8111556]</ref>. The partly formed C-C length is 2.27 Å, which is shorter than two times the the van der Waals radius of a carbon atom. Therefore the carbon atoms are within a close enough distance to each other to share electron density and yet the bond has not formed at this point.

From looking at the animations for the imaginary frequency above it can be seen that the formation of the two bonds between the termini of the reactants is synchronous. This is expected of the Diels-Alder reaction since the mechanism is concerted. However the lowest real frequency is asynchronous with respect to the two bonds forming between the termini.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12: Frontier molecular orbitals of the transition state '''
| '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| LUMO || [[Image:LUMO1TSdh08.jpg|100x100px]] || [[Image:LUMO2TSdh08.jpg|100x100px]]|| Symmetric || +0.023
|-
| HOMO || [[Image:HOMO1TSdh08.jpg|100x100px]] || [[Image:HOMO2TSdh08.jpg|100x100px]]|| Anti-symmetric || -0.323
|}

From examining the images from table 12 above it can be deduced that the LUMO of the transition state originates from the cis-butadiene's LUMO and the ethylene's HOMO, which represents the σ*-antibonding orbitals. However the HOMO of the transition state originates from the cis-butadiene's HOMO and the ethylene's LUMO, which represents the σ-bonding orbitals within the two single C-C bonds that are formed within the reaction.

==='''cis-butadiene and ethylene: activation energy'''===

The cis-butadiene and ethylene were re-optimised at the DFT/B3LYP/6-31G* level of theory so the activation energy of the reaction could be calculated.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 13: Calculation for the activation energy for the reaction'''
| '''molecule''' || '''DFT/B3LYP/6-31G* energy (a. u)''' || '''Calculations''' || '''Energy '''
|-
| cis-butadiene || -155.98595591 || transition state || -234.54389573 a. u
|-
| ethylene || -78.58745828 || ΔE, activation energy || 0.02951846 a. u
|-
| cis-butadiene and ethylene combined || -234.57341419 || ΔE, activation energy || 77.50 KJ mol-1
|}

The calculated activation energy is 77.50 KJ mol-1, which is of the right magnitude when compared to the literature value (115 ± 8 KJ mol-1 at 0 K) but there is still an error of 37.50 KJ mol-1. This shows that Gaussview is not perfect but still gives an idea of the activation energy required for the reaction to take place.

==='''Cyclohexa-1,3-diene reaction with maleic anhydride'''===

The reaction scheme below shows the stereoselective Diels-Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene. The reaction is kinetically controlled with the major product being the endo product, which will be explained later on in the exercise.

[[Image:Reaction_schemedh081.jpg|center|500x500px]]

The <jmol>
<jmolAppletButton>
<uploadedFileContents>FrozenTS22dh08.mol</uploadedFileContents>
<text>exo</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_frozenTS22dh08.mol</uploadedFileContents>
<text>endo</text>
<color>white</color>
</jmolAppletButton>
</jmol> transition states were drawn on Gaussview and then the optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while the distance of the forming bonds were frozen at 2.1Å between the respective carbons. After this the structures changed slightly (<jmol>
<jmolAppletButton>
<uploadedFileContents>TS2_afteropt1_exo1dh08.mol</uploadedFileContents>
<text>exo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_TS2_afteropt1dh08.mol</uploadedFileContents>
<text>endo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol>) and the resulting files had an optimisation and frequency calculation run on it using the TA (Berny) with using the semi-empirical/AM1 level of theory while the unfrozen option as changed to "derivative". After this the endo and exo optimised transition state structures were re-optimised as they were before but using the DFT/B3LYP/6-31G* level of theory instead with the additional keyword "Opt=NoEigen".

{| class="wikitable" border="1" align="center"
|+ '''Table 14: Comparison of the exo and endo transition states '''
! rowspan="2"|-!! colspan="2"|''exo'' !! colspan="2"|''endo''
|-
| Semi-empirical/AM1 || DFT/B3LYP/6-31G* || Semi-empirical/AM1 || DFT/B3LYP/6-31G*
|-
| align="center"|Structure ||align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>After_opt2_exodh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>Endo_TS2_afteropt2dh081.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| align="center"| Energy (a. u) ||align="center"| -0.05041975 ||align="center"| -612.67931095 ||align="center"| -0.05150470 ||align="center"| -612.68339684
|-
| C-C bond forming length (Å) ||align="center"| 2.17 ||align="center"| 2.29 ||align="center"| 2.16 ||align="center"| 2.27
|-align="center"
| Imaginary frequency (cm-1) ||align="center"| -812 ||align="center"| -448 ||align="center"| -806 ||align="center"| -447
|-align="center"
| Animation ||colspan="2"|[[Image:-812_SEdh08.gif]] || colspan="2"|[[Image:-806_cm-1dh08.gif]]
|}

Again the C-C bond forming length between the termini carbons is larger (by 0.11 and 0.12 Å) for the DFT/B3LYP/6-31G* level of theory and this is due the same reasons talked about above. However the C-C bond forming length between the termini carbons are very similar in both the endo and exo transition states (0.02 Å difference).

From looking at the animations for the imaginary frequency above it can also be seen that the formation of the two bonds between the carbons in this case as well is synchronous. Again this is expected of the Diels-Alder reaction since it is concerted.

{| align=center
|+ '''Strain diagrams for the endo and exo transition states'''
| [[Image:Exostraindh08.jpg|thumb|200x200px|'''Exo TS''']]
| [[Image:Strainendodh08.jpg|thumb|200x200px|'''Endo TS''']]
|}

From studying the energies in table 14 it can be seen that the endo transition state is more stable than the exo by 4.08589 x 10-3 a. u (10.73 KJ mol-1). However by looking at the strain of both transition states one would expect the exo TS to be more stable. This is because the distance between the cyclohexa-1,3-diene ring and the -(C=O)-O-(C=O)- maleic anhydride fragment is 2.99 Å in the endo TS and 3.03 Å in the exo TS. Thus the endo form would have more strain due to steric hindrance.

However within the exo TS the vicinal hydrogens on the CH2-CH2 (carbon numbers 18 and 21) fragment on the cyclohexa-1,3-diene ring are found below the plane and next to the maleic anhydride. This leads to steric hindrance is this is where the strain originates from within the exo TS. Therefore this disfavours the exo form, which makes the endo more stable.

Another factor making the endo TS more favourable is secondary orbital overlap. These orbitals contribute to the bonds forming and they interact with each other through space. Within the endo TS below there are 2 more orbitals interacting (secondary orbitals) than within the exo TS. This therefore makes the endo TS more stable and therefore more favoured.

[[Image:Secondary_orbital_shizzledh08.jpg|500x500px|center]]

{| class="wikitable" border="1" align="center"
|+ '''Table 15: Frontier molecular orbitals of the the transition state for the cyclohexa-1,3-diene reaction '''
! rowspan="2"|Molecular Orbital !! colspan="4"|''exo'' !! colspan="4"|''endo''
|- align="center"
| '''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''||'''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''
|- align="center"
| LUMO || [[Image:LUMO1exoh081.jpg|150x150px]]|| [[Image:LUMO2exodh081.jpg|150x150px]]|| Anti-symmetric || -0.078 || [[Image:LUMO1ENDOdh08.jpg|150x150px]]||[[Image:LUMO2ENDOdh08.jpg|150x150px]] || Anti-symmetric || -0.067
|- align="center"
| HOMO || [[Image:HOMO1exodh081.jpg|150x150px]]||[[Image:HOMO2exodh081.jpg|150x150px]] || Anti-symmetric|| -0.242 || [[Image:HOMO1ENDOdh08.jpg|150x150px]]|| [[Image:HOMO2ENDOdh08.jpg|150x150px]]||Anti-symmetric || -0.242
|}

Digital repositories:

exo: http://hdl.handle.net/10042/to-5596

endo: http://hdl.handle.net/10042/to-5597

From examining table 15 it is clear to see that both the HOMO and LUMO are anti-symmetric for both the transition states. The LUMO of the maleic anhydride fragment is the π* anti-bonding orbital of the C=C bond. Within the reaction then this has electrons transferred to it and the C=C bond breaks, which leaves just a C-C bond. The LUMOs for both transition states are both anti-bonding with respect to the two fragments. The HOMOs of both transitions state represent the σ-bonding that will be formed between the termini carbons on the products and reactants. This is most likely from the overlap of the cyclohexa-1,3-diene's HOMO and the maleic anhydride's LUMO.

=='''Conclusion'''==

='''References'''=
<references/>

Rep:Mod:hampluscheese

2010-11-11T12:43:13Z

Dh08: /* '''QST2 optimisation''' */

='''Module 3:Physical Computational Chemistry Lab'''=

Douglas Hunt

=='''Objectives'''==

The aim of this experiment is to

=='''The Cope Rearrangement'''==

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangement] is a [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement |3,3|-sigmatropic rearrangement] that takes place through a single [http://en.wikipedia.org/wiki/Pericyclic pericyclic], concerted [http://en.wikipedia.org/wiki/Transition_state transition state]. Within this rearrangement a σ bond is formed in concert with another being broken within a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] π system.

The Cope rearrangement we will be studying will be that of 1,5-hexadiene, which is shown below.

[[Image:Cope_rearrangement_schemedh08.jpg|300x300px|center]]

The objective is to locate the low-energy minima and transition structures on the C6H10 potential energy surface and to determine the preferred reaction mechanism by using [http://en.wikipedia.org/wiki/GAUSSIAN Gaussian], with the [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)] methods.

==='''Optimisation of 1,5-hexadiene Conformers'''===

Rough <jmol>
<jmolAppletButton>
<uploadedFileContents>Anti_2_drawndh08.mol</uploadedFileContents>
<text>anti</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche3_drawndh08.mol</uploadedFileContents>
<text>gauche</text>
<color>white</color>
</jmolAppletButton>
</jmol> conformers of 1,5-hexadiene were drawn on Gaussview and their structures were tidied by the clean function. They were then further optimised by the HF/3-21G method and were identified which conformer they were from their point group and appendix 1 on the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3 physical lab page].

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 1: Optimisation and comparison of anti and gauche 1,5-hexadiene conformers using HF/3-21G methods
| ''' Conformer''' || '''Optimised Structure''' || '''Energy (Atomic Units)''' || '''[http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1 Energies (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (Anti2 - Gauche3)''' || '''Point group'''
|-
| '''Anti2''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Anti_2_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692535 || -231.69254 ||-6.08 x 105 || 1.2606 x 10-4 a. u|| Ci
|-
| '''Gauche3''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Gauche3_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692661 || -231.69266 || -6.08 x 105 || 3.31 x 10-1 KJ mol-1 || C1
|}

From studying table 1 it can be seen that the gauche conformer is 0.33 KJ mol-1 more stable than the anti conformation. One would assume that the anti2 conformer would be more stable due to [http://en.wikipedia.org/wiki/Steric steric] factors/repulsion within the gauche3 conformation. However within the gauche3 conformer there is a favourable donation of [http://en.wikipedia.org/wiki/Electron_density electron density] from the C=C π orbital to the C-H σ* orbital from the adjacent proton, which stabilises the conformer<ref name="ref 1">B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446.[http://www.informaworld.com/smpp/content~db=all?content=10.1080/00268970110081412]</ref>.

 

The anti2 conformer was then optimised using the B3LYP 6-31G* (6-31G (d)) method.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 2: Comparison of the anti2 conformer optimisations carried out at different calculation methods
| ''' Method''' || '''Structure and bond distances (Å)''' || '''Dihedral angle (C1 to C4)''' || '''Energy (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (HF/3-21G - DFT/B3LYP/6-31G)'''
|-
| HF/3-21G || [[Image:Anti2_opt1dh08.jpg|300x300px]] || 114.6 o || -231.692535 || -6.08 x 105 || 2.919171 a. u
|-
| DFT/B3LYP/6-31G* || [[Image:Anti2_opt2DH08.jpg|300x300px]] || 118.8 o || -234.611716 || -6.16 x 105 || 7.66 x 103 KJ mol-1
|}

From examining table 2 at first it is possible to see that the bond lengths are very similar to each other and that of the literature bond lengths<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref> (C1=C2: 1.340 Å, C2-C3: 1.508 Å and C3-C4: 1.538 Å). It is also possible to see that the C=C bond length is more accurate for the DFT/B3LYP/6-31G* method by it being 0.01 Å closer to the literature value. Another main difference in terms of geometry is that the DFT/B3LYP/6-31G* method gives a larger [http://en.wikipedia.org/wiki/Dihedral_angle dihedral angle] (and most likely more accurate) of 118.8o compared to that of 114.6o.

In terms of energy the DFT/B3LYP/6-31G* method produces a conformer of a higher stability by it possessing a lower energy with a difference of 7.66 x 103 KJ mol-1.

==='''Frequency analysis of 1,5-hexadiene Conformers'''===

[[Image:Vibrations_anti2dh08.jpg|right|300x300px]]

For the frequency analysis the DFT/B3LYP/6-31G* method was used.

Frequency analysis is essential within this calculation due to it being the second derivative of the potential energy surface. This means that if all the frequencies are all positive the system is at a minimum, if one of them is negative the optimised system is at a transition state, and if more than one are negative then the calculation has failed to find a critical point, which means the optimisation has failed. This therefore gives us a method of seeing whether the calculation has worked.

From looking to the image on the right it can deduced that the optimisation has taken place successfully by there being no negative vibrations and therefore the conformation is at a minimum.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 3: Energies under the thermochemistry section at 298.15 K and 0 K within the frequency logfiles'''
| '''Energies''' || '''298.15 K (atomic units)''' ||| '''0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (KJ mol-1)''' || '''Discussion'''
|-
| Sum of electronic and zero-point energies || -234.469177 || -234.466673 || -2.504 x 10-3 || -6.57 || This is the potential energy at 0 K, which includes the zero-point vibrational energy (E = Eelec + ZPE).

|-
| Sum of electronic and thermal energies || -234.461835 || -234.461085 || -7.5 x 10-4|| -1.97 || This is the energy at 298.15 K (0 K for 0 K calculation) and 1 atm of pressure, which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans).

|-
| Sum of electronic and thermal enthalpies || -234.460891 || -234.460141 || -7.5 x 10-4 || -1.97 || This contains an additional correction for RT (H = E + RT), which is especially important when looking at dissociation reactions.
|-
| Sum of electronic and thermal free energies || -234.500739 || -234.495931 || 4.808 x 10-3 || -12.62 || This includes the entropic contribution to the free energy (G = H - TS).
|}

{|align=right
|+
| [[Image:Anti2_freq_IRdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 298.15 K''']]
| [[Image:Anti2_freq_0Kdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 0 K''']]
|}

From looking at table 3 above it can be deduced that the anti2 conformer is more stable at room temperature to that of 0 K with respect to all the types of energies. The spectra also seem to not change much between the two temperatures but the main change is the change in the sum of electronic and thermal free energies, which is heavily dependent of temperature by it including the entropic contribution to the free energy (G = H - '''T'''S)

 
 
 
 

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

Within the Cope Rearrangement there are two possible transition states, the chair (C2h) and the boat (C2v), which are shown within [https://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_2 appendix two ].

==='''Comparison of optimisation methods for the chair transition state'''===

An <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2hdh08.mol</uploadedFileContents>
<text>allyl fragment</text>
<color>white</color>
</jmolAppletButton>
</jmol>, CH2CHCH2 was drawn on Gaussview and was optimized through the HF/3-21G method. After this the original allyl fragment was copied and pasted into a new window and place exactly above the first fragment with a bond distance of 2.20 Å between the terminal end carbons. This therefore produced the <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2h_fulldh08.mol</uploadedFileContents>
<text>chair transition structure.</text>
<color>white</color>
</jmolAppletButton>
</jmol>

Gaussian was then set up for an optimisation for the transition state with the additional key words "Opt=NoEigen", the optimisation set to TS (Berny), the force constants set to "once" and using the HF/3-21G method.

After this the "chair" transition state was optimised an alternative way by freezing the coordinates of the two sets of terminal carbons (where the bonds form/break during the rearrangement) within the Redundant Coord Editor on Gauusview.
Following this a second calculation was run where the freeze coordinates option was changed to "Derivative" and the force constants were set to "never".

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 4: Comparison of chair transition state optimisation methods
| || '''Optimised through a TS (Berny)''' || '''Optimised through a frozen coordinate method'''
|-
| '''Optimised TS Structure''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>C2h_optfreq_TS_Bdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| '''Terminal C-C bond length (Å)''' || 2.02 || 2.02
|-
|'''Energy (atomic units)''' || -231.61932247 || -231.61932190
|-
| '''ΔE, TS (Berny) - frozen coordinate''' || 5.7 x 10-7 a.u || 1.50 x 10-3 KJ mol-1
|-
| '''Imaginary frequency (cm-1)''' || -818 || -818
|-
| '''Animation''' ||[[Image:818dh08TSB.gif|350x350px]] || [[Image:818cm-1Fdh08.gif]]
|}

The data in table 4 shows that both methods give the same results and they both give the same transition state. They both produced the same energy with a negligible energy difference between them and the bond lengths between the terminal carbons are both 2.02 Å. In addition to this the imaginary frequencies were both -818cm-1 and within the animations above it is possible to see that when one C-C bond is forming the other is breaking. This means it is a asynchronous bond formation, which is expected for the Cope Rearrangement since it is a pericyclic reaction.

The only difference between the two optimisations is the way in which the transition states were calculated. This is due to the TS (Berny) method calculating the force constant matrix within the first optimisation step and is then being recalculated as the optimisation carries on. When the frozen coordinate method freezes the reaction coordinate in order for the rest of the structure to be optimised around it, which does not require the whole hessian (force constant matrix) to be calculated but only that the reaction coordinate is differentiated (second step). The main advantage of this, is that you do not need to calculate the entire force constant matrix and just need to differentiate the reaction coordinate in order to obtain a good enough guess for the initial force constant matrix. The frozen coordinate method will therefore take less time to compute the transition state, which can be very expensive within certain systems. However to carry out this method one needs to have an idea of how far apart the terminal carbons are.

==='''Boat transition state'''===

===='''QST2 optimisation'''====

The optimisation for the boat transition state had a different method and this was the QST2 method. This calculation interpolates between two structures (product and reactant) and tries to find a transition state between them.

The reactants and products were numbered in the same way. Therefore both 1,5-hexadiene molecules had their numbering manually changed for the product molecule so it corresponded to the numbering of the reactant if it rearranged. This is illustrated in the reaction scheme below.

[[Image:Boat_rerrangmentdh08.jpg|300x300px|center]]

The chk file for the optimized anti2 conformer was opened and copied into a new window. A second window was opened within the new window and the same anti2 conformer was copied into the new window. The numbering was changed within the second molecule so the two looked like they do in the image below.

[[Image:TS_(QST2)_numbering_1dh08.jpg|400x400px|center]]

The job type was set as "Opt+Freq" and "TS (QST2)" was chosen to optimise the transition state. However this failed and produced
a structure a bit like a <jmol>
<jmolAppletButton>
<uploadedFileContents>Trialone_BPdh08.mol</uploadedFileContents>
<text>chair transition structure </text>
<color>white</color>
</jmolAppletButton>
</jmol> but more dissociated. This is because it interpolated between the two structures and translated the top allyl fragment and therefore did not consider the possibility of a rotation around the central bonds.

To get around this the C-C-C-C dihedral angle (C2-C3-C4-C5) was set to 0 o and the C-C-C angle (C2-C3-C4 and C3-C4-C5 for the product molecule) was set to 100 o for both molecules. This gave rise to the two structures below.

[[Image:TS_(QST2)_numbering_2dh08.jpg|400x400px|center]]

After this the QST2 calculation was run again and the results in table 5 were produced. The imaginary frequency at -839 cm-1 shows that while one bond is forming the other one is breaking, which is again an asynchronous bond formation. It is also possible to see that the boat transition state has a longer terminal C-C bond length by 0.12 Å to that of the chair.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5: Data for the boat transition state from the QST2 optimisation method
| Optimised TS Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.14
|-
|Energy (atomic units) || -231.60280151
|-
| Energy (KJ mol-1) || -6.08 x 105
|-
| Imaginary frequency (cm-1) || -839
|-
| Animation ||[[Image:-839dh08.gif]]
|}

===='''Intrinsic Reaction Coordinate'''====

It is not possible to predict which product the chair and boat transition state will lead to. To solve this problem within Gaussian one can use the IRC method, which allows one to follow the minimum energy path from a transition structure down to its local minimum on a potential energy surface. This creates a succession of points by taking small geometry steps in the direction to where the gradient of the energy surface is at its steepest until the minimum in energy is reached.

The chk file for the optimized chair transition structure was opened and the job type selected was "IRC", the reaction coordinate was computed in the forward direction (since molecule is symmetric) with 50 points along the IRC. This was done with two type of force constants being calculated, once and always.

Within the "once" method the output showed no bond formation process by the 26th step and therefore the calculation was incomplete, since a complete 1,5-hexadiene structure was supposed to be formed. From looking at the IRC pathway it can be seen that the minimum on the potential energy surface was not reached at that point, since the points on the pathway did not form an asymptote and the gradient did not go close enough to zero.

For the "always" method the computation of force constants were carried out at every step. This therefore lets the IRC calculation distinguish whether it is moving along the minimum energy pathway while going down the potential energy surface. While this is happening it can make changes to the pathway if necessary. Within the "always" case the bond forming arrives at 1,5-hexadiene and this is backed up by the IRC pathway reaching a minimum within its energy by the gradient going close to zero and the pathway reaches an asymptote.

The structure produced through the always method had an energy of -231.69199434 atomic units, which is very close to the energy of the gauche2 conformer in the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 appendix 1]. This suggests that the reaction takes place through the chair transition state, which later produces the gauche2 conformer of 1,5-hexadiene.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 6: Comparison of IRC calculations for the chair structure
| || '''Calculate force constants once''' || '''Calculate force constants always'''
|-
| '''Number of points along IRC''' || 26 || 47
|-
| '''Structure at the last point of IRC''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Once_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Always_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|'''Energy (atomic units)''' || -231.68906923 || -231.69199434
|-
| '''ΔE, once - always''' || 2.92511 x 10-3 a.u || 7.68 KJ mol-1
|-
| '''IRC pathway''' || [[Image:One_pathwaydh08.jpg|400x400px]] || [[Image:Always_pathwaydh08.jpg|400x400px]]
|-
| '''IRC gradient''' || [[Image:One_gradientdh08.jpg|400x400px]]|| [[Image:Always_gradientdh08.jpg|400x400px]]
|}

 
 
 

===='''Comparison of activation energies for the transition states'''====

The HF/3-21G optimised chair and boat transition structures were both reoptimised using the B3LYP/6-31G* (B3LYP/6-31G (d)) method. This gives a more accurate minimisation however this method takes longer to run. The values for the zero-point energies and thermal energies were found within the thermochemistry section of the log file.

The values for the HF/3-21G reactant (anti2) optimisation were calculated and the B3LYP/6-31G* values were taken from the "Frequency analysis of 1,5-hexadiene Conformers" earlier in this module.

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 7: Energy data for chair, boat and the anti2 conformation from the HF/3-21G and B3LYP/6-31G* methods of optimisation '''
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466677
| align="center" | -231.461320
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450936
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402342
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539524
| align="center" | -231.532562
| align="center" | -234.611716
| align="center" | -234.469177
| align="center" | -234.461835
|}
Digital repositories

Chair:
HF/3-21G: http://hdl.handle.net/10042/to-5569
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5568

Boat:
HF/3-21G: http://hdl.handle.net/10042/to-5570
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5571

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 8: Summary of activation energies for the chair and boat conformations (ΔE = chair TS or boat TS minus anti2)'''
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| align="center" | 1 a.u = 627.509 kcal mol-1
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE, chair (a.u)'''
| align="center" | 0.072847
| align="center" | 0.071242
| align="center" | 0.054248
| align="center" | 0.052827
| align="center" | -
|-
| '''ΔE, chair (kcal mol-1)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE, boat (a. u)'''
| align="center" | 0.088588
| align="center" | 0.087259
| align="center" | 0.066835
| align="center" | 0.065827
| align="center" | -
|-
| '''ΔE, boat (kcal mol-1)'''
| align="center" | 55.59
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.31
| align="center" | 44.7 ± 2.0
|}

By examining table 8 it is clear to see that the B3LYP/6-31G* method gives values which are much closer to the experimental ones. This is due to the B3LYP/6-31G* giving a higher level of optimisation than the HF/3-21G method. Table 8 also shows that the chair transition state has a lower activation energy to that of the boat transition state by 8.16 kcal mol-1. This suggests that the Cope rearrangement mechanism would most likely proceed through the chair transition state to that of the boat. Therefore the reaction would take place from the anti2 conformer to the chair transition state and then onto the gauche2 conformer within the Cope rearrangement.

It is also possible to see that the calcualted energies are very close to the literature values with the chair conformation being closer to its literature value to that of the boat.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9: Geometric parameters for the optimised chair and boat structures '''
| || ''' HF/3-21G chair TS''' || ''' B3LYP/6-31G* chair TS''' || '''HF/3-21G boat TS''' || '''B3LYP/6-31G* boat TS'''
|-
| Optimised Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Chair_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Boat_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.02 (C5-C10 or C3-C14)|| 1.97 (C5-C10 or C3-C14) || 2.14 (C3-C4 or C1-C6) || 2.21 (C3-C4 or C1-C6)
|-
| Fragment angle || 120 o (C3-C1-C5) || 120 o (C3-C1-C5) || 122 o (C4-C5-C6) || 122 o (C4-C5-C6)
|-
| Dihedral angle || 55 o (C1-C3-C14-C11)|| 54 o (C1-C3-C14-C11)|| 0 o (C2-C1-C6-C5) || 0 o (C2-C1-C6-C5)
|-
| Numbering of carbons || [[Image:Numbering_chairdh08.jpg|200x200px]]|| [[Image:Numbering_chairdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]]
|}

From studying table 9 it can be seen that the C-C bond breaking/forming length has changed between the two different types of optimisation methods. However the angles between the bonds remains unchanged.

=='''The Diels Alder Cycloaddition'''==

The Diels-Alder reaction is a cycloaddition between a conjugated diene and a dienophile (alkene system), which forms a cyclohexene system. It is often referred to as a [4s+2s] cycloaddition and proceeds through a single cyclic, concerted transition state where 2 σ bonds are formed in between the termini carbons of the 2 conjugated п systems. This is shown in the reaction scheme below where ethene is reacting with cis-butadiene.

[[Image:Reaction_scheme2dh08.jpg|300x300px|center]]

The π orbitals of the dienophile form new σ bonds with the π orbitals of the diene. The number of π electrons involved determine whether the reaction proceeds in a concerted stereospecific fashion (allowed) or not (forbidden). The HOMO/LUMO of one fragment interacts with the HOMO/LUMO of the other fragment to form two new bonding and anti-bonding MOs. The nodal properties allow one to predict how the reaction will occur.

Semi-Empirical/AM1 method and the B3LYP/6-31G* method of optimisation will be used on Gaussian to determine particular properites of the reactants to predict the way in which the reaction will proceed.

==='''cis-butadiene and ethylene: reagents'''===

<jmol>
<jmolAppletButton>
<uploadedFileContents>Cis_butadienedh08.mol</uploadedFileContents>
<text>Cis-butadiene</text>
<color>whi-te</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Ethenedh08.mol</uploadedFileContents>
<text>ethylene</text>
<color>white</color>
</jmolAppletButton>
</jmol> were optimised separately using the AM1 semi-empirical method on Gaussview.The cis-butadiene had an energy of 0.04879725 a. u (1.28 x 102 KJ mol-1) and ethylene had an energy of 0.02619027 a. u (68.76 KJ mol-1).

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10: Frontier molecular orbitals of cis-butadiene and ethylene '''
| '''Reagent''' || '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| cis-butadiene || LUMO || [[Image:LUMO1_cis_butadieneDH08.jpg|100x100px]] || [[Image:LUMO2_cis_butadienedh08.jpg|100x100px]]|| Symmetric || +0.017
|-
| cis-butadiene || HOMO || [[Image:HOMO1_cis_butadienedh08.jpg|100x100px]] || [[Image:HOMO2_cis_butadienedh08.jpg|100x100px]]|| Anti-symmetric || -0.343
|-
| ethylene || LUMO || [[Image:LUMO1_ethenedh08.jpg|100x100px]] || [[Image:LUMO2_ethenedh08.jpg|100x100px]]|| Anti-symmetric || +0.052
|-
| ethylene || HOMO || [[Image:HOMO1_ethenedh08.jpg|100x100px]] || [[Image:HOMO2_ethenedh08.jpg|100x100px]]|| Symmetric || -0.387
|}

 
The symmetries of the LUMO and HOMO orbitals for cis-butadiene are opposite of those for the ethylene. This therefore means that the only interactions that can take place are those between ethylene's LUMO and cis-butadiene's HOMO and in addition to this the interaction between cis-butadiene's LUMO and ethylene's HOMO. This is because orbital symmetry has to be conserved (the conservation of orbital symmetry states that the symmetry of an new orbital formed by the overlap of the 2 original orbitals must be the same as the symmetry of the initial orbitals).

==='''cis-butadiene and ethylene: transition state'''===

A <jmol>
<jmolAppletButton>
<uploadedFileContents>Bicyclic_startingdh08.mol</uploadedFileContents>
<text>bicyclo system</text>
<color>white</color>
</jmolAppletButton>
</jmol> was drawn and one of the -CH2-CH2- fragments were removed. Three C=C bonds were added and the previous one was taken away so a <jmol>
<jmolAppletButton>
<uploadedFileContents>Frozen_coorddh08.mol</uploadedFileContents>
<text>guess transition state</text>
<color>white</color>
</jmolAppletButton>
</jmol> was produced.

After this an optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while bonds between the termini of the fragments were frozen at 2.1Å. Following this the resulting file had two separate optimisation and frequency calculations run on it using the TS (Berny) with one using the semi-empirical/AM1 method and the other using the DFT/B3LYP/6-31G* level of theory. Both of these optimisations had the bonds unfrozen and the "derivative" option was chosen instead.

The DFT/B3LYP/6-31G* level of theory was used since it gave more accurate results in the calculations before and the energies it produces are consistent with the rest of the report (semi-empirical/AM1 method gives very different values).

The QST2 optimisation was not used since it required numbering the atoms in both the reactants and products, which took a lot of time to do.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11: Comparison of two methods of optimsations for the transition state '''
| || '''semi-empirical/AM1 method''' || '''DFT/B3LYP/6-31G* method'''
|-
| Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>SE_AM1_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>DFT_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Energy (a. u) || 0.11165608 || -234.54389573
|-
| Bond lengths || [[Image:DistancesSEdh08.jpg|150x150px]]|| [[Image:DistancesDFTdh08.jpg|150x150px]]
|-
|Imaginary frequency (cm-1) || -957 || -525
|-
|Animation || [[Image:-957dh08.gif]] || [[Image:-525dh08.gif]]
|-
| Lowest real frequency (cm-1) || 148 || 136
|-
|Animation || [[Image: 148dh08.gif]] || [[Image: 136dh08.gif]]
|}

The C-C bond forming length between the termini carbons is 0.15 Å larger for the DFT/B3LYP/6-31G* level of theory and this is because this method gives carries out the calculation to a greater degree of accuracy. This larger bond length leads a lower potential energy gradient to the transition state, which gives a lower total energy for the transition state. Therefore the given values below will be the DFT/B3LYP/6-31G* and not the semi-empirical/AM1 method.

A typical sp3 C-C bond length is 1.54 Å and a typical sp2 C=C bond length is 1.34 Å<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref>. Within the optimised transition molecule above the C-C bond lengths are 1.40 Å and the C=C bond length is 1.38 Å. This suggests that the π-bonds are breaking in order for the C-C bonds to form and additionally to this a π-bond is forming in between the two sp3 carbons.

The Van der Waals radius of a C-atom is 1.72 Å<ref name="ref 3">M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer, D. G. Truhlar, 1995, 346, pp. 63-69.[http://pubs.acs.org/doi/abs/10.1021/jp8111556]</ref>. The partly formed C-C length is 2.27 Å, which is shorter than two times the the van der Waals radius of a carbon atom. Therefore the carbon atoms are within a close enough distance to each other to share electron density and yet the bond has not formed at this point.

From looking at the animations for the imaginary frequency above it can be seen that the formation of the two bonds between the termini of the reactants is synchronous. This is expected of the Diels-Alder reaction since the mechanism is concerted. However the lowest real frequency is asynchronous with respect to the two bonds forming between the termini.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12: Frontier molecular orbitals of the transition state '''
| '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| LUMO || [[Image:LUMO1TSdh08.jpg|100x100px]] || [[Image:LUMO2TSdh08.jpg|100x100px]]|| Symmetric || +0.023
|-
| HOMO || [[Image:HOMO1TSdh08.jpg|100x100px]] || [[Image:HOMO2TSdh08.jpg|100x100px]]|| Anti-symmetric || -0.323
|}

From examining the images from table 12 above it can be deduced that the LUMO of the transition state originates from the cis-butadiene's LUMO and the ethylene's HOMO, which represents the σ*-antibonding orbitals. However the HOMO of the transition state originates from the cis-butadiene's HOMO and the ethylene's LUMO, which represents the σ-bonding orbitals within the two single C-C bonds that are formed within the reaction.

==='''cis-butadiene and ethylene: activation energy'''===

The cis-butadiene and ethylene were re-optimised at the DFT/B3LYP/6-31G* level of theory so the activation energy of the reaction could be calculated.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 13: Calculation for the activation energy for the reaction'''
| '''molecule''' || '''DFT/B3LYP/6-31G* energy (a. u)''' || '''Calculations''' || '''Energy '''
|-
| cis-butadiene || -155.98595591 || transition state || -234.54389573 a. u
|-
| ethylene || -78.58745828 || ΔE, activation energy || 0.02951846 a. u
|-
| cis-butadiene and ethylene combined || -234.57341419 || ΔE, activation energy || 77.50 KJ mol-1
|}

The calculated activation energy is 77.50 KJ mol-1, which is of the right magnitude when compared to the literature value (115 ± 8 KJ mol-1 at 0 K) but there is still an error of 37.50 KJ mol-1. This shows that Gaussview is not perfect but still gives an idea of the activation energy required for the reaction to take place.

==='''Cyclohexa-1,3-diene reaction with maleic anhydride'''===

The reaction scheme below shows the stereoselective Diels-Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene. The reaction is kinetically controlled with the major product being the endo product, which will be explained later on in the exercise.

[[Image:Reaction_schemedh081.jpg|center|500x500px]]

The <jmol>
<jmolAppletButton>
<uploadedFileContents>FrozenTS22dh08.mol</uploadedFileContents>
<text>exo</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_frozenTS22dh08.mol</uploadedFileContents>
<text>endo</text>
<color>white</color>
</jmolAppletButton>
</jmol> transition states were drawn on Gaussview and then the optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while the distance of the forming bonds were frozen at 2.1Å between the respective carbons. After this the structures changed slightly (<jmol>
<jmolAppletButton>
<uploadedFileContents>TS2_afteropt1_exo1dh08.mol</uploadedFileContents>
<text>exo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_TS2_afteropt1dh08.mol</uploadedFileContents>
<text>endo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol>) and the resulting files had an optimisation and frequency calculation run on it using the TA (Berny) with using the semi-empirical/AM1 level of theory while the unfrozen option as changed to "derivative". After this the endo and exo optimised transition state structures were re-optimised as they were before but using the DFT/B3LYP/6-31G* level of theory instead with the additional keyword "Opt=NoEigen".

{| class="wikitable" border="1" align="center"
|+ '''Table 14: Comparison of the exo and endo transition states '''
! rowspan="2"|-!! colspan="2"|''exo'' !! colspan="2"|''endo''
|-
| Semi-empirical/AM1 || DFT/B3LYP/6-31G* || Semi-empirical/AM1 || DFT/B3LYP/6-31G*
|-
| align="center"|Structure ||align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>After_opt2_exodh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>Endo_TS2_afteropt2dh081.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| align="center"| Energy (a. u) ||align="center"| -0.05041975 ||align="center"| -612.67931095 ||align="center"| -0.05150470 ||align="center"| -612.68339684
|-
| C-C bond forming length (Å) ||align="center"| 2.17 ||align="center"| 2.29 ||align="center"| 2.16 ||align="center"| 2.27
|-align="center"
| Imaginary frequency (cm-1) ||align="center"| -812 ||align="center"| -448 ||align="center"| -806 ||align="center"| -447
|-align="center"
| Animation ||colspan="2"|[[Image:-812_SEdh08.gif]] || colspan="2"|[[Image:-806_cm-1dh08.gif]]
|}

Again the C-C bond forming length between the termini carbons is larger (by 0.11 and 0.12 Å) for the DFT/B3LYP/6-31G* level of theory and this is due the same reasons talked about above. However the C-C bond forming length between the termini carbons are very similar in both the endo and exo transition states (0.02 Å difference).

From looking at the animations for the imaginary frequency above it can also be seen that the formation of the two bonds between the carbons in this case as well is synchronous. Again this is expected of the Diels-Alder reaction since it is concerted.

{| align=center
|+ '''Strain diagrams for the endo and exo transition states'''
| [[Image:Exostraindh08.jpg|thumb|200x200px|'''Exo TS''']]
| [[Image:Strainendodh08.jpg|thumb|200x200px|'''Endo TS''']]
|}

From studying the energies in table 14 it can be seen that the endo transition state is more stable than the exo by 4.08589 x 10-3 a. u (10.73 KJ mol-1). However by looking at the strain of both transition states one would expect the exo TS to be more stable. This is because the distance between the cyclohexa-1,3-diene ring and the -(C=O)-O-(C=O)- maleic anhydride fragment is 2.99 Å in the endo TS and 3.03 Å in the exo TS. Thus the endo form would have more strain due to steric hindrance.

However within the exo TS the vicinal hydrogens on the CH2-CH2 (carbon numbers 18 and 21) fragment on the cyclohexa-1,3-diene ring are found below the plane and next to the maleic anhydride. This leads to steric hindrance is this is where the strain originates from within the exo TS. Therefore this disfavours the exo form, which makes the endo more stable.

Another factor making the endo TS more favourable is secondary orbital overlap. These orbitals contribute to the bonds forming and they interact with each other through space. Within the endo TS below there are 2 more orbitals interacting (secondary orbitals) than within the exo TS. This therefore makes the endo TS more stable and therefore more favoured.

[[Image:Secondary_orbital_shizzledh08.jpg|500x500px|center]]

{| class="wikitable" border="1" align="center"
|+ '''Table 15: Frontier molecular orbitals of the the transition state for the cyclohexa-1,3-diene reaction '''
! rowspan="2"|Molecular Orbital !! colspan="4"|''exo'' !! colspan="4"|''endo''
|- align="center"
| '''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''||'''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''
|- align="center"
| LUMO || [[Image:LUMO1exoh081.jpg|150x150px]]|| [[Image:LUMO2exodh081.jpg|150x150px]]|| Anti-symmetric || -0.078 || [[Image:LUMO1ENDOdh08.jpg|150x150px]]||[[Image:LUMO2ENDOdh08.jpg|150x150px]] || Anti-symmetric || -0.067
|- align="center"
| HOMO || [[Image:HOMO1exodh081.jpg|150x150px]]||[[Image:HOMO2exodh081.jpg|150x150px]] || Anti-symmetric|| -0.242 || [[Image:HOMO1ENDOdh08.jpg|150x150px]]|| [[Image:HOMO2ENDOdh08.jpg|150x150px]]||Anti-symmetric || -0.242
|}

Digital repositories:

exo: http://hdl.handle.net/10042/to-5596

endo: http://hdl.handle.net/10042/to-5597

From examining table 15 it is clear to see that both the HOMO and LUMO are anti-symmetric for both the transition states. The LUMO of the maleic anhydride fragment is the π* anti-bonding orbital of the C=C bond. Within the reaction then this has electrons transferred to it and the C=C bond breaks, which leaves just a C-C bond. The LUMOs for both transition states are both anti-bonding with respect to the two fragments. The HOMOs of both transitions state represent the σ-bonding that will be formed between the termini carbons on the products and reactants. This is most likely from the overlap of the cyclohexa-1,3-diene's HOMO and the maleic anhydride's LUMO.

=='''Conclusion'''==

='''References'''=
<references/>

Rep:Mod:hampluscheese

2010-11-11T12:41:34Z

Dh08: /* '''Cyclohexa-1,3-diene reaction with maleic anhydride''' */

='''Module 3:Physical Computational Chemistry Lab'''=

Douglas Hunt

=='''Objectives'''==

The aim of this experiment is to

=='''The Cope Rearrangement'''==

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangement] is a [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement |3,3|-sigmatropic rearrangement] that takes place through a single [http://en.wikipedia.org/wiki/Pericyclic pericyclic], concerted [http://en.wikipedia.org/wiki/Transition_state transition state]. Within this rearrangement a σ bond is formed in concert with another being broken within a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] π system.

The Cope rearrangement we will be studying will be that of 1,5-hexadiene, which is shown below.

[[Image:Cope_rearrangement_schemedh08.jpg|300x300px|center]]

The objective is to locate the low-energy minima and transition structures on the C6H10 potential energy surface and to determine the preferred reaction mechanism by using [http://en.wikipedia.org/wiki/GAUSSIAN Gaussian], with the [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)] methods.

==='''Optimisation of 1,5-hexadiene Conformers'''===

Rough <jmol>
<jmolAppletButton>
<uploadedFileContents>Anti_2_drawndh08.mol</uploadedFileContents>
<text>anti</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche3_drawndh08.mol</uploadedFileContents>
<text>gauche</text>
<color>white</color>
</jmolAppletButton>
</jmol> conformers of 1,5-hexadiene were drawn on Gaussview and their structures were tidied by the clean function. They were then further optimised by the HF/3-21G method and were identified which conformer they were from their point group and appendix 1 on the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3 physical lab page].

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 1: Optimisation and comparison of anti and gauche 1,5-hexadiene conformers using HF/3-21G methods
| ''' Conformer''' || '''Optimised Structure''' || '''Energy (Atomic Units)''' || '''[http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1 Energies (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (Anti2 - Gauche3)''' || '''Point group'''
|-
| '''Anti2''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Anti_2_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692535 || -231.69254 ||-6.08 x 105 || 1.2606 x 10-4 a. u|| Ci
|-
| '''Gauche3''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Gauche3_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692661 || -231.69266 || -6.08 x 105 || 3.31 x 10-1 KJ mol-1 || C1
|}

From studying table 1 it can be seen that the gauche conformer is 0.33 KJ mol-1 more stable than the anti conformation. One would assume that the anti2 conformer would be more stable due to [http://en.wikipedia.org/wiki/Steric steric] factors/repulsion within the gauche3 conformation. However within the gauche3 conformer there is a favourable donation of [http://en.wikipedia.org/wiki/Electron_density electron density] from the C=C π orbital to the C-H σ* orbital from the adjacent proton, which stabilises the conformer<ref name="ref 1">B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446.[http://www.informaworld.com/smpp/content~db=all?content=10.1080/00268970110081412]</ref>.

 

The anti2 conformer was then optimised using the B3LYP 6-31G* (6-31G (d)) method.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 2: Comparison of the anti2 conformer optimisations carried out at different calculation methods
| ''' Method''' || '''Structure and bond distances (Å)''' || '''Dihedral angle (C1 to C4)''' || '''Energy (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (HF/3-21G - DFT/B3LYP/6-31G)'''
|-
| HF/3-21G || [[Image:Anti2_opt1dh08.jpg|300x300px]] || 114.6 o || -231.692535 || -6.08 x 105 || 2.919171 a. u
|-
| DFT/B3LYP/6-31G* || [[Image:Anti2_opt2DH08.jpg|300x300px]] || 118.8 o || -234.611716 || -6.16 x 105 || 7.66 x 103 KJ mol-1
|}

From examining table 2 at first it is possible to see that the bond lengths are very similar to each other and that of the literature bond lengths<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref> (C1=C2: 1.340 Å, C2-C3: 1.508 Å and C3-C4: 1.538 Å). It is also possible to see that the C=C bond length is more accurate for the DFT/B3LYP/6-31G* method by it being 0.01 Å closer to the literature value. Another main difference in terms of geometry is that the DFT/B3LYP/6-31G* method gives a larger [http://en.wikipedia.org/wiki/Dihedral_angle dihedral angle] (and most likely more accurate) of 118.8o compared to that of 114.6o.

In terms of energy the DFT/B3LYP/6-31G* method produces a conformer of a higher stability by it possessing a lower energy with a difference of 7.66 x 103 KJ mol-1.

==='''Frequency analysis of 1,5-hexadiene Conformers'''===

[[Image:Vibrations_anti2dh08.jpg|right|300x300px]]

For the frequency analysis the DFT/B3LYP/6-31G* method was used.

Frequency analysis is essential within this calculation due to it being the second derivative of the potential energy surface. This means that if all the frequencies are all positive the system is at a minimum, if one of them is negative the optimised system is at a transition state, and if more than one are negative then the calculation has failed to find a critical point, which means the optimisation has failed. This therefore gives us a method of seeing whether the calculation has worked.

From looking to the image on the right it can deduced that the optimisation has taken place successfully by there being no negative vibrations and therefore the conformation is at a minimum.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 3: Energies under the thermochemistry section at 298.15 K and 0 K within the frequency logfiles'''
| '''Energies''' || '''298.15 K (atomic units)''' ||| '''0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (KJ mol-1)''' || '''Discussion'''
|-
| Sum of electronic and zero-point energies || -234.469177 || -234.466673 || -2.504 x 10-3 || -6.57 || This is the potential energy at 0 K, which includes the zero-point vibrational energy (E = Eelec + ZPE).

|-
| Sum of electronic and thermal energies || -234.461835 || -234.461085 || -7.5 x 10-4|| -1.97 || This is the energy at 298.15 K (0 K for 0 K calculation) and 1 atm of pressure, which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans).

|-
| Sum of electronic and thermal enthalpies || -234.460891 || -234.460141 || -7.5 x 10-4 || -1.97 || This contains an additional correction for RT (H = E + RT), which is especially important when looking at dissociation reactions.
|-
| Sum of electronic and thermal free energies || -234.500739 || -234.495931 || 4.808 x 10-3 || -12.62 || This includes the entropic contribution to the free energy (G = H - TS).
|}

{|align=right
|+
| [[Image:Anti2_freq_IRdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 298.15 K''']]
| [[Image:Anti2_freq_0Kdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 0 K''']]
|}

From looking at table 3 above it can be deduced that the anti2 conformer is more stable at room temperature to that of 0 K with respect to all the types of energies. The spectra also seem to not change much between the two temperatures but the main change is the change in the sum of electronic and thermal free energies, which is heavily dependent of temperature by it including the entropic contribution to the free energy (G = H - '''T'''S)

 
 
 
 

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

Within the Cope Rearrangement there are two possible transition states, the chair (C2h) and the boat (C2v), which are shown within [https://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_2 appendix two ].

==='''Comparison of optimisation methods for the chair transition state'''===

An <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2hdh08.mol</uploadedFileContents>
<text>allyl fragment</text>
<color>white</color>
</jmolAppletButton>
</jmol>, CH2CHCH2 was drawn on Gaussview and was optimized through the HF/3-21G method. After this the original allyl fragment was copied and pasted into a new window and place exactly above the first fragment with a bond distance of 2.20 Å between the terminal end carbons. This therefore produced the <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2h_fulldh08.mol</uploadedFileContents>
<text>chair transition structure.</text>
<color>white</color>
</jmolAppletButton>
</jmol>

Gaussian was then set up for an optimisation for the transition state with the additional key words "Opt=NoEigen", the optimisation set to TS (Berny), the force constants set to "once" and using the HF/3-21G method.

After this the "chair" transition state was optimised an alternative way by freezing the coordinates of the two sets of terminal carbons (where the bonds form/break during the rearrangement) within the Redundant Coord Editor on Gauusview.
Following this a second calculation was run where the freeze coordinates option was changed to "Derivative" and the force constants were set to "never".

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 4: Comparison of chair transition state optimisation methods
| || '''Optimised through a TS (Berny)''' || '''Optimised through a frozen coordinate method'''
|-
| '''Optimised TS Structure''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>C2h_optfreq_TS_Bdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| '''Terminal C-C bond length (Å)''' || 2.02 || 2.02
|-
|'''Energy (atomic units)''' || -231.61932247 || -231.61932190
|-
| '''ΔE, TS (Berny) - frozen coordinate''' || 5.7 x 10-7 a.u || 1.50 x 10-3 KJ mol-1
|-
| '''Imaginary frequency (cm-1)''' || -818 || -818
|-
| '''Animation''' ||[[Image:818dh08TSB.gif|350x350px]] || [[Image:818cm-1Fdh08.gif]]
|}

The data in table 4 shows that both methods give the same results and they both give the same transition state. They both produced the same energy with a negligible energy difference between them and the bond lengths between the terminal carbons are both 2.02 Å. In addition to this the imaginary frequencies were both -818cm-1 and within the animations above it is possible to see that when one C-C bond is forming the other is breaking. This means it is a asynchronous bond formation, which is expected for the Cope Rearrangement since it is a pericyclic reaction.

The only difference between the two optimisations is the way in which the transition states were calculated. This is due to the TS (Berny) method calculating the force constant matrix within the first optimisation step and is then being recalculated as the optimisation carries on. When the frozen coordinate method freezes the reaction coordinate in order for the rest of the structure to be optimised around it, which does not require the whole hessian (force constant matrix) to be calculated but only that the reaction coordinate is differentiated (second step). The main advantage of this, is that you do not need to calculate the entire force constant matrix and just need to differentiate the reaction coordinate in order to obtain a good enough guess for the initial force constant matrix. The frozen coordinate method will therefore take less time to compute the transition state, which can be very expensive within certain systems. However to carry out this method one needs to have an idea of how far apart the terminal carbons are.

==='''Boat transition state'''===

===='''QST2 optimisation'''====

The optimisation for the boat transition state had a different method and this was the QST2 method. This calculation interpolates between two structures (product and reactant) and tries to find a transition state between them.

The reactants and products were numbered in the same way. Therefore both 1,5-hexadiene molecules had their numbering manually changed for the product molecule so it corresponded to the numbering of the reactant if it rearranged. This is illustrated in the reaction scheme below.

[[Image:Boat_rerrangmentdh08.jpg|300x300px|center]]

The chk file for the optimized anti2 conformer was opened and copied into a new window. A second window was opened within the new window and the same anti2 conformer was copied into the new window. The numbering was changed within the second molecule so the two looked like they do in the image below.

[[Image:TS_(QST2)_numbering_1dh08.jpg|400x400px|center]]

The job type was set as "Opt+Freq" and "TS (QST2)" was chosen to optimise the transition state. However this failed and produced
a structure a bit like a <jmol>
<jmolAppletButton>
<uploadedFileContents>Trialone_BPdh08.mol</uploadedFileContents>
<text>chair transition structure </text>
<color>white</color>
</jmolAppletButton>
</jmol> but more dissociated. This is because it interpolated between the two structures and translated the top allyl fragment and therefore did not consider the possibility of a rotation around the central bonds.

To get around this the C-C-C-C dihedral angle (C2-C3-C4-C5) was set to 0 o and the C-C-C angle (C2-C3-C4 and C3-C4-C5 for the product molecule) was set to 100 o for both molecules. This gave rise to the two structures below.

[[Image:TS_(QST2)_numbering_2dh08.jpg|400x400px|center]]

After this the QST2 calculation was run again and the results in table 5 were produced. The imaginary frequency at -839 cm-1 shows that while one bond is forming the other one is breaking, which is again an asynchronous bond formation.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5: Data for the boat transition state from the QST2 optimisation method
| Optimised TS Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.14
|-
|Energy (atomic units) || -231.60280151
|-
| Energy (KJ mol-1) || -6.08 x 105
|-
| Imaginary frequency (cm-1) || -839
|-
| Animation ||[[Image:-839dh08.gif]]
|}

===='''Intrinsic Reaction Coordinate'''====

It is not possible to predict which product the chair and boat transition state will lead to. To solve this problem within Gaussian one can use the IRC method, which allows one to follow the minimum energy path from a transition structure down to its local minimum on a potential energy surface. This creates a succession of points by taking small geometry steps in the direction to where the gradient of the energy surface is at its steepest until the minimum in energy is reached.

The chk file for the optimized chair transition structure was opened and the job type selected was "IRC", the reaction coordinate was computed in the forward direction (since molecule is symmetric) with 50 points along the IRC. This was done with two type of force constants being calculated, once and always.

Within the "once" method the output showed no bond formation process by the 26th step and therefore the calculation was incomplete, since a complete 1,5-hexadiene structure was supposed to be formed. From looking at the IRC pathway it can be seen that the minimum on the potential energy surface was not reached at that point, since the points on the pathway did not form an asymptote and the gradient did not go close enough to zero.

For the "always" method the computation of force constants were carried out at every step. This therefore lets the IRC calculation distinguish whether it is moving along the minimum energy pathway while going down the potential energy surface. While this is happening it can make changes to the pathway if necessary. Within the "always" case the bond forming arrives at 1,5-hexadiene and this is backed up by the IRC pathway reaching a minimum within its energy by the gradient going close to zero and the pathway reaches an asymptote.

The structure produced through the always method had an energy of -231.69199434 atomic units, which is very close to the energy of the gauche2 conformer in the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 appendix 1]. This suggests that the reaction takes place through the chair transition state, which later produces the gauche2 conformer of 1,5-hexadiene.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 6: Comparison of IRC calculations for the chair structure
| || '''Calculate force constants once''' || '''Calculate force constants always'''
|-
| '''Number of points along IRC''' || 26 || 47
|-
| '''Structure at the last point of IRC''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Once_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Always_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|'''Energy (atomic units)''' || -231.68906923 || -231.69199434
|-
| '''ΔE, once - always''' || 2.92511 x 10-3 a.u || 7.68 KJ mol-1
|-
| '''IRC pathway''' || [[Image:One_pathwaydh08.jpg|400x400px]] || [[Image:Always_pathwaydh08.jpg|400x400px]]
|-
| '''IRC gradient''' || [[Image:One_gradientdh08.jpg|400x400px]]|| [[Image:Always_gradientdh08.jpg|400x400px]]
|}

 
 
 

===='''Comparison of activation energies for the transition states'''====

The HF/3-21G optimised chair and boat transition structures were both reoptimised using the B3LYP/6-31G* (B3LYP/6-31G (d)) method. This gives a more accurate minimisation however this method takes longer to run. The values for the zero-point energies and thermal energies were found within the thermochemistry section of the log file.

The values for the HF/3-21G reactant (anti2) optimisation were calculated and the B3LYP/6-31G* values were taken from the "Frequency analysis of 1,5-hexadiene Conformers" earlier in this module.

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 7: Energy data for chair, boat and the anti2 conformation from the HF/3-21G and B3LYP/6-31G* methods of optimisation '''
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466677
| align="center" | -231.461320
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450936
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402342
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539524
| align="center" | -231.532562
| align="center" | -234.611716
| align="center" | -234.469177
| align="center" | -234.461835
|}
Digital repositories

Chair:
HF/3-21G: http://hdl.handle.net/10042/to-5569
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5568

Boat:
HF/3-21G: http://hdl.handle.net/10042/to-5570
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5571

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 8: Summary of activation energies for the chair and boat conformations (ΔE = chair TS or boat TS minus anti2)'''
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| align="center" | 1 a.u = 627.509 kcal mol-1
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE, chair (a.u)'''
| align="center" | 0.072847
| align="center" | 0.071242
| align="center" | 0.054248
| align="center" | 0.052827
| align="center" | -
|-
| '''ΔE, chair (kcal mol-1)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE, boat (a. u)'''
| align="center" | 0.088588
| align="center" | 0.087259
| align="center" | 0.066835
| align="center" | 0.065827
| align="center" | -
|-
| '''ΔE, boat (kcal mol-1)'''
| align="center" | 55.59
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.31
| align="center" | 44.7 ± 2.0
|}

By examining table 8 it is clear to see that the B3LYP/6-31G* method gives values which are much closer to the experimental ones. This is due to the B3LYP/6-31G* giving a higher level of optimisation than the HF/3-21G method. Table 8 also shows that the chair transition state has a lower activation energy to that of the boat transition state by 8.16 kcal mol-1. This suggests that the Cope rearrangement mechanism would most likely proceed through the chair transition state to that of the boat. Therefore the reaction would take place from the anti2 conformer to the chair transition state and then onto the gauche2 conformer within the Cope rearrangement.

It is also possible to see that the calcualted energies are very close to the literature values with the chair conformation being closer to its literature value to that of the boat.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9: Geometric parameters for the optimised chair and boat structures '''
| || ''' HF/3-21G chair TS''' || ''' B3LYP/6-31G* chair TS''' || '''HF/3-21G boat TS''' || '''B3LYP/6-31G* boat TS'''
|-
| Optimised Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Chair_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Boat_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.02 (C5-C10 or C3-C14)|| 1.97 (C5-C10 or C3-C14) || 2.14 (C3-C4 or C1-C6) || 2.21 (C3-C4 or C1-C6)
|-
| Fragment angle || 120 o (C3-C1-C5) || 120 o (C3-C1-C5) || 122 o (C4-C5-C6) || 122 o (C4-C5-C6)
|-
| Dihedral angle || 55 o (C1-C3-C14-C11)|| 54 o (C1-C3-C14-C11)|| 0 o (C2-C1-C6-C5) || 0 o (C2-C1-C6-C5)
|-
| Numbering of carbons || [[Image:Numbering_chairdh08.jpg|200x200px]]|| [[Image:Numbering_chairdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]]
|}

From studying table 9 it can be seen that the C-C bond breaking/forming length has changed between the two different types of optimisation methods. However the angles between the bonds remains unchanged.

=='''The Diels Alder Cycloaddition'''==

The Diels-Alder reaction is a cycloaddition between a conjugated diene and a dienophile (alkene system), which forms a cyclohexene system. It is often referred to as a [4s+2s] cycloaddition and proceeds through a single cyclic, concerted transition state where 2 σ bonds are formed in between the termini carbons of the 2 conjugated п systems. This is shown in the reaction scheme below where ethene is reacting with cis-butadiene.

[[Image:Reaction_scheme2dh08.jpg|300x300px|center]]

The π orbitals of the dienophile form new σ bonds with the π orbitals of the diene. The number of π electrons involved determine whether the reaction proceeds in a concerted stereospecific fashion (allowed) or not (forbidden). The HOMO/LUMO of one fragment interacts with the HOMO/LUMO of the other fragment to form two new bonding and anti-bonding MOs. The nodal properties allow one to predict how the reaction will occur.

Semi-Empirical/AM1 method and the B3LYP/6-31G* method of optimisation will be used on Gaussian to determine particular properites of the reactants to predict the way in which the reaction will proceed.

==='''cis-butadiene and ethylene: reagents'''===

<jmol>
<jmolAppletButton>
<uploadedFileContents>Cis_butadienedh08.mol</uploadedFileContents>
<text>Cis-butadiene</text>
<color>whi-te</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Ethenedh08.mol</uploadedFileContents>
<text>ethylene</text>
<color>white</color>
</jmolAppletButton>
</jmol> were optimised separately using the AM1 semi-empirical method on Gaussview.The cis-butadiene had an energy of 0.04879725 a. u (1.28 x 102 KJ mol-1) and ethylene had an energy of 0.02619027 a. u (68.76 KJ mol-1).

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10: Frontier molecular orbitals of cis-butadiene and ethylene '''
| '''Reagent''' || '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| cis-butadiene || LUMO || [[Image:LUMO1_cis_butadieneDH08.jpg|100x100px]] || [[Image:LUMO2_cis_butadienedh08.jpg|100x100px]]|| Symmetric || +0.017
|-
| cis-butadiene || HOMO || [[Image:HOMO1_cis_butadienedh08.jpg|100x100px]] || [[Image:HOMO2_cis_butadienedh08.jpg|100x100px]]|| Anti-symmetric || -0.343
|-
| ethylene || LUMO || [[Image:LUMO1_ethenedh08.jpg|100x100px]] || [[Image:LUMO2_ethenedh08.jpg|100x100px]]|| Anti-symmetric || +0.052
|-
| ethylene || HOMO || [[Image:HOMO1_ethenedh08.jpg|100x100px]] || [[Image:HOMO2_ethenedh08.jpg|100x100px]]|| Symmetric || -0.387
|}

 
The symmetries of the LUMO and HOMO orbitals for cis-butadiene are opposite of those for the ethylene. This therefore means that the only interactions that can take place are those between ethylene's LUMO and cis-butadiene's HOMO and in addition to this the interaction between cis-butadiene's LUMO and ethylene's HOMO. This is because orbital symmetry has to be conserved (the conservation of orbital symmetry states that the symmetry of an new orbital formed by the overlap of the 2 original orbitals must be the same as the symmetry of the initial orbitals).

==='''cis-butadiene and ethylene: transition state'''===

A <jmol>
<jmolAppletButton>
<uploadedFileContents>Bicyclic_startingdh08.mol</uploadedFileContents>
<text>bicyclo system</text>
<color>white</color>
</jmolAppletButton>
</jmol> was drawn and one of the -CH2-CH2- fragments were removed. Three C=C bonds were added and the previous one was taken away so a <jmol>
<jmolAppletButton>
<uploadedFileContents>Frozen_coorddh08.mol</uploadedFileContents>
<text>guess transition state</text>
<color>white</color>
</jmolAppletButton>
</jmol> was produced.

After this an optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while bonds between the termini of the fragments were frozen at 2.1Å. Following this the resulting file had two separate optimisation and frequency calculations run on it using the TS (Berny) with one using the semi-empirical/AM1 method and the other using the DFT/B3LYP/6-31G* level of theory. Both of these optimisations had the bonds unfrozen and the "derivative" option was chosen instead.

The DFT/B3LYP/6-31G* level of theory was used since it gave more accurate results in the calculations before and the energies it produces are consistent with the rest of the report (semi-empirical/AM1 method gives very different values).

The QST2 optimisation was not used since it required numbering the atoms in both the reactants and products, which took a lot of time to do.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11: Comparison of two methods of optimsations for the transition state '''
| || '''semi-empirical/AM1 method''' || '''DFT/B3LYP/6-31G* method'''
|-
| Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>SE_AM1_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>DFT_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Energy (a. u) || 0.11165608 || -234.54389573
|-
| Bond lengths || [[Image:DistancesSEdh08.jpg|150x150px]]|| [[Image:DistancesDFTdh08.jpg|150x150px]]
|-
|Imaginary frequency (cm-1) || -957 || -525
|-
|Animation || [[Image:-957dh08.gif]] || [[Image:-525dh08.gif]]
|-
| Lowest real frequency (cm-1) || 148 || 136
|-
|Animation || [[Image: 148dh08.gif]] || [[Image: 136dh08.gif]]
|}

The C-C bond forming length between the termini carbons is 0.15 Å larger for the DFT/B3LYP/6-31G* level of theory and this is because this method gives carries out the calculation to a greater degree of accuracy. This larger bond length leads a lower potential energy gradient to the transition state, which gives a lower total energy for the transition state. Therefore the given values below will be the DFT/B3LYP/6-31G* and not the semi-empirical/AM1 method.

A typical sp3 C-C bond length is 1.54 Å and a typical sp2 C=C bond length is 1.34 Å<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref>. Within the optimised transition molecule above the C-C bond lengths are 1.40 Å and the C=C bond length is 1.38 Å. This suggests that the π-bonds are breaking in order for the C-C bonds to form and additionally to this a π-bond is forming in between the two sp3 carbons.

The Van der Waals radius of a C-atom is 1.72 Å<ref name="ref 3">M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer, D. G. Truhlar, 1995, 346, pp. 63-69.[http://pubs.acs.org/doi/abs/10.1021/jp8111556]</ref>. The partly formed C-C length is 2.27 Å, which is shorter than two times the the van der Waals radius of a carbon atom. Therefore the carbon atoms are within a close enough distance to each other to share electron density and yet the bond has not formed at this point.

From looking at the animations for the imaginary frequency above it can be seen that the formation of the two bonds between the termini of the reactants is synchronous. This is expected of the Diels-Alder reaction since the mechanism is concerted. However the lowest real frequency is asynchronous with respect to the two bonds forming between the termini.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12: Frontier molecular orbitals of the transition state '''
| '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| LUMO || [[Image:LUMO1TSdh08.jpg|100x100px]] || [[Image:LUMO2TSdh08.jpg|100x100px]]|| Symmetric || +0.023
|-
| HOMO || [[Image:HOMO1TSdh08.jpg|100x100px]] || [[Image:HOMO2TSdh08.jpg|100x100px]]|| Anti-symmetric || -0.323
|}

From examining the images from table 12 above it can be deduced that the LUMO of the transition state originates from the cis-butadiene's LUMO and the ethylene's HOMO, which represents the σ*-antibonding orbitals. However the HOMO of the transition state originates from the cis-butadiene's HOMO and the ethylene's LUMO, which represents the σ-bonding orbitals within the two single C-C bonds that are formed within the reaction.

==='''cis-butadiene and ethylene: activation energy'''===

The cis-butadiene and ethylene were re-optimised at the DFT/B3LYP/6-31G* level of theory so the activation energy of the reaction could be calculated.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 13: Calculation for the activation energy for the reaction'''
| '''molecule''' || '''DFT/B3LYP/6-31G* energy (a. u)''' || '''Calculations''' || '''Energy '''
|-
| cis-butadiene || -155.98595591 || transition state || -234.54389573 a. u
|-
| ethylene || -78.58745828 || ΔE, activation energy || 0.02951846 a. u
|-
| cis-butadiene and ethylene combined || -234.57341419 || ΔE, activation energy || 77.50 KJ mol-1
|}

The calculated activation energy is 77.50 KJ mol-1, which is of the right magnitude when compared to the literature value (115 ± 8 KJ mol-1 at 0 K) but there is still an error of 37.50 KJ mol-1. This shows that Gaussview is not perfect but still gives an idea of the activation energy required for the reaction to take place.

==='''Cyclohexa-1,3-diene reaction with maleic anhydride'''===

The reaction scheme below shows the stereoselective Diels-Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene. The reaction is kinetically controlled with the major product being the endo product, which will be explained later on in the exercise.

[[Image:Reaction_schemedh081.jpg|center|500x500px]]

The <jmol>
<jmolAppletButton>
<uploadedFileContents>FrozenTS22dh08.mol</uploadedFileContents>
<text>exo</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_frozenTS22dh08.mol</uploadedFileContents>
<text>endo</text>
<color>white</color>
</jmolAppletButton>
</jmol> transition states were drawn on Gaussview and then the optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while the distance of the forming bonds were frozen at 2.1Å between the respective carbons. After this the structures changed slightly (<jmol>
<jmolAppletButton>
<uploadedFileContents>TS2_afteropt1_exo1dh08.mol</uploadedFileContents>
<text>exo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_TS2_afteropt1dh08.mol</uploadedFileContents>
<text>endo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol>) and the resulting files had an optimisation and frequency calculation run on it using the TA (Berny) with using the semi-empirical/AM1 level of theory while the unfrozen option as changed to "derivative". After this the endo and exo optimised transition state structures were re-optimised as they were before but using the DFT/B3LYP/6-31G* level of theory instead with the additional keyword "Opt=NoEigen".

{| class="wikitable" border="1" align="center"
|+ '''Table 14: Comparison of the exo and endo transition states '''
! rowspan="2"|-!! colspan="2"|''exo'' !! colspan="2"|''endo''
|-
| Semi-empirical/AM1 || DFT/B3LYP/6-31G* || Semi-empirical/AM1 || DFT/B3LYP/6-31G*
|-
| align="center"|Structure ||align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>After_opt2_exodh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>Endo_TS2_afteropt2dh081.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| align="center"| Energy (a. u) ||align="center"| -0.05041975 ||align="center"| -612.67931095 ||align="center"| -0.05150470 ||align="center"| -612.68339684
|-
| C-C bond forming length (Å) ||align="center"| 2.17 ||align="center"| 2.29 ||align="center"| 2.16 ||align="center"| 2.27
|-align="center"
| Imaginary frequency (cm-1) ||align="center"| -812 ||align="center"| -448 ||align="center"| -806 ||align="center"| -447
|-align="center"
| Animation ||colspan="2"|[[Image:-812_SEdh08.gif]] || colspan="2"|[[Image:-806_cm-1dh08.gif]]
|}

Again the C-C bond forming length between the termini carbons is larger (by 0.11 and 0.12 Å) for the DFT/B3LYP/6-31G* level of theory and this is due the same reasons talked about above. However the C-C bond forming length between the termini carbons are very similar in both the endo and exo transition states (0.02 Å difference).

From looking at the animations for the imaginary frequency above it can also be seen that the formation of the two bonds between the carbons in this case as well is synchronous. Again this is expected of the Diels-Alder reaction since it is concerted.

{| align=center
|+ '''Strain diagrams for the endo and exo transition states'''
| [[Image:Exostraindh08.jpg|thumb|200x200px|'''Exo TS''']]
| [[Image:Strainendodh08.jpg|thumb|200x200px|'''Endo TS''']]
|}

From studying the energies in table 14 it can be seen that the endo transition state is more stable than the exo by 4.08589 x 10-3 a. u (10.73 KJ mol-1). However by looking at the strain of both transition states one would expect the exo TS to be more stable. This is because the distance between the cyclohexa-1,3-diene ring and the -(C=O)-O-(C=O)- maleic anhydride fragment is 2.99 Å in the endo TS and 3.03 Å in the exo TS. Thus the endo form would have more strain due to steric hindrance.

However within the exo TS the vicinal hydrogens on the CH2-CH2 (carbon numbers 18 and 21) fragment on the cyclohexa-1,3-diene ring are found below the plane and next to the maleic anhydride. This leads to steric hindrance is this is where the strain originates from within the exo TS. Therefore this disfavours the exo form, which makes the endo more stable.

Another factor making the endo TS more favourable is secondary orbital overlap. These orbitals contribute to the bonds forming and they interact with each other through space. Within the endo TS below there are 2 more orbitals interacting (secondary orbitals) than within the exo TS. This therefore makes the endo TS more stable and therefore more favoured.

[[Image:Secondary_orbital_shizzledh08.jpg|500x500px|center]]

{| class="wikitable" border="1" align="center"
|+ '''Table 15: Frontier molecular orbitals of the the transition state for the cyclohexa-1,3-diene reaction '''
! rowspan="2"|Molecular Orbital !! colspan="4"|''exo'' !! colspan="4"|''endo''
|- align="center"
| '''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''||'''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''
|- align="center"
| LUMO || [[Image:LUMO1exoh081.jpg|150x150px]]|| [[Image:LUMO2exodh081.jpg|150x150px]]|| Anti-symmetric || -0.078 || [[Image:LUMO1ENDOdh08.jpg|150x150px]]||[[Image:LUMO2ENDOdh08.jpg|150x150px]] || Anti-symmetric || -0.067
|- align="center"
| HOMO || [[Image:HOMO1exodh081.jpg|150x150px]]||[[Image:HOMO2exodh081.jpg|150x150px]] || Anti-symmetric|| -0.242 || [[Image:HOMO1ENDOdh08.jpg|150x150px]]|| [[Image:HOMO2ENDOdh08.jpg|150x150px]]||Anti-symmetric || -0.242
|}

Digital repositories:

exo: http://hdl.handle.net/10042/to-5596

endo: http://hdl.handle.net/10042/to-5597

From examining table 15 it is clear to see that both the HOMO and LUMO are anti-symmetric for both the transition states. The LUMO of the maleic anhydride fragment is the π* anti-bonding orbital of the C=C bond. Within the reaction then this has electrons transferred to it and the C=C bond breaks, which leaves just a C-C bond. The LUMOs for both transition states are both anti-bonding with respect to the two fragments. The HOMOs of both transitions state represent the σ-bonding that will be formed between the termini carbons on the products and reactants. This is most likely from the overlap of the cyclohexa-1,3-diene's HOMO and the maleic anhydride's LUMO.

=='''Conclusion'''==

='''References'''=
<references/>

Rep:Mod:hampluscheese

2010-11-11T12:32:33Z

Dh08: /* '''Cyclohexa-1,3-diene reaction with maleic anhydride''' */

='''Module 3:Physical Computational Chemistry Lab'''=

Douglas Hunt

=='''Objectives'''==

The aim of this experiment is to

=='''The Cope Rearrangement'''==

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangement] is a [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement |3,3|-sigmatropic rearrangement] that takes place through a single [http://en.wikipedia.org/wiki/Pericyclic pericyclic], concerted [http://en.wikipedia.org/wiki/Transition_state transition state]. Within this rearrangement a σ bond is formed in concert with another being broken within a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] π system.

The Cope rearrangement we will be studying will be that of 1,5-hexadiene, which is shown below.

[[Image:Cope_rearrangement_schemedh08.jpg|300x300px|center]]

The objective is to locate the low-energy minima and transition structures on the C6H10 potential energy surface and to determine the preferred reaction mechanism by using [http://en.wikipedia.org/wiki/GAUSSIAN Gaussian], with the [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)] methods.

==='''Optimisation of 1,5-hexadiene Conformers'''===

Rough <jmol>
<jmolAppletButton>
<uploadedFileContents>Anti_2_drawndh08.mol</uploadedFileContents>
<text>anti</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche3_drawndh08.mol</uploadedFileContents>
<text>gauche</text>
<color>white</color>
</jmolAppletButton>
</jmol> conformers of 1,5-hexadiene were drawn on Gaussview and their structures were tidied by the clean function. They were then further optimised by the HF/3-21G method and were identified which conformer they were from their point group and appendix 1 on the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3 physical lab page].

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 1: Optimisation and comparison of anti and gauche 1,5-hexadiene conformers using HF/3-21G methods
| ''' Conformer''' || '''Optimised Structure''' || '''Energy (Atomic Units)''' || '''[http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1 Energies (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (Anti2 - Gauche3)''' || '''Point group'''
|-
| '''Anti2''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Anti_2_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692535 || -231.69254 ||-6.08 x 105 || 1.2606 x 10-4 a. u|| Ci
|-
| '''Gauche3''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Gauche3_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692661 || -231.69266 || -6.08 x 105 || 3.31 x 10-1 KJ mol-1 || C1
|}

From studying table 1 it can be seen that the gauche conformer is 0.33 KJ mol-1 more stable than the anti conformation. One would assume that the anti2 conformer would be more stable due to [http://en.wikipedia.org/wiki/Steric steric] factors/repulsion within the gauche3 conformation. However within the gauche3 conformer there is a favourable donation of [http://en.wikipedia.org/wiki/Electron_density electron density] from the C=C π orbital to the C-H σ* orbital from the adjacent proton, which stabilises the conformer<ref name="ref 1">B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446.[http://www.informaworld.com/smpp/content~db=all?content=10.1080/00268970110081412]</ref>.

 

The anti2 conformer was then optimised using the B3LYP 6-31G* (6-31G (d)) method.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 2: Comparison of the anti2 conformer optimisations carried out at different calculation methods
| ''' Method''' || '''Structure and bond distances (Å)''' || '''Dihedral angle (C1 to C4)''' || '''Energy (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (HF/3-21G - DFT/B3LYP/6-31G)'''
|-
| HF/3-21G || [[Image:Anti2_opt1dh08.jpg|300x300px]] || 114.6 o || -231.692535 || -6.08 x 105 || 2.919171 a. u
|-
| DFT/B3LYP/6-31G* || [[Image:Anti2_opt2DH08.jpg|300x300px]] || 118.8 o || -234.611716 || -6.16 x 105 || 7.66 x 103 KJ mol-1
|}

From examining table 2 at first it is possible to see that the bond lengths are very similar to each other and that of the literature bond lengths<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref> (C1=C2: 1.340 Å, C2-C3: 1.508 Å and C3-C4: 1.538 Å). It is also possible to see that the C=C bond length is more accurate for the DFT/B3LYP/6-31G* method by it being 0.01 Å closer to the literature value. Another main difference in terms of geometry is that the DFT/B3LYP/6-31G* method gives a larger [http://en.wikipedia.org/wiki/Dihedral_angle dihedral angle] (and most likely more accurate) of 118.8o compared to that of 114.6o.

In terms of energy the DFT/B3LYP/6-31G* method produces a conformer of a higher stability by it possessing a lower energy with a difference of 7.66 x 103 KJ mol-1.

==='''Frequency analysis of 1,5-hexadiene Conformers'''===

[[Image:Vibrations_anti2dh08.jpg|right|300x300px]]

For the frequency analysis the DFT/B3LYP/6-31G* method was used.

Frequency analysis is essential within this calculation due to it being the second derivative of the potential energy surface. This means that if all the frequencies are all positive the system is at a minimum, if one of them is negative the optimised system is at a transition state, and if more than one are negative then the calculation has failed to find a critical point, which means the optimisation has failed. This therefore gives us a method of seeing whether the calculation has worked.

From looking to the image on the right it can deduced that the optimisation has taken place successfully by there being no negative vibrations and therefore the conformation is at a minimum.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 3: Energies under the thermochemistry section at 298.15 K and 0 K within the frequency logfiles'''
| '''Energies''' || '''298.15 K (atomic units)''' ||| '''0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (KJ mol-1)''' || '''Discussion'''
|-
| Sum of electronic and zero-point energies || -234.469177 || -234.466673 || -2.504 x 10-3 || -6.57 || This is the potential energy at 0 K, which includes the zero-point vibrational energy (E = Eelec + ZPE).

|-
| Sum of electronic and thermal energies || -234.461835 || -234.461085 || -7.5 x 10-4|| -1.97 || This is the energy at 298.15 K (0 K for 0 K calculation) and 1 atm of pressure, which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans).

|-
| Sum of electronic and thermal enthalpies || -234.460891 || -234.460141 || -7.5 x 10-4 || -1.97 || This contains an additional correction for RT (H = E + RT), which is especially important when looking at dissociation reactions.
|-
| Sum of electronic and thermal free energies || -234.500739 || -234.495931 || 4.808 x 10-3 || -12.62 || This includes the entropic contribution to the free energy (G = H - TS).
|}

{|align=right
|+
| [[Image:Anti2_freq_IRdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 298.15 K''']]
| [[Image:Anti2_freq_0Kdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 0 K''']]
|}

From looking at table 3 above it can be deduced that the anti2 conformer is more stable at room temperature to that of 0 K with respect to all the types of energies. The spectra also seem to not change much between the two temperatures but the main change is the change in the sum of electronic and thermal free energies, which is heavily dependent of temperature by it including the entropic contribution to the free energy (G = H - '''T'''S)

 
 
 
 

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

Within the Cope Rearrangement there are two possible transition states, the chair (C2h) and the boat (C2v), which are shown within [https://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_2 appendix two ].

==='''Comparison of optimisation methods for the chair transition state'''===

An <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2hdh08.mol</uploadedFileContents>
<text>allyl fragment</text>
<color>white</color>
</jmolAppletButton>
</jmol>, CH2CHCH2 was drawn on Gaussview and was optimized through the HF/3-21G method. After this the original allyl fragment was copied and pasted into a new window and place exactly above the first fragment with a bond distance of 2.20 Å between the terminal end carbons. This therefore produced the <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2h_fulldh08.mol</uploadedFileContents>
<text>chair transition structure.</text>
<color>white</color>
</jmolAppletButton>
</jmol>

Gaussian was then set up for an optimisation for the transition state with the additional key words "Opt=NoEigen", the optimisation set to TS (Berny), the force constants set to "once" and using the HF/3-21G method.

After this the "chair" transition state was optimised an alternative way by freezing the coordinates of the two sets of terminal carbons (where the bonds form/break during the rearrangement) within the Redundant Coord Editor on Gauusview.
Following this a second calculation was run where the freeze coordinates option was changed to "Derivative" and the force constants were set to "never".

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 4: Comparison of chair transition state optimisation methods
| || '''Optimised through a TS (Berny)''' || '''Optimised through a frozen coordinate method'''
|-
| '''Optimised TS Structure''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>C2h_optfreq_TS_Bdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| '''Terminal C-C bond length (Å)''' || 2.02 || 2.02
|-
|'''Energy (atomic units)''' || -231.61932247 || -231.61932190
|-
| '''ΔE, TS (Berny) - frozen coordinate''' || 5.7 x 10-7 a.u || 1.50 x 10-3 KJ mol-1
|-
| '''Imaginary frequency (cm-1)''' || -818 || -818
|-
| '''Animation''' ||[[Image:818dh08TSB.gif|350x350px]] || [[Image:818cm-1Fdh08.gif]]
|}

The data in table 4 shows that both methods give the same results and they both give the same transition state. They both produced the same energy with a negligible energy difference between them and the bond lengths between the terminal carbons are both 2.02 Å. In addition to this the imaginary frequencies were both -818cm-1 and within the animations above it is possible to see that when one C-C bond is forming the other is breaking. This means it is a asynchronous bond formation, which is expected for the Cope Rearrangement since it is a pericyclic reaction.

The only difference between the two optimisations is the way in which the transition states were calculated. This is due to the TS (Berny) method calculating the force constant matrix within the first optimisation step and is then being recalculated as the optimisation carries on. When the frozen coordinate method freezes the reaction coordinate in order for the rest of the structure to be optimised around it, which does not require the whole hessian (force constant matrix) to be calculated but only that the reaction coordinate is differentiated (second step). The main advantage of this, is that you do not need to calculate the entire force constant matrix and just need to differentiate the reaction coordinate in order to obtain a good enough guess for the initial force constant matrix. The frozen coordinate method will therefore take less time to compute the transition state, which can be very expensive within certain systems. However to carry out this method one needs to have an idea of how far apart the terminal carbons are.

==='''Boat transition state'''===

===='''QST2 optimisation'''====

The optimisation for the boat transition state had a different method and this was the QST2 method. This calculation interpolates between two structures (product and reactant) and tries to find a transition state between them.

The reactants and products were numbered in the same way. Therefore both 1,5-hexadiene molecules had their numbering manually changed for the product molecule so it corresponded to the numbering of the reactant if it rearranged. This is illustrated in the reaction scheme below.

[[Image:Boat_rerrangmentdh08.jpg|300x300px|center]]

The chk file for the optimized anti2 conformer was opened and copied into a new window. A second window was opened within the new window and the same anti2 conformer was copied into the new window. The numbering was changed within the second molecule so the two looked like they do in the image below.

[[Image:TS_(QST2)_numbering_1dh08.jpg|400x400px|center]]

The job type was set as "Opt+Freq" and "TS (QST2)" was chosen to optimise the transition state. However this failed and produced
a structure a bit like a <jmol>
<jmolAppletButton>
<uploadedFileContents>Trialone_BPdh08.mol</uploadedFileContents>
<text>chair transition structure </text>
<color>white</color>
</jmolAppletButton>
</jmol> but more dissociated. This is because it interpolated between the two structures and translated the top allyl fragment and therefore did not consider the possibility of a rotation around the central bonds.

To get around this the C-C-C-C dihedral angle (C2-C3-C4-C5) was set to 0 o and the C-C-C angle (C2-C3-C4 and C3-C4-C5 for the product molecule) was set to 100 o for both molecules. This gave rise to the two structures below.

[[Image:TS_(QST2)_numbering_2dh08.jpg|400x400px|center]]

After this the QST2 calculation was run again and the results in table 5 were produced. The imaginary frequency at -839 cm-1 shows that while one bond is forming the other one is breaking, which is again an asynchronous bond formation.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5: Data for the boat transition state from the QST2 optimisation method
| Optimised TS Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.14
|-
|Energy (atomic units) || -231.60280151
|-
| Energy (KJ mol-1) || -6.08 x 105
|-
| Imaginary frequency (cm-1) || -839
|-
| Animation ||[[Image:-839dh08.gif]]
|}

===='''Intrinsic Reaction Coordinate'''====

It is not possible to predict which product the chair and boat transition state will lead to. To solve this problem within Gaussian one can use the IRC method, which allows one to follow the minimum energy path from a transition structure down to its local minimum on a potential energy surface. This creates a succession of points by taking small geometry steps in the direction to where the gradient of the energy surface is at its steepest until the minimum in energy is reached.

The chk file for the optimized chair transition structure was opened and the job type selected was "IRC", the reaction coordinate was computed in the forward direction (since molecule is symmetric) with 50 points along the IRC. This was done with two type of force constants being calculated, once and always.

Within the "once" method the output showed no bond formation process by the 26th step and therefore the calculation was incomplete, since a complete 1,5-hexadiene structure was supposed to be formed. From looking at the IRC pathway it can be seen that the minimum on the potential energy surface was not reached at that point, since the points on the pathway did not form an asymptote and the gradient did not go close enough to zero.

For the "always" method the computation of force constants were carried out at every step. This therefore lets the IRC calculation distinguish whether it is moving along the minimum energy pathway while going down the potential energy surface. While this is happening it can make changes to the pathway if necessary. Within the "always" case the bond forming arrives at 1,5-hexadiene and this is backed up by the IRC pathway reaching a minimum within its energy by the gradient going close to zero and the pathway reaches an asymptote.

The structure produced through the always method had an energy of -231.69199434 atomic units, which is very close to the energy of the gauche2 conformer in the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 appendix 1]. This suggests that the reaction takes place through the chair transition state, which later produces the gauche2 conformer of 1,5-hexadiene.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 6: Comparison of IRC calculations for the chair structure
| || '''Calculate force constants once''' || '''Calculate force constants always'''
|-
| '''Number of points along IRC''' || 26 || 47
|-
| '''Structure at the last point of IRC''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Once_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Always_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|'''Energy (atomic units)''' || -231.68906923 || -231.69199434
|-
| '''ΔE, once - always''' || 2.92511 x 10-3 a.u || 7.68 KJ mol-1
|-
| '''IRC pathway''' || [[Image:One_pathwaydh08.jpg|400x400px]] || [[Image:Always_pathwaydh08.jpg|400x400px]]
|-
| '''IRC gradient''' || [[Image:One_gradientdh08.jpg|400x400px]]|| [[Image:Always_gradientdh08.jpg|400x400px]]
|}

 
 
 

===='''Comparison of activation energies for the transition states'''====

The HF/3-21G optimised chair and boat transition structures were both reoptimised using the B3LYP/6-31G* (B3LYP/6-31G (d)) method. This gives a more accurate minimisation however this method takes longer to run. The values for the zero-point energies and thermal energies were found within the thermochemistry section of the log file.

The values for the HF/3-21G reactant (anti2) optimisation were calculated and the B3LYP/6-31G* values were taken from the "Frequency analysis of 1,5-hexadiene Conformers" earlier in this module.

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 7: Energy data for chair, boat and the anti2 conformation from the HF/3-21G and B3LYP/6-31G* methods of optimisation '''
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466677
| align="center" | -231.461320
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450936
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402342
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539524
| align="center" | -231.532562
| align="center" | -234.611716
| align="center" | -234.469177
| align="center" | -234.461835
|}
Digital repositories

Chair:
HF/3-21G: http://hdl.handle.net/10042/to-5569
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5568

Boat:
HF/3-21G: http://hdl.handle.net/10042/to-5570
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5571

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 8: Summary of activation energies for the chair and boat conformations (ΔE = chair TS or boat TS minus anti2)'''
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| align="center" | 1 a.u = 627.509 kcal mol-1
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE, chair (a.u)'''
| align="center" | 0.072847
| align="center" | 0.071242
| align="center" | 0.054248
| align="center" | 0.052827
| align="center" | -
|-
| '''ΔE, chair (kcal mol-1)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE, boat (a. u)'''
| align="center" | 0.088588
| align="center" | 0.087259
| align="center" | 0.066835
| align="center" | 0.065827
| align="center" | -
|-
| '''ΔE, boat (kcal mol-1)'''
| align="center" | 55.59
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.31
| align="center" | 44.7 ± 2.0
|}

By examining table 8 it is clear to see that the B3LYP/6-31G* method gives values which are much closer to the experimental ones. This is due to the B3LYP/6-31G* giving a higher level of optimisation than the HF/3-21G method. Table 8 also shows that the chair transition state has a lower activation energy to that of the boat transition state by 8.16 kcal mol-1. This suggests that the Cope rearrangement mechanism would most likely proceed through the chair transition state to that of the boat. Therefore the reaction would take place from the anti2 conformer to the chair transition state and then onto the gauche2 conformer within the Cope rearrangement.

It is also possible to see that the calcualted energies are very close to the literature values with the chair conformation being closer to its literature value to that of the boat.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9: Geometric parameters for the optimised chair and boat structures '''
| || ''' HF/3-21G chair TS''' || ''' B3LYP/6-31G* chair TS''' || '''HF/3-21G boat TS''' || '''B3LYP/6-31G* boat TS'''
|-
| Optimised Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Chair_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Boat_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.02 (C5-C10 or C3-C14)|| 1.97 (C5-C10 or C3-C14) || 2.14 (C3-C4 or C1-C6) || 2.21 (C3-C4 or C1-C6)
|-
| Fragment angle || 120 o (C3-C1-C5) || 120 o (C3-C1-C5) || 122 o (C4-C5-C6) || 122 o (C4-C5-C6)
|-
| Dihedral angle || 55 o (C1-C3-C14-C11)|| 54 o (C1-C3-C14-C11)|| 0 o (C2-C1-C6-C5) || 0 o (C2-C1-C6-C5)
|-
| Numbering of carbons || [[Image:Numbering_chairdh08.jpg|200x200px]]|| [[Image:Numbering_chairdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]]
|}

From studying table 9 it can be seen that the C-C bond breaking/forming length has changed between the two different types of optimisation methods. However the angles between the bonds remains unchanged.

=='''The Diels Alder Cycloaddition'''==

The Diels-Alder reaction is a cycloaddition between a conjugated diene and a dienophile (alkene system), which forms a cyclohexene system. It is often referred to as a [4s+2s] cycloaddition and proceeds through a single cyclic, concerted transition state where 2 σ bonds are formed in between the termini carbons of the 2 conjugated п systems. This is shown in the reaction scheme below where ethene is reacting with cis-butadiene.

[[Image:Reaction_scheme2dh08.jpg|300x300px|center]]

The π orbitals of the dienophile form new σ bonds with the π orbitals of the diene. The number of π electrons involved determine whether the reaction proceeds in a concerted stereospecific fashion (allowed) or not (forbidden). The HOMO/LUMO of one fragment interacts with the HOMO/LUMO of the other fragment to form two new bonding and anti-bonding MOs. The nodal properties allow one to predict how the reaction will occur.

Semi-Empirical/AM1 method and the B3LYP/6-31G* method of optimisation will be used on Gaussian to determine particular properites of the reactants to predict the way in which the reaction will proceed.

==='''cis-butadiene and ethylene: reagents'''===

<jmol>
<jmolAppletButton>
<uploadedFileContents>Cis_butadienedh08.mol</uploadedFileContents>
<text>Cis-butadiene</text>
<color>whi-te</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Ethenedh08.mol</uploadedFileContents>
<text>ethylene</text>
<color>white</color>
</jmolAppletButton>
</jmol> were optimised separately using the AM1 semi-empirical method on Gaussview.The cis-butadiene had an energy of 0.04879725 a. u (1.28 x 102 KJ mol-1) and ethylene had an energy of 0.02619027 a. u (68.76 KJ mol-1).

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10: Frontier molecular orbitals of cis-butadiene and ethylene '''
| '''Reagent''' || '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| cis-butadiene || LUMO || [[Image:LUMO1_cis_butadieneDH08.jpg|100x100px]] || [[Image:LUMO2_cis_butadienedh08.jpg|100x100px]]|| Symmetric || +0.017
|-
| cis-butadiene || HOMO || [[Image:HOMO1_cis_butadienedh08.jpg|100x100px]] || [[Image:HOMO2_cis_butadienedh08.jpg|100x100px]]|| Anti-symmetric || -0.343
|-
| ethylene || LUMO || [[Image:LUMO1_ethenedh08.jpg|100x100px]] || [[Image:LUMO2_ethenedh08.jpg|100x100px]]|| Anti-symmetric || +0.052
|-
| ethylene || HOMO || [[Image:HOMO1_ethenedh08.jpg|100x100px]] || [[Image:HOMO2_ethenedh08.jpg|100x100px]]|| Symmetric || -0.387
|}

 
The symmetries of the LUMO and HOMO orbitals for cis-butadiene are opposite of those for the ethylene. This therefore means that the only interactions that can take place are those between ethylene's LUMO and cis-butadiene's HOMO and in addition to this the interaction between cis-butadiene's LUMO and ethylene's HOMO. This is because orbital symmetry has to be conserved (the conservation of orbital symmetry states that the symmetry of an new orbital formed by the overlap of the 2 original orbitals must be the same as the symmetry of the initial orbitals).

==='''cis-butadiene and ethylene: transition state'''===

A <jmol>
<jmolAppletButton>
<uploadedFileContents>Bicyclic_startingdh08.mol</uploadedFileContents>
<text>bicyclo system</text>
<color>white</color>
</jmolAppletButton>
</jmol> was drawn and one of the -CH2-CH2- fragments were removed. Three C=C bonds were added and the previous one was taken away so a <jmol>
<jmolAppletButton>
<uploadedFileContents>Frozen_coorddh08.mol</uploadedFileContents>
<text>guess transition state</text>
<color>white</color>
</jmolAppletButton>
</jmol> was produced.

After this an optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while bonds between the termini of the fragments were frozen at 2.1Å. Following this the resulting file had two separate optimisation and frequency calculations run on it using the TS (Berny) with one using the semi-empirical/AM1 method and the other using the DFT/B3LYP/6-31G* level of theory. Both of these optimisations had the bonds unfrozen and the "derivative" option was chosen instead.

The DFT/B3LYP/6-31G* level of theory was used since it gave more accurate results in the calculations before and the energies it produces are consistent with the rest of the report (semi-empirical/AM1 method gives very different values).

The QST2 optimisation was not used since it required numbering the atoms in both the reactants and products, which took a lot of time to do.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11: Comparison of two methods of optimsations for the transition state '''
| || '''semi-empirical/AM1 method''' || '''DFT/B3LYP/6-31G* method'''
|-
| Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>SE_AM1_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>DFT_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Energy (a. u) || 0.11165608 || -234.54389573
|-
| Bond lengths || [[Image:DistancesSEdh08.jpg|150x150px]]|| [[Image:DistancesDFTdh08.jpg|150x150px]]
|-
|Imaginary frequency (cm-1) || -957 || -525
|-
|Animation || [[Image:-957dh08.gif]] || [[Image:-525dh08.gif]]
|-
| Lowest real frequency (cm-1) || 148 || 136
|-
|Animation || [[Image: 148dh08.gif]] || [[Image: 136dh08.gif]]
|}

The C-C bond forming length between the termini carbons is 0.15 Å larger for the DFT/B3LYP/6-31G* level of theory and this is because this method gives carries out the calculation to a greater degree of accuracy. This larger bond length leads a lower potential energy gradient to the transition state, which gives a lower total energy for the transition state. Therefore the given values below will be the DFT/B3LYP/6-31G* and not the semi-empirical/AM1 method.

A typical sp3 C-C bond length is 1.54 Å and a typical sp2 C=C bond length is 1.34 Å<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref>. Within the optimised transition molecule above the C-C bond lengths are 1.40 Å and the C=C bond length is 1.38 Å. This suggests that the π-bonds are breaking in order for the C-C bonds to form and additionally to this a π-bond is forming in between the two sp3 carbons.

The Van der Waals radius of a C-atom is 1.72 Å<ref name="ref 3">M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer, D. G. Truhlar, 1995, 346, pp. 63-69.[http://pubs.acs.org/doi/abs/10.1021/jp8111556]</ref>. The partly formed C-C length is 2.27 Å, which is shorter than two times the the van der Waals radius of a carbon atom. Therefore the carbon atoms are within a close enough distance to each other to share electron density and yet the bond has not formed at this point.

From looking at the animations for the imaginary frequency above it can be seen that the formation of the two bonds between the termini of the reactants is synchronous. This is expected of the Diels-Alder reaction since the mechanism is concerted. However the lowest real frequency is asynchronous with respect to the two bonds forming between the termini.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12: Frontier molecular orbitals of the transition state '''
| '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| LUMO || [[Image:LUMO1TSdh08.jpg|100x100px]] || [[Image:LUMO2TSdh08.jpg|100x100px]]|| Symmetric || +0.023
|-
| HOMO || [[Image:HOMO1TSdh08.jpg|100x100px]] || [[Image:HOMO2TSdh08.jpg|100x100px]]|| Anti-symmetric || -0.323
|}

From examining the images from table 12 above it can be deduced that the LUMO of the transition state originates from the cis-butadiene's LUMO and the ethylene's HOMO, which represents the σ*-antibonding orbitals. However the HOMO of the transition state originates from the cis-butadiene's HOMO and the ethylene's LUMO, which represents the σ-bonding orbitals within the two single C-C bonds that are formed within the reaction.

==='''cis-butadiene and ethylene: activation energy'''===

The cis-butadiene and ethylene were re-optimised at the DFT/B3LYP/6-31G* level of theory so the activation energy of the reaction could be calculated.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 13: Calculation for the activation energy for the reaction'''
| '''molecule''' || '''DFT/B3LYP/6-31G* energy (a. u)''' || '''Calculations''' || '''Energy '''
|-
| cis-butadiene || -155.98595591 || transition state || -234.54389573 a. u
|-
| ethylene || -78.58745828 || ΔE, activation energy || 0.02951846 a. u
|-
| cis-butadiene and ethylene combined || -234.57341419 || ΔE, activation energy || 77.50 KJ mol-1
|}

The calculated activation energy is 77.50 KJ mol-1, which is of the right magnitude when compared to the literature value (115 ± 8 KJ mol-1 at 0 K) but there is still an error of 37.50 KJ mol-1. This shows that Gaussview is not perfect but still gives an idea of the activation energy required for the reaction to take place.

==='''Cyclohexa-1,3-diene reaction with maleic anhydride'''===

The reaction scheme below shows the stereoselective Diels-Alder cycloaddition of maleic anhydride with cyclohexa-1,3-diene. The reaction is kinetically controlled with the major product being the endo product, which will be explained later on in the exercise.

[[Image:Reaction_schemedh081.jpg|center|500x500px]]

The <jmol>
<jmolAppletButton>
<uploadedFileContents>FrozenTS22dh08.mol</uploadedFileContents>
<text>exo</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_frozenTS22dh08.mol</uploadedFileContents>
<text>endo</text>
<color>white</color>
</jmolAppletButton>
</jmol> transition states were drawn on Gaussview and then the optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while the distance of the forming bonds were frozen at 2.1Å between the respective carbons. After this the structures changed slightly (<jmol>
<jmolAppletButton>
<uploadedFileContents>TS2_afteropt1_exo1dh08.mol</uploadedFileContents>
<text>exo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_TS2_afteropt1dh08.mol</uploadedFileContents>
<text>endo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol>) and the resulting files had an optimisation and frequency calculation run on it using the TA (Berny) with using the semi-empirical/AM1 level of theory while the unfrozen option as changed to "derivative". After this the endo and exo optimised transition state structures were re-optimised as they were before but using the DFT/B3LYP/6-31G* level of theory instead with the additional keyword "Opt=NoEigen".

{| class="wikitable" border="1" align="center"
|+ '''Table 14: Comparison of the exo and endo transition states '''
! rowspan="2"|-!! colspan="2"|''exo'' !! colspan="2"|''endo''
|-
| Semi-empirical/AM1 || DFT/B3LYP/6-31G* || Semi-empirical/AM1 || DFT/B3LYP/6-31G*
|-
| align="center"|Structure ||align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>After_opt2_exodh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>Endo_TS2_afteropt2dh081.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| align="center"| Energy (a. u) ||align="center"| -0.05041975 ||align="center"| -612.67931095 ||align="center"| -0.05150470 ||align="center"| -612.68339684
|-
| C-C bond forming length (Å) ||align="center"| 2.17 ||align="center"| 2.29 ||align="center"| 2.16 ||align="center"| 2.27
|-align="center"
| Imaginary frequency (cm-1) ||align="center"| -812 ||align="center"| -448 ||align="center"| -806 ||align="center"| -447
|-align="center"
| Animation ||colspan="2"|[[Image:-812_SEdh08.gif]] || colspan="2"|[[Image:-806_cm-1dh08.gif]]
|}

Again the C-C bond forming length between the termini carbons is larger (by 0.11 and 0.12 Å) for the DFT/B3LYP/6-31G* level of theory and this is due the same reasons talked about above. However the C-C bond forming length between the termini carbons are very similar in both the endo and exo transition states (0.02 Å difference).

From looking at the animations for the imaginary frequency above it can also be seen that the formation of the two bonds between the carbons in this case as well is synchronous. Again this is expected of the Diels-Alder reaction since it is concerted.

{| align=center
|+ '''Strain diagrams for the endo and exo transition states'''
| [[Image:Exostraindh08.jpg|thumb|200x200px|'''Exo TS''']]
| [[Image:Strainendodh08.jpg|thumb|200x200px|'''Endo TS''']]
|}

From studying the energies in table 13 it can be seen that the endo transition state is more stable than the exo by 4.08589 x 10-3 a. u (10.73 KJ mol-1). However by looking at the strain of both transition states one would expect the exo TS to be more stable. This is because the distance between the cyclohexa-1,3-diene ring and the -(C=O)-O-(C=O)- maleic anhydride fragment is 2.99 Å in the endo TS and 3.03 Å in the exo TS. Thus the endo form would have more strain due to steric hindrance.

However within the exo TS the vicinal hydrogens on the CH2-CH2 (carbon numbers 18 and 21) fragment on the cyclohexa-1,3-diene ring are found below the plane and next to the maleic anhydride. This leads to steric hindrance is this is where the strain originates from within the exo TS. Therefore this disfavours the exo form, which makes the endo more stable.

Another factor making the endo TS more favourable is secondary orbital overlap. These orbitals contribute to the bonds forming and they interact with each other through space. Within the endo TS below there are 2 more orbitals interacting (secondary orbitals) than within the exo TS. This therefore makes the endo TS more stable and therefore more favoured.

[[Image:Secondary_orbital_shizzledh08.jpg|500x500px|center]]

{| class="wikitable" border="1" align="center"
|+ '''Table 15: Frontier molecular orbitals of the the transition state for the cyclohexa-1,3-diene reaction '''
! rowspan="2"|Molecular Orbital !! colspan="4"|''exo'' !! colspan="4"|''endo''
|- align="center"
| '''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''||'''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''
|- align="center"
| LUMO || [[Image:LUMO1exoh081.jpg|150x150px]]|| [[Image:LUMO2exodh081.jpg|150x150px]]|| Anti-symmetric || -0.078 || [[Image:LUMO1ENDOdh08.jpg|150x150px]]||[[Image:LUMO2ENDOdh08.jpg|150x150px]] || Anti-symmetric || -0.067
|- align="center"
| HOMO || [[Image:HOMO1exodh081.jpg|150x150px]]||[[Image:HOMO2exodh081.jpg|150x150px]] || Anti-symmetric|| -0.242 || [[Image:HOMO1ENDOdh08.jpg|150x150px]]|| [[Image:HOMO2ENDOdh08.jpg|150x150px]]||Anti-symmetric || -0.242
|}

Digital repositories:

exo: http://hdl.handle.net/10042/to-5596

endo: http://hdl.handle.net/10042/to-5597

From examining table 14 it is clear to see that both the HOMO and LUMO are anti-symmetric for both the transition states. The LUMO of the maleic anhydride fragment is the π* anti-bonding orbital of the C=C bond. Within the reaction then this has electrons transferred to it and the C=C bond breaks, which leaves just a C-C bond. The LUMOs for both transition states are both anti-bonding with respect to the two fragments. The HOMOs of both transitions state represent the σ-bonding that will be formed between the termini carbons on the products and reactants. This is most likely from the overlap of the cyclohexa-1,3-diene's HOMO and the maleic anhydride's LUMO.

=='''Conclusion'''==

='''References'''=
<references/>

Rep:Mod:hampluscheese

2010-11-11T12:27:54Z

Dh08: /* '''Cyclohexa-1,3-diene reaction with maleic anhydride''' */

='''Module 3:Physical Computational Chemistry Lab'''=

Douglas Hunt

=='''Objectives'''==

The aim of this experiment is to

=='''The Cope Rearrangement'''==

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangement] is a [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement |3,3|-sigmatropic rearrangement] that takes place through a single [http://en.wikipedia.org/wiki/Pericyclic pericyclic], concerted [http://en.wikipedia.org/wiki/Transition_state transition state]. Within this rearrangement a σ bond is formed in concert with another being broken within a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] π system.

The Cope rearrangement we will be studying will be that of 1,5-hexadiene, which is shown below.

[[Image:Cope_rearrangement_schemedh08.jpg|300x300px|center]]

The objective is to locate the low-energy minima and transition structures on the C6H10 potential energy surface and to determine the preferred reaction mechanism by using [http://en.wikipedia.org/wiki/GAUSSIAN Gaussian], with the [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)] methods.

==='''Optimisation of 1,5-hexadiene Conformers'''===

Rough <jmol>
<jmolAppletButton>
<uploadedFileContents>Anti_2_drawndh08.mol</uploadedFileContents>
<text>anti</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche3_drawndh08.mol</uploadedFileContents>
<text>gauche</text>
<color>white</color>
</jmolAppletButton>
</jmol> conformers of 1,5-hexadiene were drawn on Gaussview and their structures were tidied by the clean function. They were then further optimised by the HF/3-21G method and were identified which conformer they were from their point group and appendix 1 on the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3 physical lab page].

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 1: Optimisation and comparison of anti and gauche 1,5-hexadiene conformers using HF/3-21G methods
| ''' Conformer''' || '''Optimised Structure''' || '''Energy (Atomic Units)''' || '''[http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1 Energies (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (Anti2 - Gauche3)''' || '''Point group'''
|-
| '''Anti2''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Anti_2_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692535 || -231.69254 ||-6.08 x 105 || 1.2606 x 10-4 a. u|| Ci
|-
| '''Gauche3''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Gauche3_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692661 || -231.69266 || -6.08 x 105 || 3.31 x 10-1 KJ mol-1 || C1
|}

From studying table 1 it can be seen that the gauche conformer is 0.33 KJ mol-1 more stable than the anti conformation. One would assume that the anti2 conformer would be more stable due to [http://en.wikipedia.org/wiki/Steric steric] factors/repulsion within the gauche3 conformation. However within the gauche3 conformer there is a favourable donation of [http://en.wikipedia.org/wiki/Electron_density electron density] from the C=C π orbital to the C-H σ* orbital from the adjacent proton, which stabilises the conformer<ref name="ref 1">B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446.[http://www.informaworld.com/smpp/content~db=all?content=10.1080/00268970110081412]</ref>.

 

The anti2 conformer was then optimised using the B3LYP 6-31G* (6-31G (d)) method.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 2: Comparison of the anti2 conformer optimisations carried out at different calculation methods
| ''' Method''' || '''Structure and bond distances (Å)''' || '''Dihedral angle (C1 to C4)''' || '''Energy (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (HF/3-21G - DFT/B3LYP/6-31G)'''
|-
| HF/3-21G || [[Image:Anti2_opt1dh08.jpg|300x300px]] || 114.6 o || -231.692535 || -6.08 x 105 || 2.919171 a. u
|-
| DFT/B3LYP/6-31G* || [[Image:Anti2_opt2DH08.jpg|300x300px]] || 118.8 o || -234.611716 || -6.16 x 105 || 7.66 x 103 KJ mol-1
|}

From examining table 2 at first it is possible to see that the bond lengths are very similar to each other and that of the literature bond lengths<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref> (C1=C2: 1.340 Å, C2-C3: 1.508 Å and C3-C4: 1.538 Å). It is also possible to see that the C=C bond length is more accurate for the DFT/B3LYP/6-31G* method by it being 0.01 Å closer to the literature value. Another main difference in terms of geometry is that the DFT/B3LYP/6-31G* method gives a larger [http://en.wikipedia.org/wiki/Dihedral_angle dihedral angle] (and most likely more accurate) of 118.8o compared to that of 114.6o.

In terms of energy the DFT/B3LYP/6-31G* method produces a conformer of a higher stability by it possessing a lower energy with a difference of 7.66 x 103 KJ mol-1.

==='''Frequency analysis of 1,5-hexadiene Conformers'''===

[[Image:Vibrations_anti2dh08.jpg|right|300x300px]]

For the frequency analysis the DFT/B3LYP/6-31G* method was used.

Frequency analysis is essential within this calculation due to it being the second derivative of the potential energy surface. This means that if all the frequencies are all positive the system is at a minimum, if one of them is negative the optimised system is at a transition state, and if more than one are negative then the calculation has failed to find a critical point, which means the optimisation has failed. This therefore gives us a method of seeing whether the calculation has worked.

From looking to the image on the right it can deduced that the optimisation has taken place successfully by there being no negative vibrations and therefore the conformation is at a minimum.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 3: Energies under the thermochemistry section at 298.15 K and 0 K within the frequency logfiles'''
| '''Energies''' || '''298.15 K (atomic units)''' ||| '''0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (KJ mol-1)''' || '''Discussion'''
|-
| Sum of electronic and zero-point energies || -234.469177 || -234.466673 || -2.504 x 10-3 || -6.57 || This is the potential energy at 0 K, which includes the zero-point vibrational energy (E = Eelec + ZPE).

|-
| Sum of electronic and thermal energies || -234.461835 || -234.461085 || -7.5 x 10-4|| -1.97 || This is the energy at 298.15 K (0 K for 0 K calculation) and 1 atm of pressure, which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans).

|-
| Sum of electronic and thermal enthalpies || -234.460891 || -234.460141 || -7.5 x 10-4 || -1.97 || This contains an additional correction for RT (H = E + RT), which is especially important when looking at dissociation reactions.
|-
| Sum of electronic and thermal free energies || -234.500739 || -234.495931 || 4.808 x 10-3 || -12.62 || This includes the entropic contribution to the free energy (G = H - TS).
|}

{|align=right
|+
| [[Image:Anti2_freq_IRdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 298.15 K''']]
| [[Image:Anti2_freq_0Kdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 0 K''']]
|}

From looking at table 3 above it can be deduced that the anti2 conformer is more stable at room temperature to that of 0 K with respect to all the types of energies. The spectra also seem to not change much between the two temperatures but the main change is the change in the sum of electronic and thermal free energies, which is heavily dependent of temperature by it including the entropic contribution to the free energy (G = H - '''T'''S)

 
 
 
 

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

Within the Cope Rearrangement there are two possible transition states, the chair (C2h) and the boat (C2v), which are shown within [https://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_2 appendix two ].

==='''Comparison of optimisation methods for the chair transition state'''===

An <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2hdh08.mol</uploadedFileContents>
<text>allyl fragment</text>
<color>white</color>
</jmolAppletButton>
</jmol>, CH2CHCH2 was drawn on Gaussview and was optimized through the HF/3-21G method. After this the original allyl fragment was copied and pasted into a new window and place exactly above the first fragment with a bond distance of 2.20 Å between the terminal end carbons. This therefore produced the <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2h_fulldh08.mol</uploadedFileContents>
<text>chair transition structure.</text>
<color>white</color>
</jmolAppletButton>
</jmol>

Gaussian was then set up for an optimisation for the transition state with the additional key words "Opt=NoEigen", the optimisation set to TS (Berny), the force constants set to "once" and using the HF/3-21G method.

After this the "chair" transition state was optimised an alternative way by freezing the coordinates of the two sets of terminal carbons (where the bonds form/break during the rearrangement) within the Redundant Coord Editor on Gauusview.
Following this a second calculation was run where the freeze coordinates option was changed to "Derivative" and the force constants were set to "never".

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 4: Comparison of chair transition state optimisation methods
| || '''Optimised through a TS (Berny)''' || '''Optimised through a frozen coordinate method'''
|-
| '''Optimised TS Structure''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>C2h_optfreq_TS_Bdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| '''Terminal C-C bond length (Å)''' || 2.02 || 2.02
|-
|'''Energy (atomic units)''' || -231.61932247 || -231.61932190
|-
| '''ΔE, TS (Berny) - frozen coordinate''' || 5.7 x 10-7 a.u || 1.50 x 10-3 KJ mol-1
|-
| '''Imaginary frequency (cm-1)''' || -818 || -818
|-
| '''Animation''' ||[[Image:818dh08TSB.gif|350x350px]] || [[Image:818cm-1Fdh08.gif]]
|}

The data in table 4 shows that both methods give the same results and they both give the same transition state. They both produced the same energy with a negligible energy difference between them and the bond lengths between the terminal carbons are both 2.02 Å. In addition to this the imaginary frequencies were both -818cm-1 and within the animations above it is possible to see that when one C-C bond is forming the other is breaking. This means it is a asynchronous bond formation, which is expected for the Cope Rearrangement since it is a pericyclic reaction.

The only difference between the two optimisations is the way in which the transition states were calculated. This is due to the TS (Berny) method calculating the force constant matrix within the first optimisation step and is then being recalculated as the optimisation carries on. When the frozen coordinate method freezes the reaction coordinate in order for the rest of the structure to be optimised around it, which does not require the whole hessian (force constant matrix) to be calculated but only that the reaction coordinate is differentiated (second step). The main advantage of this, is that you do not need to calculate the entire force constant matrix and just need to differentiate the reaction coordinate in order to obtain a good enough guess for the initial force constant matrix. The frozen coordinate method will therefore take less time to compute the transition state, which can be very expensive within certain systems. However to carry out this method one needs to have an idea of how far apart the terminal carbons are.

==='''Boat transition state'''===

===='''QST2 optimisation'''====

The optimisation for the boat transition state had a different method and this was the QST2 method. This calculation interpolates between two structures (product and reactant) and tries to find a transition state between them.

The reactants and products were numbered in the same way. Therefore both 1,5-hexadiene molecules had their numbering manually changed for the product molecule so it corresponded to the numbering of the reactant if it rearranged. This is illustrated in the reaction scheme below.

[[Image:Boat_rerrangmentdh08.jpg|300x300px|center]]

The chk file for the optimized anti2 conformer was opened and copied into a new window. A second window was opened within the new window and the same anti2 conformer was copied into the new window. The numbering was changed within the second molecule so the two looked like they do in the image below.

[[Image:TS_(QST2)_numbering_1dh08.jpg|400x400px|center]]

The job type was set as "Opt+Freq" and "TS (QST2)" was chosen to optimise the transition state. However this failed and produced
a structure a bit like a <jmol>
<jmolAppletButton>
<uploadedFileContents>Trialone_BPdh08.mol</uploadedFileContents>
<text>chair transition structure </text>
<color>white</color>
</jmolAppletButton>
</jmol> but more dissociated. This is because it interpolated between the two structures and translated the top allyl fragment and therefore did not consider the possibility of a rotation around the central bonds.

To get around this the C-C-C-C dihedral angle (C2-C3-C4-C5) was set to 0 o and the C-C-C angle (C2-C3-C4 and C3-C4-C5 for the product molecule) was set to 100 o for both molecules. This gave rise to the two structures below.

[[Image:TS_(QST2)_numbering_2dh08.jpg|400x400px|center]]

After this the QST2 calculation was run again and the results in table 5 were produced. The imaginary frequency at -839 cm-1 shows that while one bond is forming the other one is breaking, which is again an asynchronous bond formation.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5: Data for the boat transition state from the QST2 optimisation method
| Optimised TS Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.14
|-
|Energy (atomic units) || -231.60280151
|-
| Energy (KJ mol-1) || -6.08 x 105
|-
| Imaginary frequency (cm-1) || -839
|-
| Animation ||[[Image:-839dh08.gif]]
|}

===='''Intrinsic Reaction Coordinate'''====

It is not possible to predict which product the chair and boat transition state will lead to. To solve this problem within Gaussian one can use the IRC method, which allows one to follow the minimum energy path from a transition structure down to its local minimum on a potential energy surface. This creates a succession of points by taking small geometry steps in the direction to where the gradient of the energy surface is at its steepest until the minimum in energy is reached.

The chk file for the optimized chair transition structure was opened and the job type selected was "IRC", the reaction coordinate was computed in the forward direction (since molecule is symmetric) with 50 points along the IRC. This was done with two type of force constants being calculated, once and always.

Within the "once" method the output showed no bond formation process by the 26th step and therefore the calculation was incomplete, since a complete 1,5-hexadiene structure was supposed to be formed. From looking at the IRC pathway it can be seen that the minimum on the potential energy surface was not reached at that point, since the points on the pathway did not form an asymptote and the gradient did not go close enough to zero.

For the "always" method the computation of force constants were carried out at every step. This therefore lets the IRC calculation distinguish whether it is moving along the minimum energy pathway while going down the potential energy surface. While this is happening it can make changes to the pathway if necessary. Within the "always" case the bond forming arrives at 1,5-hexadiene and this is backed up by the IRC pathway reaching a minimum within its energy by the gradient going close to zero and the pathway reaches an asymptote.

The structure produced through the always method had an energy of -231.69199434 atomic units, which is very close to the energy of the gauche2 conformer in the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 appendix 1]. This suggests that the reaction takes place through the chair transition state, which later produces the gauche2 conformer of 1,5-hexadiene.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 6: Comparison of IRC calculations for the chair structure
| || '''Calculate force constants once''' || '''Calculate force constants always'''
|-
| '''Number of points along IRC''' || 26 || 47
|-
| '''Structure at the last point of IRC''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Once_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Always_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|'''Energy (atomic units)''' || -231.68906923 || -231.69199434
|-
| '''ΔE, once - always''' || 2.92511 x 10-3 a.u || 7.68 KJ mol-1
|-
| '''IRC pathway''' || [[Image:One_pathwaydh08.jpg|400x400px]] || [[Image:Always_pathwaydh08.jpg|400x400px]]
|-
| '''IRC gradient''' || [[Image:One_gradientdh08.jpg|400x400px]]|| [[Image:Always_gradientdh08.jpg|400x400px]]
|}

 
 
 

===='''Comparison of activation energies for the transition states'''====

The HF/3-21G optimised chair and boat transition structures were both reoptimised using the B3LYP/6-31G* (B3LYP/6-31G (d)) method. This gives a more accurate minimisation however this method takes longer to run. The values for the zero-point energies and thermal energies were found within the thermochemistry section of the log file.

The values for the HF/3-21G reactant (anti2) optimisation were calculated and the B3LYP/6-31G* values were taken from the "Frequency analysis of 1,5-hexadiene Conformers" earlier in this module.

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 7: Energy data for chair, boat and the anti2 conformation from the HF/3-21G and B3LYP/6-31G* methods of optimisation '''
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466677
| align="center" | -231.461320
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450936
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402342
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539524
| align="center" | -231.532562
| align="center" | -234.611716
| align="center" | -234.469177
| align="center" | -234.461835
|}
Digital repositories

Chair:
HF/3-21G: http://hdl.handle.net/10042/to-5569
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5568

Boat:
HF/3-21G: http://hdl.handle.net/10042/to-5570
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5571

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 8: Summary of activation energies for the chair and boat conformations (ΔE = chair TS or boat TS minus anti2)'''
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| align="center" | 1 a.u = 627.509 kcal mol-1
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE, chair (a.u)'''
| align="center" | 0.072847
| align="center" | 0.071242
| align="center" | 0.054248
| align="center" | 0.052827
| align="center" | -
|-
| '''ΔE, chair (kcal mol-1)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE, boat (a. u)'''
| align="center" | 0.088588
| align="center" | 0.087259
| align="center" | 0.066835
| align="center" | 0.065827
| align="center" | -
|-
| '''ΔE, boat (kcal mol-1)'''
| align="center" | 55.59
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.31
| align="center" | 44.7 ± 2.0
|}

By examining table 8 it is clear to see that the B3LYP/6-31G* method gives values which are much closer to the experimental ones. This is due to the B3LYP/6-31G* giving a higher level of optimisation than the HF/3-21G method. Table 8 also shows that the chair transition state has a lower activation energy to that of the boat transition state by 8.16 kcal mol-1. This suggests that the Cope rearrangement mechanism would most likely proceed through the chair transition state to that of the boat. Therefore the reaction would take place from the anti2 conformer to the chair transition state and then onto the gauche2 conformer within the Cope rearrangement.

It is also possible to see that the calcualted energies are very close to the literature values with the chair conformation being closer to its literature value to that of the boat.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9: Geometric parameters for the optimised chair and boat structures '''
| || ''' HF/3-21G chair TS''' || ''' B3LYP/6-31G* chair TS''' || '''HF/3-21G boat TS''' || '''B3LYP/6-31G* boat TS'''
|-
| Optimised Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Chair_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Boat_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.02 (C5-C10 or C3-C14)|| 1.97 (C5-C10 or C3-C14) || 2.14 (C3-C4 or C1-C6) || 2.21 (C3-C4 or C1-C6)
|-
| Fragment angle || 120 o (C3-C1-C5) || 120 o (C3-C1-C5) || 122 o (C4-C5-C6) || 122 o (C4-C5-C6)
|-
| Dihedral angle || 55 o (C1-C3-C14-C11)|| 54 o (C1-C3-C14-C11)|| 0 o (C2-C1-C6-C5) || 0 o (C2-C1-C6-C5)
|-
| Numbering of carbons || [[Image:Numbering_chairdh08.jpg|200x200px]]|| [[Image:Numbering_chairdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]]
|}

From studying table 9 it can be seen that the C-C bond breaking/forming length has changed between the two different types of optimisation methods. However the angles between the bonds remains unchanged.

=='''The Diels Alder Cycloaddition'''==

The Diels-Alder reaction is a cycloaddition between a conjugated diene and a dienophile (alkene system), which forms a cyclohexene system. It is often referred to as a [4s+2s] cycloaddition and proceeds through a single cyclic, concerted transition state where 2 σ bonds are formed in between the termini carbons of the 2 conjugated п systems. This is shown in the reaction scheme below where ethene is reacting with cis-butadiene.

[[Image:Reaction_scheme2dh08.jpg|300x300px|center]]

The π orbitals of the dienophile form new σ bonds with the π orbitals of the diene. The number of π electrons involved determine whether the reaction proceeds in a concerted stereospecific fashion (allowed) or not (forbidden). The HOMO/LUMO of one fragment interacts with the HOMO/LUMO of the other fragment to form two new bonding and anti-bonding MOs. The nodal properties allow one to predict how the reaction will occur.

Semi-Empirical/AM1 method and the B3LYP/6-31G* method of optimisation will be used on Gaussian to determine particular properites of the reactants to predict the way in which the reaction will proceed.

==='''cis-butadiene and ethylene: reagents'''===

<jmol>
<jmolAppletButton>
<uploadedFileContents>Cis_butadienedh08.mol</uploadedFileContents>
<text>Cis-butadiene</text>
<color>whi-te</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Ethenedh08.mol</uploadedFileContents>
<text>ethylene</text>
<color>white</color>
</jmolAppletButton>
</jmol> were optimised separately using the AM1 semi-empirical method on Gaussview.The cis-butadiene had an energy of 0.04879725 a. u (1.28 x 102 KJ mol-1) and ethylene had an energy of 0.02619027 a. u (68.76 KJ mol-1).

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10: Frontier molecular orbitals of cis-butadiene and ethylene '''
| '''Reagent''' || '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| cis-butadiene || LUMO || [[Image:LUMO1_cis_butadieneDH08.jpg|100x100px]] || [[Image:LUMO2_cis_butadienedh08.jpg|100x100px]]|| Symmetric || +0.017
|-
| cis-butadiene || HOMO || [[Image:HOMO1_cis_butadienedh08.jpg|100x100px]] || [[Image:HOMO2_cis_butadienedh08.jpg|100x100px]]|| Anti-symmetric || -0.343
|-
| ethylene || LUMO || [[Image:LUMO1_ethenedh08.jpg|100x100px]] || [[Image:LUMO2_ethenedh08.jpg|100x100px]]|| Anti-symmetric || +0.052
|-
| ethylene || HOMO || [[Image:HOMO1_ethenedh08.jpg|100x100px]] || [[Image:HOMO2_ethenedh08.jpg|100x100px]]|| Symmetric || -0.387
|}

 
The symmetries of the LUMO and HOMO orbitals for cis-butadiene are opposite of those for the ethylene. This therefore means that the only interactions that can take place are those between ethylene's LUMO and cis-butadiene's HOMO and in addition to this the interaction between cis-butadiene's LUMO and ethylene's HOMO. This is because orbital symmetry has to be conserved (the conservation of orbital symmetry states that the symmetry of an new orbital formed by the overlap of the 2 original orbitals must be the same as the symmetry of the initial orbitals).

==='''cis-butadiene and ethylene: transition state'''===

A <jmol>
<jmolAppletButton>
<uploadedFileContents>Bicyclic_startingdh08.mol</uploadedFileContents>
<text>bicyclo system</text>
<color>white</color>
</jmolAppletButton>
</jmol> was drawn and one of the -CH2-CH2- fragments were removed. Three C=C bonds were added and the previous one was taken away so a <jmol>
<jmolAppletButton>
<uploadedFileContents>Frozen_coorddh08.mol</uploadedFileContents>
<text>guess transition state</text>
<color>white</color>
</jmolAppletButton>
</jmol> was produced.

After this an optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while bonds between the termini of the fragments were frozen at 2.1Å. Following this the resulting file had two separate optimisation and frequency calculations run on it using the TS (Berny) with one using the semi-empirical/AM1 method and the other using the DFT/B3LYP/6-31G* level of theory. Both of these optimisations had the bonds unfrozen and the "derivative" option was chosen instead.

The DFT/B3LYP/6-31G* level of theory was used since it gave more accurate results in the calculations before and the energies it produces are consistent with the rest of the report (semi-empirical/AM1 method gives very different values).

The QST2 optimisation was not used since it required numbering the atoms in both the reactants and products, which took a lot of time to do.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11: Comparison of two methods of optimsations for the transition state '''
| || '''semi-empirical/AM1 method''' || '''DFT/B3LYP/6-31G* method'''
|-
| Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>SE_AM1_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>DFT_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Energy (a. u) || 0.11165608 || -234.54389573
|-
| Bond lengths || [[Image:DistancesSEdh08.jpg|150x150px]]|| [[Image:DistancesDFTdh08.jpg|150x150px]]
|-
|Imaginary frequency (cm-1) || -957 || -525
|-
|Animation || [[Image:-957dh08.gif]] || [[Image:-525dh08.gif]]
|-
| Lowest real frequency (cm-1) || 148 || 136
|-
|Animation || [[Image: 148dh08.gif]] || [[Image: 136dh08.gif]]
|}

The C-C bond forming length between the termini carbons is 0.15 Å larger for the DFT/B3LYP/6-31G* level of theory and this is because this method gives carries out the calculation to a greater degree of accuracy. This larger bond length leads a lower potential energy gradient to the transition state, which gives a lower total energy for the transition state. Therefore the given values below will be the DFT/B3LYP/6-31G* and not the semi-empirical/AM1 method.

A typical sp3 C-C bond length is 1.54 Å and a typical sp2 C=C bond length is 1.34 Å<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref>. Within the optimised transition molecule above the C-C bond lengths are 1.40 Å and the C=C bond length is 1.38 Å. This suggests that the π-bonds are breaking in order for the C-C bonds to form and additionally to this a π-bond is forming in between the two sp3 carbons.

The Van der Waals radius of a C-atom is 1.72 Å<ref name="ref 3">M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer, D. G. Truhlar, 1995, 346, pp. 63-69.[http://pubs.acs.org/doi/abs/10.1021/jp8111556]</ref>. The partly formed C-C length is 2.27 Å, which is shorter than two times the the van der Waals radius of a carbon atom. Therefore the carbon atoms are within a close enough distance to each other to share electron density and yet the bond has not formed at this point.

From looking at the animations for the imaginary frequency above it can be seen that the formation of the two bonds between the termini of the reactants is synchronous. This is expected of the Diels-Alder reaction since the mechanism is concerted. However the lowest real frequency is asynchronous with respect to the two bonds forming between the termini.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12: Frontier molecular orbitals of the transition state '''
| '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| LUMO || [[Image:LUMO1TSdh08.jpg|100x100px]] || [[Image:LUMO2TSdh08.jpg|100x100px]]|| Symmetric || +0.023
|-
| HOMO || [[Image:HOMO1TSdh08.jpg|100x100px]] || [[Image:HOMO2TSdh08.jpg|100x100px]]|| Anti-symmetric || -0.323
|}

From examining the images from table 12 above it can be deduced that the LUMO of the transition state originates from the cis-butadiene's LUMO and the ethylene's HOMO, which represents the σ*-antibonding orbitals. However the HOMO of the transition state originates from the cis-butadiene's HOMO and the ethylene's LUMO, which represents the σ-bonding orbitals within the two single C-C bonds that are formed within the reaction.

==='''cis-butadiene and ethylene: activation energy'''===

The cis-butadiene and ethylene were re-optimised at the DFT/B3LYP/6-31G* level of theory so the activation energy of the reaction could be calculated.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 13: Calculation for the activation energy for the reaction'''
| '''molecule''' || '''DFT/B3LYP/6-31G* energy (a. u)''' || '''Calculations''' || '''Energy '''
|-
| cis-butadiene || -155.98595591 || transition state || -234.54389573 a. u
|-
| ethylene || -78.58745828 || ΔE, activation energy || 0.02951846 a. u
|-
| cis-butadiene and ethylene combined || -234.57341419 || ΔE, activation energy || 77.50 KJ mol-1
|}

The calculated activation energy is 77.50 KJ mol-1, which is of the right magnitude when compared to the literature value (115 ± 8 KJ mol-1 at 0 K) but there is still an error of 37.50 KJ mol-1. This shows that Gaussview is not perfect but still gives an idea of the activation energy required for the reaction to take place.

==='''Cyclohexa-1,3-diene reaction with maleic anhydride'''===

[[Image:Reaction_schemedh081.jpg|center|500x500px]]

The <jmol>
<jmolAppletButton>
<uploadedFileContents>FrozenTS22dh08.mol</uploadedFileContents>
<text>exo</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_frozenTS22dh08.mol</uploadedFileContents>
<text>endo</text>
<color>white</color>
</jmolAppletButton>
</jmol> transition states were drawn on Gaussview and then the optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while the distance of the forming bonds were frozen at 2.1Å between the respective carbons. After this the structures changed slightly (<jmol>
<jmolAppletButton>
<uploadedFileContents>TS2_afteropt1_exo1dh08.mol</uploadedFileContents>
<text>exo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_TS2_afteropt1dh08.mol</uploadedFileContents>
<text>endo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol>) and the resulting files had an optimisation and frequency calculation run on it using the TA (Berny) with using the semi-empirical/AM1 level of theory while the unfrozen option as changed to "derivative". After this the endo and exo optimised transition state structures were re-optimised as they were before but using the DFT/B3LYP/6-31G* level of theory instead with the additional keyword "Opt=NoEigen".

{| class="wikitable" border="1" align="center"
|+ '''Table 14: Comparison of the exo and endo transition states '''
! rowspan="2"|-!! colspan="2"|''exo'' !! colspan="2"|''endo''
|-
| Semi-empirical/AM1 || DFT/B3LYP/6-31G* || Semi-empirical/AM1 || DFT/B3LYP/6-31G*
|-
| align="center"|Structure ||align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>After_opt2_exodh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>Endo_TS2_afteropt2dh081.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| align="center"| Energy (a. u) ||align="center"| -0.05041975 ||align="center"| -612.67931095 ||align="center"| -0.05150470 ||align="center"| -612.68339684
|-
| C-C bond forming length (Å) ||align="center"| 2.17 ||align="center"| 2.29 ||align="center"| 2.16 ||align="center"| 2.27
|-align="center"
| Imaginary frequency (cm-1) ||align="center"| -812 ||align="center"| -448 ||align="center"| -806 ||align="center"| -447
|-align="center"
| Animation ||colspan="2"|[[Image:-812_SEdh08.gif]] || colspan="2"|[[Image:-806_cm-1dh08.gif]]
|}

Again the C-C bond forming length between the termini carbons is larger (by 0.11 and 0.12 Å) for the DFT/B3LYP/6-31G* level of theory and this is due the same reasons talked about above. However the C-C bond forming length between the termini carbons are very similar in both the endo and exo transition states (0.02 Å difference).

From looking at the animations for the imaginary frequency above it can also be seen that the formation of the two bonds between the carbons in this case as well is synchronous. Again this is expected of the Diels-Alder reaction since it is concerted.

{| align=center
|+ '''Strain diagrams for the endo and exo transition states'''
| [[Image:Exostraindh08.jpg|thumb|200x200px|'''Exo TS''']]
| [[Image:Strainendodh08.jpg|thumb|200x200px|'''Endo TS''']]
|}

From studying the energies in table 13 it can be seen that the endo transition state is more stable than the exo by 4.08589 x 10-3 a. u (10.73 KJ mol-1). However by looking at the strain of both transition states one would expect the exo TS to be more stable. This is because the distance between the cyclohexa-1,3-diene ring and the -(C=O)-O-(C=O)- maleic anhydride fragment is 2.99 Å in the endo TS and 3.03 Å in the exo TS. Thus the endo form would have more strain due to steric hindrance.

However within the exo TS the vicinal hydrogens on the CH2-CH2 (carbon numbers 18 and 21) fragment on the cyclohexa-1,3-diene ring are found below the plane and next to the maleic anhydride. This leads to steric hindrance is this is where the strain originates from within the exo TS. Therefore this disfavours the exo form, which makes the endo more stable.

Another factor making the endo TS more favourable is secondary orbital overlap. These orbitals contribute to the bonds forming and they interact with each other through space. Within the endo TS below there are 2 more orbitals interacting (secondary orbitals) than within the exo TS. This therefore makes the endo TS more stable and therefore more favoured.

[[Image:Secondary_orbital_shizzledh08.jpg|500x500px|center]]

{| class="wikitable" border="1" align="center"
|+ '''Table 15: Frontier molecular orbitals of the the transition state for the cyclohexa-1,3-diene reaction '''
! rowspan="2"|Molecular Orbital !! colspan="4"|''exo'' !! colspan="4"|''endo''
|- align="center"
| '''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''||'''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''
|- align="center"
| LUMO || [[Image:LUMO1exoh081.jpg|150x150px]]|| [[Image:LUMO2exodh081.jpg|150x150px]]|| Anti-symmetric || -0.078 || [[Image:LUMO1ENDOdh08.jpg|150x150px]]||[[Image:LUMO2ENDOdh08.jpg|150x150px]] || Anti-symmetric || -0.067
|- align="center"
| HOMO || [[Image:HOMO1exodh081.jpg|150x150px]]||[[Image:HOMO2exodh081.jpg|150x150px]] || Anti-symmetric|| -0.242 || [[Image:HOMO1ENDOdh08.jpg|150x150px]]|| [[Image:HOMO2ENDOdh08.jpg|150x150px]]||Anti-symmetric || -0.242
|}

Digital repositories:

exo: http://hdl.handle.net/10042/to-5596

endo: http://hdl.handle.net/10042/to-5597

From examining table 14 it is clear to see that both the HOMO and LUMO are anti-symmetric for both the transition states. The LUMO of the maleic anhydride fragment is the π* anti-bonding orbital of the C=C bond. Within the reaction then this has electrons transferred to it and the C=C bond breaks, which leaves just a C-C bond. The LUMOs for both transition states are both anti-bonding with respect to the two fragments. The HOMOs of both transitions state represent the σ-bonding that will be formed between the termini carbons on the products and reactants. This is most likely from the overlap of the cyclohexa-1,3-diene's HOMO and the maleic anhydride's LUMO.

=='''Conclusion'''==

='''References'''=
<references/>

File:Reaction schemedh081.jpg

2010-11-11T12:27:21Z

Dh08:

Rep:Mod:hampluscheese

2010-11-11T12:07:26Z

Dh08: /* '''Comparison of activation energies for the transition states''' */

='''Module 3:Physical Computational Chemistry Lab'''=

Douglas Hunt

=='''Objectives'''==

The aim of this experiment is to

=='''The Cope Rearrangement'''==

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangement] is a [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement |3,3|-sigmatropic rearrangement] that takes place through a single [http://en.wikipedia.org/wiki/Pericyclic pericyclic], concerted [http://en.wikipedia.org/wiki/Transition_state transition state]. Within this rearrangement a σ bond is formed in concert with another being broken within a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] π system.

The Cope rearrangement we will be studying will be that of 1,5-hexadiene, which is shown below.

[[Image:Cope_rearrangement_schemedh08.jpg|300x300px|center]]

The objective is to locate the low-energy minima and transition structures on the C6H10 potential energy surface and to determine the preferred reaction mechanism by using [http://en.wikipedia.org/wiki/GAUSSIAN Gaussian], with the [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)] methods.

==='''Optimisation of 1,5-hexadiene Conformers'''===

Rough <jmol>
<jmolAppletButton>
<uploadedFileContents>Anti_2_drawndh08.mol</uploadedFileContents>
<text>anti</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche3_drawndh08.mol</uploadedFileContents>
<text>gauche</text>
<color>white</color>
</jmolAppletButton>
</jmol> conformers of 1,5-hexadiene were drawn on Gaussview and their structures were tidied by the clean function. They were then further optimised by the HF/3-21G method and were identified which conformer they were from their point group and appendix 1 on the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3 physical lab page].

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 1: Optimisation and comparison of anti and gauche 1,5-hexadiene conformers using HF/3-21G methods
| ''' Conformer''' || '''Optimised Structure''' || '''Energy (Atomic Units)''' || '''[http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1 Energies (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (Anti2 - Gauche3)''' || '''Point group'''
|-
| '''Anti2''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Anti_2_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692535 || -231.69254 ||-6.08 x 105 || 1.2606 x 10-4 a. u|| Ci
|-
| '''Gauche3''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Gauche3_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692661 || -231.69266 || -6.08 x 105 || 3.31 x 10-1 KJ mol-1 || C1
|}

From studying table 1 it can be seen that the gauche conformer is 0.33 KJ mol-1 more stable than the anti conformation. One would assume that the anti2 conformer would be more stable due to [http://en.wikipedia.org/wiki/Steric steric] factors/repulsion within the gauche3 conformation. However within the gauche3 conformer there is a favourable donation of [http://en.wikipedia.org/wiki/Electron_density electron density] from the C=C π orbital to the C-H σ* orbital from the adjacent proton, which stabilises the conformer<ref name="ref 1">B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446.[http://www.informaworld.com/smpp/content~db=all?content=10.1080/00268970110081412]</ref>.

 

The anti2 conformer was then optimised using the B3LYP 6-31G* (6-31G (d)) method.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 2: Comparison of the anti2 conformer optimisations carried out at different calculation methods
| ''' Method''' || '''Structure and bond distances (Å)''' || '''Dihedral angle (C1 to C4)''' || '''Energy (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (HF/3-21G - DFT/B3LYP/6-31G)'''
|-
| HF/3-21G || [[Image:Anti2_opt1dh08.jpg|300x300px]] || 114.6 o || -231.692535 || -6.08 x 105 || 2.919171 a. u
|-
| DFT/B3LYP/6-31G* || [[Image:Anti2_opt2DH08.jpg|300x300px]] || 118.8 o || -234.611716 || -6.16 x 105 || 7.66 x 103 KJ mol-1
|}

From examining table 2 at first it is possible to see that the bond lengths are very similar to each other and that of the literature bond lengths<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref> (C1=C2: 1.340 Å, C2-C3: 1.508 Å and C3-C4: 1.538 Å). It is also possible to see that the C=C bond length is more accurate for the DFT/B3LYP/6-31G* method by it being 0.01 Å closer to the literature value. Another main difference in terms of geometry is that the DFT/B3LYP/6-31G* method gives a larger [http://en.wikipedia.org/wiki/Dihedral_angle dihedral angle] (and most likely more accurate) of 118.8o compared to that of 114.6o.

In terms of energy the DFT/B3LYP/6-31G* method produces a conformer of a higher stability by it possessing a lower energy with a difference of 7.66 x 103 KJ mol-1.

==='''Frequency analysis of 1,5-hexadiene Conformers'''===

[[Image:Vibrations_anti2dh08.jpg|right|300x300px]]

For the frequency analysis the DFT/B3LYP/6-31G* method was used.

Frequency analysis is essential within this calculation due to it being the second derivative of the potential energy surface. This means that if all the frequencies are all positive the system is at a minimum, if one of them is negative the optimised system is at a transition state, and if more than one are negative then the calculation has failed to find a critical point, which means the optimisation has failed. This therefore gives us a method of seeing whether the calculation has worked.

From looking to the image on the right it can deduced that the optimisation has taken place successfully by there being no negative vibrations and therefore the conformation is at a minimum.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 3: Energies under the thermochemistry section at 298.15 K and 0 K within the frequency logfiles'''
| '''Energies''' || '''298.15 K (atomic units)''' ||| '''0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (KJ mol-1)''' || '''Discussion'''
|-
| Sum of electronic and zero-point energies || -234.469177 || -234.466673 || -2.504 x 10-3 || -6.57 || This is the potential energy at 0 K, which includes the zero-point vibrational energy (E = Eelec + ZPE).

|-
| Sum of electronic and thermal energies || -234.461835 || -234.461085 || -7.5 x 10-4|| -1.97 || This is the energy at 298.15 K (0 K for 0 K calculation) and 1 atm of pressure, which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans).

|-
| Sum of electronic and thermal enthalpies || -234.460891 || -234.460141 || -7.5 x 10-4 || -1.97 || This contains an additional correction for RT (H = E + RT), which is especially important when looking at dissociation reactions.
|-
| Sum of electronic and thermal free energies || -234.500739 || -234.495931 || 4.808 x 10-3 || -12.62 || This includes the entropic contribution to the free energy (G = H - TS).
|}

{|align=right
|+
| [[Image:Anti2_freq_IRdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 298.15 K''']]
| [[Image:Anti2_freq_0Kdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 0 K''']]
|}

From looking at table 3 above it can be deduced that the anti2 conformer is more stable at room temperature to that of 0 K with respect to all the types of energies. The spectra also seem to not change much between the two temperatures but the main change is the change in the sum of electronic and thermal free energies, which is heavily dependent of temperature by it including the entropic contribution to the free energy (G = H - '''T'''S)

 
 
 
 

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

Within the Cope Rearrangement there are two possible transition states, the chair (C2h) and the boat (C2v), which are shown within [https://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_2 appendix two ].

==='''Comparison of optimisation methods for the chair transition state'''===

An <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2hdh08.mol</uploadedFileContents>
<text>allyl fragment</text>
<color>white</color>
</jmolAppletButton>
</jmol>, CH2CHCH2 was drawn on Gaussview and was optimized through the HF/3-21G method. After this the original allyl fragment was copied and pasted into a new window and place exactly above the first fragment with a bond distance of 2.20 Å between the terminal end carbons. This therefore produced the <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2h_fulldh08.mol</uploadedFileContents>
<text>chair transition structure.</text>
<color>white</color>
</jmolAppletButton>
</jmol>

Gaussian was then set up for an optimisation for the transition state with the additional key words "Opt=NoEigen", the optimisation set to TS (Berny), the force constants set to "once" and using the HF/3-21G method.

After this the "chair" transition state was optimised an alternative way by freezing the coordinates of the two sets of terminal carbons (where the bonds form/break during the rearrangement) within the Redundant Coord Editor on Gauusview.
Following this a second calculation was run where the freeze coordinates option was changed to "Derivative" and the force constants were set to "never".

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 4: Comparison of chair transition state optimisation methods
| || '''Optimised through a TS (Berny)''' || '''Optimised through a frozen coordinate method'''
|-
| '''Optimised TS Structure''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>C2h_optfreq_TS_Bdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| '''Terminal C-C bond length (Å)''' || 2.02 || 2.02
|-
|'''Energy (atomic units)''' || -231.61932247 || -231.61932190
|-
| '''ΔE, TS (Berny) - frozen coordinate''' || 5.7 x 10-7 a.u || 1.50 x 10-3 KJ mol-1
|-
| '''Imaginary frequency (cm-1)''' || -818 || -818
|-
| '''Animation''' ||[[Image:818dh08TSB.gif|350x350px]] || [[Image:818cm-1Fdh08.gif]]
|}

The data in table 4 shows that both methods give the same results and they both give the same transition state. They both produced the same energy with a negligible energy difference between them and the bond lengths between the terminal carbons are both 2.02 Å. In addition to this the imaginary frequencies were both -818cm-1 and within the animations above it is possible to see that when one C-C bond is forming the other is breaking. This means it is a asynchronous bond formation, which is expected for the Cope Rearrangement since it is a pericyclic reaction.

The only difference between the two optimisations is the way in which the transition states were calculated. This is due to the TS (Berny) method calculating the force constant matrix within the first optimisation step and is then being recalculated as the optimisation carries on. When the frozen coordinate method freezes the reaction coordinate in order for the rest of the structure to be optimised around it, which does not require the whole hessian (force constant matrix) to be calculated but only that the reaction coordinate is differentiated (second step). The main advantage of this, is that you do not need to calculate the entire force constant matrix and just need to differentiate the reaction coordinate in order to obtain a good enough guess for the initial force constant matrix. The frozen coordinate method will therefore take less time to compute the transition state, which can be very expensive within certain systems. However to carry out this method one needs to have an idea of how far apart the terminal carbons are.

==='''Boat transition state'''===

===='''QST2 optimisation'''====

The optimisation for the boat transition state had a different method and this was the QST2 method. This calculation interpolates between two structures (product and reactant) and tries to find a transition state between them.

The reactants and products were numbered in the same way. Therefore both 1,5-hexadiene molecules had their numbering manually changed for the product molecule so it corresponded to the numbering of the reactant if it rearranged. This is illustrated in the reaction scheme below.

[[Image:Boat_rerrangmentdh08.jpg|300x300px|center]]

The chk file for the optimized anti2 conformer was opened and copied into a new window. A second window was opened within the new window and the same anti2 conformer was copied into the new window. The numbering was changed within the second molecule so the two looked like they do in the image below.

[[Image:TS_(QST2)_numbering_1dh08.jpg|400x400px|center]]

The job type was set as "Opt+Freq" and "TS (QST2)" was chosen to optimise the transition state. However this failed and produced
a structure a bit like a <jmol>
<jmolAppletButton>
<uploadedFileContents>Trialone_BPdh08.mol</uploadedFileContents>
<text>chair transition structure </text>
<color>white</color>
</jmolAppletButton>
</jmol> but more dissociated. This is because it interpolated between the two structures and translated the top allyl fragment and therefore did not consider the possibility of a rotation around the central bonds.

To get around this the C-C-C-C dihedral angle (C2-C3-C4-C5) was set to 0 o and the C-C-C angle (C2-C3-C4 and C3-C4-C5 for the product molecule) was set to 100 o for both molecules. This gave rise to the two structures below.

[[Image:TS_(QST2)_numbering_2dh08.jpg|400x400px|center]]

After this the QST2 calculation was run again and the results in table 5 were produced. The imaginary frequency at -839 cm-1 shows that while one bond is forming the other one is breaking, which is again an asynchronous bond formation.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5: Data for the boat transition state from the QST2 optimisation method
| Optimised TS Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.14
|-
|Energy (atomic units) || -231.60280151
|-
| Energy (KJ mol-1) || -6.08 x 105
|-
| Imaginary frequency (cm-1) || -839
|-
| Animation ||[[Image:-839dh08.gif]]
|}

===='''Intrinsic Reaction Coordinate'''====

It is not possible to predict which product the chair and boat transition state will lead to. To solve this problem within Gaussian one can use the IRC method, which allows one to follow the minimum energy path from a transition structure down to its local minimum on a potential energy surface. This creates a succession of points by taking small geometry steps in the direction to where the gradient of the energy surface is at its steepest until the minimum in energy is reached.

The chk file for the optimized chair transition structure was opened and the job type selected was "IRC", the reaction coordinate was computed in the forward direction (since molecule is symmetric) with 50 points along the IRC. This was done with two type of force constants being calculated, once and always.

Within the "once" method the output showed no bond formation process by the 26th step and therefore the calculation was incomplete, since a complete 1,5-hexadiene structure was supposed to be formed. From looking at the IRC pathway it can be seen that the minimum on the potential energy surface was not reached at that point, since the points on the pathway did not form an asymptote and the gradient did not go close enough to zero.

For the "always" method the computation of force constants were carried out at every step. This therefore lets the IRC calculation distinguish whether it is moving along the minimum energy pathway while going down the potential energy surface. While this is happening it can make changes to the pathway if necessary. Within the "always" case the bond forming arrives at 1,5-hexadiene and this is backed up by the IRC pathway reaching a minimum within its energy by the gradient going close to zero and the pathway reaches an asymptote.

The structure produced through the always method had an energy of -231.69199434 atomic units, which is very close to the energy of the gauche2 conformer in the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 appendix 1]. This suggests that the reaction takes place through the chair transition state, which later produces the gauche2 conformer of 1,5-hexadiene.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 6: Comparison of IRC calculations for the chair structure
| || '''Calculate force constants once''' || '''Calculate force constants always'''
|-
| '''Number of points along IRC''' || 26 || 47
|-
| '''Structure at the last point of IRC''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Once_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Always_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|'''Energy (atomic units)''' || -231.68906923 || -231.69199434
|-
| '''ΔE, once - always''' || 2.92511 x 10-3 a.u || 7.68 KJ mol-1
|-
| '''IRC pathway''' || [[Image:One_pathwaydh08.jpg|400x400px]] || [[Image:Always_pathwaydh08.jpg|400x400px]]
|-
| '''IRC gradient''' || [[Image:One_gradientdh08.jpg|400x400px]]|| [[Image:Always_gradientdh08.jpg|400x400px]]
|}

 
 
 

===='''Comparison of activation energies for the transition states'''====

The HF/3-21G optimised chair and boat transition structures were both reoptimised using the B3LYP/6-31G* (B3LYP/6-31G (d)) method. This gives a more accurate minimisation however this method takes longer to run. The values for the zero-point energies and thermal energies were found within the thermochemistry section of the log file.

The values for the HF/3-21G reactant (anti2) optimisation were calculated and the B3LYP/6-31G* values were taken from the "Frequency analysis of 1,5-hexadiene Conformers" earlier in this module.

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 7: Energy data for chair, boat and the anti2 conformation from the HF/3-21G and B3LYP/6-31G* methods of optimisation '''
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466677
| align="center" | -231.461320
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450936
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402342
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539524
| align="center" | -231.532562
| align="center" | -234.611716
| align="center" | -234.469177
| align="center" | -234.461835
|}
Digital repositories

Chair:
HF/3-21G: http://hdl.handle.net/10042/to-5569
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5568

Boat:
HF/3-21G: http://hdl.handle.net/10042/to-5570
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5571

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 8: Summary of activation energies for the chair and boat conformations (ΔE = chair TS or boat TS minus anti2)'''
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| align="center" | 1 a.u = 627.509 kcal mol-1
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE, chair (a.u)'''
| align="center" | 0.072847
| align="center" | 0.071242
| align="center" | 0.054248
| align="center" | 0.052827
| align="center" | -
|-
| '''ΔE, chair (kcal mol-1)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE, boat (a. u)'''
| align="center" | 0.088588
| align="center" | 0.087259
| align="center" | 0.066835
| align="center" | 0.065827
| align="center" | -
|-
| '''ΔE, boat (kcal mol-1)'''
| align="center" | 55.59
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.31
| align="center" | 44.7 ± 2.0
|}

By examining table 8 it is clear to see that the B3LYP/6-31G* method gives values which are much closer to the experimental ones. This is due to the B3LYP/6-31G* giving a higher level of optimisation than the HF/3-21G method. Table 8 also shows that the chair transition state has a lower activation energy to that of the boat transition state by 8.16 kcal mol-1. This suggests that the Cope rearrangement mechanism would most likely proceed through the chair transition state to that of the boat. Therefore the reaction would take place from the anti2 conformer to the chair transition state and then onto the gauche2 conformer within the Cope rearrangement.

It is also possible to see that the calcualted energies are very close to the literature values with the chair conformation being closer to its literature value to that of the boat.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9: Geometric parameters for the optimised chair and boat structures '''
| || ''' HF/3-21G chair TS''' || ''' B3LYP/6-31G* chair TS''' || '''HF/3-21G boat TS''' || '''B3LYP/6-31G* boat TS'''
|-
| Optimised Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Chair_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Boat_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.02 (C5-C10 or C3-C14)|| 1.97 (C5-C10 or C3-C14) || 2.14 (C3-C4 or C1-C6) || 2.21 (C3-C4 or C1-C6)
|-
| Fragment angle || 120 o (C3-C1-C5) || 120 o (C3-C1-C5) || 122 o (C4-C5-C6) || 122 o (C4-C5-C6)
|-
| Dihedral angle || 55 o (C1-C3-C14-C11)|| 54 o (C1-C3-C14-C11)|| 0 o (C2-C1-C6-C5) || 0 o (C2-C1-C6-C5)
|-
| Numbering of carbons || [[Image:Numbering_chairdh08.jpg|200x200px]]|| [[Image:Numbering_chairdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]]
|}

From studying table 9 it can be seen that the C-C bond breaking/forming length has changed between the two different types of optimisation methods. However the angles between the bonds remains unchanged.

=='''The Diels Alder Cycloaddition'''==

The Diels-Alder reaction is a cycloaddition between a conjugated diene and a dienophile (alkene system), which forms a cyclohexene system. It is often referred to as a [4s+2s] cycloaddition and proceeds through a single cyclic, concerted transition state where 2 σ bonds are formed in between the termini carbons of the 2 conjugated п systems. This is shown in the reaction scheme below where ethene is reacting with cis-butadiene.

[[Image:Reaction_scheme2dh08.jpg|300x300px|center]]

The π orbitals of the dienophile form new σ bonds with the π orbitals of the diene. The number of π electrons involved determine whether the reaction proceeds in a concerted stereospecific fashion (allowed) or not (forbidden). The HOMO/LUMO of one fragment interacts with the HOMO/LUMO of the other fragment to form two new bonding and anti-bonding MOs. The nodal properties allow one to predict how the reaction will occur.

Semi-Empirical/AM1 method and the B3LYP/6-31G* method of optimisation will be used on Gaussian to determine particular properites of the reactants to predict the way in which the reaction will proceed.

==='''cis-butadiene and ethylene: reagents'''===

<jmol>
<jmolAppletButton>
<uploadedFileContents>Cis_butadienedh08.mol</uploadedFileContents>
<text>Cis-butadiene</text>
<color>whi-te</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Ethenedh08.mol</uploadedFileContents>
<text>ethylene</text>
<color>white</color>
</jmolAppletButton>
</jmol> were optimised separately using the AM1 semi-empirical method on Gaussview.The cis-butadiene had an energy of 0.04879725 a. u (1.28 x 102 KJ mol-1) and ethylene had an energy of 0.02619027 a. u (68.76 KJ mol-1).

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10: Frontier molecular orbitals of cis-butadiene and ethylene '''
| '''Reagent''' || '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| cis-butadiene || LUMO || [[Image:LUMO1_cis_butadieneDH08.jpg|100x100px]] || [[Image:LUMO2_cis_butadienedh08.jpg|100x100px]]|| Symmetric || +0.017
|-
| cis-butadiene || HOMO || [[Image:HOMO1_cis_butadienedh08.jpg|100x100px]] || [[Image:HOMO2_cis_butadienedh08.jpg|100x100px]]|| Anti-symmetric || -0.343
|-
| ethylene || LUMO || [[Image:LUMO1_ethenedh08.jpg|100x100px]] || [[Image:LUMO2_ethenedh08.jpg|100x100px]]|| Anti-symmetric || +0.052
|-
| ethylene || HOMO || [[Image:HOMO1_ethenedh08.jpg|100x100px]] || [[Image:HOMO2_ethenedh08.jpg|100x100px]]|| Symmetric || -0.387
|}

 
The symmetries of the LUMO and HOMO orbitals for cis-butadiene are opposite of those for the ethylene. This therefore means that the only interactions that can take place are those between ethylene's LUMO and cis-butadiene's HOMO and in addition to this the interaction between cis-butadiene's LUMO and ethylene's HOMO. This is because orbital symmetry has to be conserved (the conservation of orbital symmetry states that the symmetry of an new orbital formed by the overlap of the 2 original orbitals must be the same as the symmetry of the initial orbitals).

==='''cis-butadiene and ethylene: transition state'''===

A <jmol>
<jmolAppletButton>
<uploadedFileContents>Bicyclic_startingdh08.mol</uploadedFileContents>
<text>bicyclo system</text>
<color>white</color>
</jmolAppletButton>
</jmol> was drawn and one of the -CH2-CH2- fragments were removed. Three C=C bonds were added and the previous one was taken away so a <jmol>
<jmolAppletButton>
<uploadedFileContents>Frozen_coorddh08.mol</uploadedFileContents>
<text>guess transition state</text>
<color>white</color>
</jmolAppletButton>
</jmol> was produced.

After this an optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while bonds between the termini of the fragments were frozen at 2.1Å. Following this the resulting file had two separate optimisation and frequency calculations run on it using the TS (Berny) with one using the semi-empirical/AM1 method and the other using the DFT/B3LYP/6-31G* level of theory. Both of these optimisations had the bonds unfrozen and the "derivative" option was chosen instead.

The DFT/B3LYP/6-31G* level of theory was used since it gave more accurate results in the calculations before and the energies it produces are consistent with the rest of the report (semi-empirical/AM1 method gives very different values).

The QST2 optimisation was not used since it required numbering the atoms in both the reactants and products, which took a lot of time to do.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11: Comparison of two methods of optimsations for the transition state '''
| || '''semi-empirical/AM1 method''' || '''DFT/B3LYP/6-31G* method'''
|-
| Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>SE_AM1_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>DFT_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Energy (a. u) || 0.11165608 || -234.54389573
|-
| Bond lengths || [[Image:DistancesSEdh08.jpg|150x150px]]|| [[Image:DistancesDFTdh08.jpg|150x150px]]
|-
|Imaginary frequency (cm-1) || -957 || -525
|-
|Animation || [[Image:-957dh08.gif]] || [[Image:-525dh08.gif]]
|-
| Lowest real frequency (cm-1) || 148 || 136
|-
|Animation || [[Image: 148dh08.gif]] || [[Image: 136dh08.gif]]
|}

The C-C bond forming length between the termini carbons is 0.15 Å larger for the DFT/B3LYP/6-31G* level of theory and this is because this method gives carries out the calculation to a greater degree of accuracy. This larger bond length leads a lower potential energy gradient to the transition state, which gives a lower total energy for the transition state. Therefore the given values below will be the DFT/B3LYP/6-31G* and not the semi-empirical/AM1 method.

A typical sp3 C-C bond length is 1.54 Å and a typical sp2 C=C bond length is 1.34 Å<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref>. Within the optimised transition molecule above the C-C bond lengths are 1.40 Å and the C=C bond length is 1.38 Å. This suggests that the π-bonds are breaking in order for the C-C bonds to form and additionally to this a π-bond is forming in between the two sp3 carbons.

The Van der Waals radius of a C-atom is 1.72 Å<ref name="ref 3">M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer, D. G. Truhlar, 1995, 346, pp. 63-69.[http://pubs.acs.org/doi/abs/10.1021/jp8111556]</ref>. The partly formed C-C length is 2.27 Å, which is shorter than two times the the van der Waals radius of a carbon atom. Therefore the carbon atoms are within a close enough distance to each other to share electron density and yet the bond has not formed at this point.

From looking at the animations for the imaginary frequency above it can be seen that the formation of the two bonds between the termini of the reactants is synchronous. This is expected of the Diels-Alder reaction since the mechanism is concerted. However the lowest real frequency is asynchronous with respect to the two bonds forming between the termini.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12: Frontier molecular orbitals of the transition state '''
| '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| LUMO || [[Image:LUMO1TSdh08.jpg|100x100px]] || [[Image:LUMO2TSdh08.jpg|100x100px]]|| Symmetric || +0.023
|-
| HOMO || [[Image:HOMO1TSdh08.jpg|100x100px]] || [[Image:HOMO2TSdh08.jpg|100x100px]]|| Anti-symmetric || -0.323
|}

From examining the images from table 12 above it can be deduced that the LUMO of the transition state originates from the cis-butadiene's LUMO and the ethylene's HOMO, which represents the σ*-antibonding orbitals. However the HOMO of the transition state originates from the cis-butadiene's HOMO and the ethylene's LUMO, which represents the σ-bonding orbitals within the two single C-C bonds that are formed within the reaction.

==='''cis-butadiene and ethylene: activation energy'''===

The cis-butadiene and ethylene were re-optimised at the DFT/B3LYP/6-31G* level of theory so the activation energy of the reaction could be calculated.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 13: Calculation for the activation energy for the reaction'''
| '''molecule''' || '''DFT/B3LYP/6-31G* energy (a. u)''' || '''Calculations''' || '''Energy '''
|-
| cis-butadiene || -155.98595591 || transition state || -234.54389573 a. u
|-
| ethylene || -78.58745828 || ΔE, activation energy || 0.02951846 a. u
|-
| cis-butadiene and ethylene combined || -234.57341419 || ΔE, activation energy || 77.50 KJ mol-1
|}

The calculated activation energy is 77.50 KJ mol-1, which is of the right magnitude when compared to the literature value (115 ± 8 KJ mol-1 at 0 K) but there is still an error of 37.50 KJ mol-1. This shows that Gaussview is not perfect but still gives an idea of the activation energy required for the reaction to take place.

==='''Cyclohexa-1,3-diene reaction with maleic anhydride'''===

The <jmol>
<jmolAppletButton>
<uploadedFileContents>FrozenTS22dh08.mol</uploadedFileContents>
<text>exo</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_frozenTS22dh08.mol</uploadedFileContents>
<text>endo</text>
<color>white</color>
</jmolAppletButton>
</jmol> transition states were drawn on Gaussview and then the optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while the distance of the forming bonds were frozen at 2.1Å between the respective carbons. After this the structures changed slightly (<jmol>
<jmolAppletButton>
<uploadedFileContents>TS2_afteropt1_exo1dh08.mol</uploadedFileContents>
<text>exo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_TS2_afteropt1dh08.mol</uploadedFileContents>
<text>endo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol>) and the resulting files had an optimisation and frequency calculation run on it using the TA (Berny) with using the semi-empirical/AM1 level of theory while the unfrozen option as changed to "derivative". After this the endo and exo optimised transition state structures were re-optimised as they were before but using the DFT/B3LYP/6-31G* level of theory instead with the additional keyword "Opt=NoEigen".

{| class="wikitable" border="1" align="center"
|+ '''Table 14: Comparison of the exo and endo transition states '''
! rowspan="2"|-!! colspan="2"|''exo'' !! colspan="2"|''endo''
|-
| Semi-empirical/AM1 || DFT/B3LYP/6-31G* || Semi-empirical/AM1 || DFT/B3LYP/6-31G*
|-
| align="center"|Structure ||align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>After_opt2_exodh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>Endo_TS2_afteropt2dh081.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| align="center"| Energy (a. u) ||align="center"| -0.05041975 ||align="center"| -612.67931095 ||align="center"| -0.05150470 ||align="center"| -612.68339684
|-
| C-C bond forming length (Å) ||align="center"| 2.17 ||align="center"| 2.29 ||align="center"| 2.16 ||align="center"| 2.27
|-align="center"
| Imaginary frequency (cm-1) ||align="center"| -812 ||align="center"| -448 ||align="center"| -806 ||align="center"| -447
|-align="center"
| Animation ||colspan="2"|[[Image:-812_SEdh08.gif]] || colspan="2"|[[Image:-806_cm-1dh08.gif]]
|}

Again the C-C bond forming length between the termini carbons is larger (by 0.11 and 0.12 Å) for the DFT/B3LYP/6-31G* level of theory and this is due the same reasons talked about above. However the C-C bond forming length between the termini carbons are very similar in both the endo and exo transition states (0.02 Å difference).

From looking at the animations for the imaginary frequency above it can also be seen that the formation of the two bonds between the carbons in this case as well is synchronous. Again this is expected of the Diels-Alder reaction since it is concerted.

{| align=center
|+ '''Strain diagrams for the endo and exo transition states'''
| [[Image:Exostraindh08.jpg|thumb|200x200px|'''Exo TS''']]
| [[Image:Strainendodh08.jpg|thumb|200x200px|'''Endo TS''']]
|}

From studying the energies in table 13 it can be seen that the endo transition state is more stable than the exo by 4.08589 x 10-3 a. u (10.73 KJ mol-1). However by looking at the strain of both transition states one would expect the exo TS to be more stable. This is because the distance between the cyclohexa-1,3-diene ring and the -(C=O)-O-(C=O)- maleic anhydride fragment is 2.99 Å in the endo TS and 3.03 Å in the exo TS. Thus the endo form would have more strain due to steric hindrance.

However within the exo TS the vicinal hydrogens on the CH2-CH2 (carbon numbers 18 and 21) fragment on the cyclohexa-1,3-diene ring are found below the plane and next to the maleic anhydride. This leads to steric hindrance is this is where the strain originates from within the exo TS. Therefore this disfavours the exo form, which makes the endo more stable.

Another factor making the endo TS more favourable is secondary orbital overlap. These orbitals contribute to the bonds forming and they interact with each other through space. Within the endo TS below there are 2 more orbitals interacting (secondary orbitals) than within the exo TS. This therefore makes the endo TS more stable and therefore more favoured.

[[Image:Secondary_orbital_shizzledh08.jpg|500x500px|center]]

{| class="wikitable" border="1" align="center"
|+ '''Table 15: Frontier molecular orbitals of the the transition state for the cyclohexa-1,3-diene reaction '''
! rowspan="2"|Molecular Orbital !! colspan="4"|''exo'' !! colspan="4"|''endo''
|- align="center"
| '''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''||'''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''
|- align="center"
| LUMO || [[Image:LUMO1exoh081.jpg|150x150px]]|| [[Image:LUMO2exodh081.jpg|150x150px]]|| Anti-symmetric || -0.078 || [[Image:LUMO1ENDOdh08.jpg|150x150px]]||[[Image:LUMO2ENDOdh08.jpg|150x150px]] || Anti-symmetric || -0.067
|- align="center"
| HOMO || [[Image:HOMO1exodh081.jpg|150x150px]]||[[Image:HOMO2exodh081.jpg|150x150px]] || Anti-symmetric|| -0.242 || [[Image:HOMO1ENDOdh08.jpg|150x150px]]|| [[Image:HOMO2ENDOdh08.jpg|150x150px]]||Anti-symmetric || -0.242
|}

Digital repositories:

exo: http://hdl.handle.net/10042/to-5596

endo: http://hdl.handle.net/10042/to-5597

From examining table 14 it is clear to see that both the HOMO and LUMO are anti-symmetric for both the transition states. The LUMO of the maleic anhydride fragment is the π* anti-bonding orbital of the C=C bond. Within the reaction then this has electrons transferred to it and the C=C bond breaks, which leaves just a C-C bond. The LUMOs for both transition states are both anti-bonding with respect to the two fragments. The HOMOs of both transitions state represent the σ-bonding that will be formed between the termini carbons on the products and reactants. This is most likely from the overlap of the cyclohexa-1,3-diene's HOMO and the maleic anhydride's LUMO.

=='''Conclusion'''==

='''References'''=
<references/>

Rep:Mod:hampluscheese

2010-11-11T12:05:44Z

Dh08: /* '''cis-butadiene and ethylene: activation energy''' */

='''Module 3:Physical Computational Chemistry Lab'''=

Douglas Hunt

=='''Objectives'''==

The aim of this experiment is to

=='''The Cope Rearrangement'''==

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangement] is a [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement |3,3|-sigmatropic rearrangement] that takes place through a single [http://en.wikipedia.org/wiki/Pericyclic pericyclic], concerted [http://en.wikipedia.org/wiki/Transition_state transition state]. Within this rearrangement a σ bond is formed in concert with another being broken within a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] π system.

The Cope rearrangement we will be studying will be that of 1,5-hexadiene, which is shown below.

[[Image:Cope_rearrangement_schemedh08.jpg|300x300px|center]]

The objective is to locate the low-energy minima and transition structures on the C6H10 potential energy surface and to determine the preferred reaction mechanism by using [http://en.wikipedia.org/wiki/GAUSSIAN Gaussian], with the [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)] methods.

==='''Optimisation of 1,5-hexadiene Conformers'''===

Rough <jmol>
<jmolAppletButton>
<uploadedFileContents>Anti_2_drawndh08.mol</uploadedFileContents>
<text>anti</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche3_drawndh08.mol</uploadedFileContents>
<text>gauche</text>
<color>white</color>
</jmolAppletButton>
</jmol> conformers of 1,5-hexadiene were drawn on Gaussview and their structures were tidied by the clean function. They were then further optimised by the HF/3-21G method and were identified which conformer they were from their point group and appendix 1 on the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3 physical lab page].

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 1: Optimisation and comparison of anti and gauche 1,5-hexadiene conformers using HF/3-21G methods
| ''' Conformer''' || '''Optimised Structure''' || '''Energy (Atomic Units)''' || '''[http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1 Energies (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (Anti2 - Gauche3)''' || '''Point group'''
|-
| '''Anti2''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Anti_2_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692535 || -231.69254 ||-6.08 x 105 || 1.2606 x 10-4 a. u|| Ci
|-
| '''Gauche3''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Gauche3_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692661 || -231.69266 || -6.08 x 105 || 3.31 x 10-1 KJ mol-1 || C1
|}

From studying table 1 it can be seen that the gauche conformer is 0.33 KJ mol-1 more stable than the anti conformation. One would assume that the anti2 conformer would be more stable due to [http://en.wikipedia.org/wiki/Steric steric] factors/repulsion within the gauche3 conformation. However within the gauche3 conformer there is a favourable donation of [http://en.wikipedia.org/wiki/Electron_density electron density] from the C=C π orbital to the C-H σ* orbital from the adjacent proton, which stabilises the conformer<ref name="ref 1">B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446.[http://www.informaworld.com/smpp/content~db=all?content=10.1080/00268970110081412]</ref>.

 

The anti2 conformer was then optimised using the B3LYP 6-31G* (6-31G (d)) method.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 2: Comparison of the anti2 conformer optimisations carried out at different calculation methods
| ''' Method''' || '''Structure and bond distances (Å)''' || '''Dihedral angle (C1 to C4)''' || '''Energy (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (HF/3-21G - DFT/B3LYP/6-31G)'''
|-
| HF/3-21G || [[Image:Anti2_opt1dh08.jpg|300x300px]] || 114.6 o || -231.692535 || -6.08 x 105 || 2.919171 a. u
|-
| DFT/B3LYP/6-31G* || [[Image:Anti2_opt2DH08.jpg|300x300px]] || 118.8 o || -234.611716 || -6.16 x 105 || 7.66 x 103 KJ mol-1
|}

From examining table 2 at first it is possible to see that the bond lengths are very similar to each other and that of the literature bond lengths<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref> (C1=C2: 1.340 Å, C2-C3: 1.508 Å and C3-C4: 1.538 Å). It is also possible to see that the C=C bond length is more accurate for the DFT/B3LYP/6-31G* method by it being 0.01 Å closer to the literature value. Another main difference in terms of geometry is that the DFT/B3LYP/6-31G* method gives a larger [http://en.wikipedia.org/wiki/Dihedral_angle dihedral angle] (and most likely more accurate) of 118.8o compared to that of 114.6o.

In terms of energy the DFT/B3LYP/6-31G* method produces a conformer of a higher stability by it possessing a lower energy with a difference of 7.66 x 103 KJ mol-1.

==='''Frequency analysis of 1,5-hexadiene Conformers'''===

[[Image:Vibrations_anti2dh08.jpg|right|300x300px]]

For the frequency analysis the DFT/B3LYP/6-31G* method was used.

Frequency analysis is essential within this calculation due to it being the second derivative of the potential energy surface. This means that if all the frequencies are all positive the system is at a minimum, if one of them is negative the optimised system is at a transition state, and if more than one are negative then the calculation has failed to find a critical point, which means the optimisation has failed. This therefore gives us a method of seeing whether the calculation has worked.

From looking to the image on the right it can deduced that the optimisation has taken place successfully by there being no negative vibrations and therefore the conformation is at a minimum.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 3: Energies under the thermochemistry section at 298.15 K and 0 K within the frequency logfiles'''
| '''Energies''' || '''298.15 K (atomic units)''' ||| '''0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (KJ mol-1)''' || '''Discussion'''
|-
| Sum of electronic and zero-point energies || -234.469177 || -234.466673 || -2.504 x 10-3 || -6.57 || This is the potential energy at 0 K, which includes the zero-point vibrational energy (E = Eelec + ZPE).

|-
| Sum of electronic and thermal energies || -234.461835 || -234.461085 || -7.5 x 10-4|| -1.97 || This is the energy at 298.15 K (0 K for 0 K calculation) and 1 atm of pressure, which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans).

|-
| Sum of electronic and thermal enthalpies || -234.460891 || -234.460141 || -7.5 x 10-4 || -1.97 || This contains an additional correction for RT (H = E + RT), which is especially important when looking at dissociation reactions.
|-
| Sum of electronic and thermal free energies || -234.500739 || -234.495931 || 4.808 x 10-3 || -12.62 || This includes the entropic contribution to the free energy (G = H - TS).
|}

{|align=right
|+
| [[Image:Anti2_freq_IRdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 298.15 K''']]
| [[Image:Anti2_freq_0Kdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 0 K''']]
|}

From looking at table 3 above it can be deduced that the anti2 conformer is more stable at room temperature to that of 0 K with respect to all the types of energies. The spectra also seem to not change much between the two temperatures but the main change is the change in the sum of electronic and thermal free energies, which is heavily dependent of temperature by it including the entropic contribution to the free energy (G = H - '''T'''S)

 
 
 
 

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

Within the Cope Rearrangement there are two possible transition states, the chair (C2h) and the boat (C2v), which are shown within [https://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_2 appendix two ].

==='''Comparison of optimisation methods for the chair transition state'''===

An <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2hdh08.mol</uploadedFileContents>
<text>allyl fragment</text>
<color>white</color>
</jmolAppletButton>
</jmol>, CH2CHCH2 was drawn on Gaussview and was optimized through the HF/3-21G method. After this the original allyl fragment was copied and pasted into a new window and place exactly above the first fragment with a bond distance of 2.20 Å between the terminal end carbons. This therefore produced the <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2h_fulldh08.mol</uploadedFileContents>
<text>chair transition structure.</text>
<color>white</color>
</jmolAppletButton>
</jmol>

Gaussian was then set up for an optimisation for the transition state with the additional key words "Opt=NoEigen", the optimisation set to TS (Berny), the force constants set to "once" and using the HF/3-21G method.

After this the "chair" transition state was optimised an alternative way by freezing the coordinates of the two sets of terminal carbons (where the bonds form/break during the rearrangement) within the Redundant Coord Editor on Gauusview.
Following this a second calculation was run where the freeze coordinates option was changed to "Derivative" and the force constants were set to "never".

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 4: Comparison of chair transition state optimisation methods
| || '''Optimised through a TS (Berny)''' || '''Optimised through a frozen coordinate method'''
|-
| '''Optimised TS Structure''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>C2h_optfreq_TS_Bdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| '''Terminal C-C bond length (Å)''' || 2.02 || 2.02
|-
|'''Energy (atomic units)''' || -231.61932247 || -231.61932190
|-
| '''ΔE, TS (Berny) - frozen coordinate''' || 5.7 x 10-7 a.u || 1.50 x 10-3 KJ mol-1
|-
| '''Imaginary frequency (cm-1)''' || -818 || -818
|-
| '''Animation''' ||[[Image:818dh08TSB.gif|350x350px]] || [[Image:818cm-1Fdh08.gif]]
|}

The data in table 4 shows that both methods give the same results and they both give the same transition state. They both produced the same energy with a negligible energy difference between them and the bond lengths between the terminal carbons are both 2.02 Å. In addition to this the imaginary frequencies were both -818cm-1 and within the animations above it is possible to see that when one C-C bond is forming the other is breaking. This means it is a asynchronous bond formation, which is expected for the Cope Rearrangement since it is a pericyclic reaction.

The only difference between the two optimisations is the way in which the transition states were calculated. This is due to the TS (Berny) method calculating the force constant matrix within the first optimisation step and is then being recalculated as the optimisation carries on. When the frozen coordinate method freezes the reaction coordinate in order for the rest of the structure to be optimised around it, which does not require the whole hessian (force constant matrix) to be calculated but only that the reaction coordinate is differentiated (second step). The main advantage of this, is that you do not need to calculate the entire force constant matrix and just need to differentiate the reaction coordinate in order to obtain a good enough guess for the initial force constant matrix. The frozen coordinate method will therefore take less time to compute the transition state, which can be very expensive within certain systems. However to carry out this method one needs to have an idea of how far apart the terminal carbons are.

==='''Boat transition state'''===

===='''QST2 optimisation'''====

The optimisation for the boat transition state had a different method and this was the QST2 method. This calculation interpolates between two structures (product and reactant) and tries to find a transition state between them.

The reactants and products were numbered in the same way. Therefore both 1,5-hexadiene molecules had their numbering manually changed for the product molecule so it corresponded to the numbering of the reactant if it rearranged. This is illustrated in the reaction scheme below.

[[Image:Boat_rerrangmentdh08.jpg|300x300px|center]]

The chk file for the optimized anti2 conformer was opened and copied into a new window. A second window was opened within the new window and the same anti2 conformer was copied into the new window. The numbering was changed within the second molecule so the two looked like they do in the image below.

[[Image:TS_(QST2)_numbering_1dh08.jpg|400x400px|center]]

The job type was set as "Opt+Freq" and "TS (QST2)" was chosen to optimise the transition state. However this failed and produced
a structure a bit like a <jmol>
<jmolAppletButton>
<uploadedFileContents>Trialone_BPdh08.mol</uploadedFileContents>
<text>chair transition structure </text>
<color>white</color>
</jmolAppletButton>
</jmol> but more dissociated. This is because it interpolated between the two structures and translated the top allyl fragment and therefore did not consider the possibility of a rotation around the central bonds.

To get around this the C-C-C-C dihedral angle (C2-C3-C4-C5) was set to 0 o and the C-C-C angle (C2-C3-C4 and C3-C4-C5 for the product molecule) was set to 100 o for both molecules. This gave rise to the two structures below.

[[Image:TS_(QST2)_numbering_2dh08.jpg|400x400px|center]]

After this the QST2 calculation was run again and the results in table 5 were produced. The imaginary frequency at -839 cm-1 shows that while one bond is forming the other one is breaking, which is again an asynchronous bond formation.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5: Data for the boat transition state from the QST2 optimisation method
| Optimised TS Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.14
|-
|Energy (atomic units) || -231.60280151
|-
| Energy (KJ mol-1) || -6.08 x 105
|-
| Imaginary frequency (cm-1) || -839
|-
| Animation ||[[Image:-839dh08.gif]]
|}

===='''Intrinsic Reaction Coordinate'''====

It is not possible to predict which product the chair and boat transition state will lead to. To solve this problem within Gaussian one can use the IRC method, which allows one to follow the minimum energy path from a transition structure down to its local minimum on a potential energy surface. This creates a succession of points by taking small geometry steps in the direction to where the gradient of the energy surface is at its steepest until the minimum in energy is reached.

The chk file for the optimized chair transition structure was opened and the job type selected was "IRC", the reaction coordinate was computed in the forward direction (since molecule is symmetric) with 50 points along the IRC. This was done with two type of force constants being calculated, once and always.

Within the "once" method the output showed no bond formation process by the 26th step and therefore the calculation was incomplete, since a complete 1,5-hexadiene structure was supposed to be formed. From looking at the IRC pathway it can be seen that the minimum on the potential energy surface was not reached at that point, since the points on the pathway did not form an asymptote and the gradient did not go close enough to zero.

For the "always" method the computation of force constants were carried out at every step. This therefore lets the IRC calculation distinguish whether it is moving along the minimum energy pathway while going down the potential energy surface. While this is happening it can make changes to the pathway if necessary. Within the "always" case the bond forming arrives at 1,5-hexadiene and this is backed up by the IRC pathway reaching a minimum within its energy by the gradient going close to zero and the pathway reaches an asymptote.

The structure produced through the always method had an energy of -231.69199434 atomic units, which is very close to the energy of the gauche2 conformer in the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 appendix 1]. This suggests that the reaction takes place through the chair transition state, which later produces the gauche2 conformer of 1,5-hexadiene.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 6: Comparison of IRC calculations for the chair structure
| || '''Calculate force constants once''' || '''Calculate force constants always'''
|-
| '''Number of points along IRC''' || 26 || 47
|-
| '''Structure at the last point of IRC''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Once_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Always_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|'''Energy (atomic units)''' || -231.68906923 || -231.69199434
|-
| '''ΔE, once - always''' || 2.92511 x 10-3 a.u || 7.68 KJ mol-1
|-
| '''IRC pathway''' || [[Image:One_pathwaydh08.jpg|400x400px]] || [[Image:Always_pathwaydh08.jpg|400x400px]]
|-
| '''IRC gradient''' || [[Image:One_gradientdh08.jpg|400x400px]]|| [[Image:Always_gradientdh08.jpg|400x400px]]
|}

 
 
 

===='''Comparison of activation energies for the transition states'''====

The HF/3-21G optimised chair and boat transition structures were both reoptimised using the B3LYP/6-31G* (B3LYP/6-31G (d)) method. This gives a more accurate minimisation however this method takes longer to run. The values for the zero-point energies and thermal energies were found within the thermochemistry section of the log file.

The values for the HF/3-21G reactant (anti2) optimisation were calculated and the B3LYP/6-31G* values were taken from the "Frequency analysis of 1,5-hexadiene Conformers" earlier in this module.

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 7: Energy data for chair, boat and the anti2 conformation from the HF/3-21G and B3LYP/6-31G* methods of optimisation '''
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466677
| align="center" | -231.461320
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450936
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402342
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539524
| align="center" | -231.532562
| align="center" | -234.611716
| align="center" | -234.469177
| align="center" | -234.461835
|}
Digital repositories

Chair:
HF/3-21G: http://hdl.handle.net/10042/to-5569
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5568

Boat:
HF/3-21G: http://hdl.handle.net/10042/to-5570
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5571

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 8: Summary of activation energies for the chair and boat conformations (ΔE = chair TS or boat TS minus anti2)'''
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| align="center" | 1 a.u = 627.509 kcal mol-1
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE, chair (a.u)'''
| align="center" | 0.072847
| align="center" | 0.071242
| align="center" | 0.054248
| align="center" | 0.052827
| align="center" | -
|-
| '''ΔE, chair (kcal mol-1)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE, boat (a. u)'''
| align="center" | 0.088588
| align="center" | 0.087259
| align="center" | 0.066835
| align="center" | 0.065827
| align="center" | -
|-
| '''ΔE, boat (kcal mol-1)'''
| align="center" | 55.59
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.31
| align="center" | 44.7 ± 2.0
|}

By examining table 8 it is clear to see that the B3LYP/6-31G* method gives values which are much closer to the experimental ones. This is due to the B3LYP/6-31G* giving a higher level of optimisation than the HF/3-21G method. Table 8 also shows that the chair transition state has a lower activation energy to that of the boat transition state by 8.16 kcal mol-1. This suggests that the Cope rearrangement mechanism would most likely proceed through the chair transition state to that of the boat. Therefore the reaction would take place from the anti2 conformer to the chair transition state and then onto the gauche2 conformer within the Cope rearrangement.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9: Geometric parameters for the optimised chair and boat structures '''
| || ''' HF/3-21G chair TS''' || ''' B3LYP/6-31G* chair TS''' || '''HF/3-21G boat TS''' || '''B3LYP/6-31G* boat TS'''
|-
| Optimised Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Chair_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Boat_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.02 (C5-C10 or C3-C14)|| 1.97 (C5-C10 or C3-C14) || 2.14 (C3-C4 or C1-C6) || 2.21 (C3-C4 or C1-C6)
|-
| Fragment angle || 120 o (C3-C1-C5) || 120 o (C3-C1-C5) || 122 o (C4-C5-C6) || 122 o (C4-C5-C6)
|-
| Dihedral angle || 55 o (C1-C3-C14-C11)|| 54 o (C1-C3-C14-C11)|| 0 o (C2-C1-C6-C5) || 0 o (C2-C1-C6-C5)
|-
| Numbering of carbons || [[Image:Numbering_chairdh08.jpg|200x200px]]|| [[Image:Numbering_chairdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]]
|}

From studying table 9 it can be seen that the C-C bond breaking/forming length has changed between the two different types of optimisation methods. However the angles between the bonds remains unchanged.

=='''The Diels Alder Cycloaddition'''==

The Diels-Alder reaction is a cycloaddition between a conjugated diene and a dienophile (alkene system), which forms a cyclohexene system. It is often referred to as a [4s+2s] cycloaddition and proceeds through a single cyclic, concerted transition state where 2 σ bonds are formed in between the termini carbons of the 2 conjugated п systems. This is shown in the reaction scheme below where ethene is reacting with cis-butadiene.

[[Image:Reaction_scheme2dh08.jpg|300x300px|center]]

The π orbitals of the dienophile form new σ bonds with the π orbitals of the diene. The number of π electrons involved determine whether the reaction proceeds in a concerted stereospecific fashion (allowed) or not (forbidden). The HOMO/LUMO of one fragment interacts with the HOMO/LUMO of the other fragment to form two new bonding and anti-bonding MOs. The nodal properties allow one to predict how the reaction will occur.

Semi-Empirical/AM1 method and the B3LYP/6-31G* method of optimisation will be used on Gaussian to determine particular properites of the reactants to predict the way in which the reaction will proceed.

==='''cis-butadiene and ethylene: reagents'''===

<jmol>
<jmolAppletButton>
<uploadedFileContents>Cis_butadienedh08.mol</uploadedFileContents>
<text>Cis-butadiene</text>
<color>whi-te</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Ethenedh08.mol</uploadedFileContents>
<text>ethylene</text>
<color>white</color>
</jmolAppletButton>
</jmol> were optimised separately using the AM1 semi-empirical method on Gaussview.The cis-butadiene had an energy of 0.04879725 a. u (1.28 x 102 KJ mol-1) and ethylene had an energy of 0.02619027 a. u (68.76 KJ mol-1).

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10: Frontier molecular orbitals of cis-butadiene and ethylene '''
| '''Reagent''' || '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| cis-butadiene || LUMO || [[Image:LUMO1_cis_butadieneDH08.jpg|100x100px]] || [[Image:LUMO2_cis_butadienedh08.jpg|100x100px]]|| Symmetric || +0.017
|-
| cis-butadiene || HOMO || [[Image:HOMO1_cis_butadienedh08.jpg|100x100px]] || [[Image:HOMO2_cis_butadienedh08.jpg|100x100px]]|| Anti-symmetric || -0.343
|-
| ethylene || LUMO || [[Image:LUMO1_ethenedh08.jpg|100x100px]] || [[Image:LUMO2_ethenedh08.jpg|100x100px]]|| Anti-symmetric || +0.052
|-
| ethylene || HOMO || [[Image:HOMO1_ethenedh08.jpg|100x100px]] || [[Image:HOMO2_ethenedh08.jpg|100x100px]]|| Symmetric || -0.387
|}

 
The symmetries of the LUMO and HOMO orbitals for cis-butadiene are opposite of those for the ethylene. This therefore means that the only interactions that can take place are those between ethylene's LUMO and cis-butadiene's HOMO and in addition to this the interaction between cis-butadiene's LUMO and ethylene's HOMO. This is because orbital symmetry has to be conserved (the conservation of orbital symmetry states that the symmetry of an new orbital formed by the overlap of the 2 original orbitals must be the same as the symmetry of the initial orbitals).

==='''cis-butadiene and ethylene: transition state'''===

A <jmol>
<jmolAppletButton>
<uploadedFileContents>Bicyclic_startingdh08.mol</uploadedFileContents>
<text>bicyclo system</text>
<color>white</color>
</jmolAppletButton>
</jmol> was drawn and one of the -CH2-CH2- fragments were removed. Three C=C bonds were added and the previous one was taken away so a <jmol>
<jmolAppletButton>
<uploadedFileContents>Frozen_coorddh08.mol</uploadedFileContents>
<text>guess transition state</text>
<color>white</color>
</jmolAppletButton>
</jmol> was produced.

After this an optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while bonds between the termini of the fragments were frozen at 2.1Å. Following this the resulting file had two separate optimisation and frequency calculations run on it using the TS (Berny) with one using the semi-empirical/AM1 method and the other using the DFT/B3LYP/6-31G* level of theory. Both of these optimisations had the bonds unfrozen and the "derivative" option was chosen instead.

The DFT/B3LYP/6-31G* level of theory was used since it gave more accurate results in the calculations before and the energies it produces are consistent with the rest of the report (semi-empirical/AM1 method gives very different values).

The QST2 optimisation was not used since it required numbering the atoms in both the reactants and products, which took a lot of time to do.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11: Comparison of two methods of optimsations for the transition state '''
| || '''semi-empirical/AM1 method''' || '''DFT/B3LYP/6-31G* method'''
|-
| Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>SE_AM1_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>DFT_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Energy (a. u) || 0.11165608 || -234.54389573
|-
| Bond lengths || [[Image:DistancesSEdh08.jpg|150x150px]]|| [[Image:DistancesDFTdh08.jpg|150x150px]]
|-
|Imaginary frequency (cm-1) || -957 || -525
|-
|Animation || [[Image:-957dh08.gif]] || [[Image:-525dh08.gif]]
|-
| Lowest real frequency (cm-1) || 148 || 136
|-
|Animation || [[Image: 148dh08.gif]] || [[Image: 136dh08.gif]]
|}

The C-C bond forming length between the termini carbons is 0.15 Å larger for the DFT/B3LYP/6-31G* level of theory and this is because this method gives carries out the calculation to a greater degree of accuracy. This larger bond length leads a lower potential energy gradient to the transition state, which gives a lower total energy for the transition state. Therefore the given values below will be the DFT/B3LYP/6-31G* and not the semi-empirical/AM1 method.

A typical sp3 C-C bond length is 1.54 Å and a typical sp2 C=C bond length is 1.34 Å<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref>. Within the optimised transition molecule above the C-C bond lengths are 1.40 Å and the C=C bond length is 1.38 Å. This suggests that the π-bonds are breaking in order for the C-C bonds to form and additionally to this a π-bond is forming in between the two sp3 carbons.

The Van der Waals radius of a C-atom is 1.72 Å<ref name="ref 3">M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer, D. G. Truhlar, 1995, 346, pp. 63-69.[http://pubs.acs.org/doi/abs/10.1021/jp8111556]</ref>. The partly formed C-C length is 2.27 Å, which is shorter than two times the the van der Waals radius of a carbon atom. Therefore the carbon atoms are within a close enough distance to each other to share electron density and yet the bond has not formed at this point.

From looking at the animations for the imaginary frequency above it can be seen that the formation of the two bonds between the termini of the reactants is synchronous. This is expected of the Diels-Alder reaction since the mechanism is concerted. However the lowest real frequency is asynchronous with respect to the two bonds forming between the termini.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12: Frontier molecular orbitals of the transition state '''
| '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| LUMO || [[Image:LUMO1TSdh08.jpg|100x100px]] || [[Image:LUMO2TSdh08.jpg|100x100px]]|| Symmetric || +0.023
|-
| HOMO || [[Image:HOMO1TSdh08.jpg|100x100px]] || [[Image:HOMO2TSdh08.jpg|100x100px]]|| Anti-symmetric || -0.323
|}

From examining the images from table 12 above it can be deduced that the LUMO of the transition state originates from the cis-butadiene's LUMO and the ethylene's HOMO, which represents the σ*-antibonding orbitals. However the HOMO of the transition state originates from the cis-butadiene's HOMO and the ethylene's LUMO, which represents the σ-bonding orbitals within the two single C-C bonds that are formed within the reaction.

==='''cis-butadiene and ethylene: activation energy'''===

The cis-butadiene and ethylene were re-optimised at the DFT/B3LYP/6-31G* level of theory so the activation energy of the reaction could be calculated.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 13: Calculation for the activation energy for the reaction'''
| '''molecule''' || '''DFT/B3LYP/6-31G* energy (a. u)''' || '''Calculations''' || '''Energy '''
|-
| cis-butadiene || -155.98595591 || transition state || -234.54389573 a. u
|-
| ethylene || -78.58745828 || ΔE, activation energy || 0.02951846 a. u
|-
| cis-butadiene and ethylene combined || -234.57341419 || ΔE, activation energy || 77.50 KJ mol-1
|}

The calculated activation energy is 77.50 KJ mol-1, which is of the right magnitude when compared to the literature value (115 ± 8 KJ mol-1 at 0 K) but there is still an error of 37.50 KJ mol-1. This shows that Gaussview is not perfect but still gives an idea of the activation energy required for the reaction to take place.

==='''Cyclohexa-1,3-diene reaction with maleic anhydride'''===

The <jmol>
<jmolAppletButton>
<uploadedFileContents>FrozenTS22dh08.mol</uploadedFileContents>
<text>exo</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_frozenTS22dh08.mol</uploadedFileContents>
<text>endo</text>
<color>white</color>
</jmolAppletButton>
</jmol> transition states were drawn on Gaussview and then the optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while the distance of the forming bonds were frozen at 2.1Å between the respective carbons. After this the structures changed slightly (<jmol>
<jmolAppletButton>
<uploadedFileContents>TS2_afteropt1_exo1dh08.mol</uploadedFileContents>
<text>exo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_TS2_afteropt1dh08.mol</uploadedFileContents>
<text>endo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol>) and the resulting files had an optimisation and frequency calculation run on it using the TA (Berny) with using the semi-empirical/AM1 level of theory while the unfrozen option as changed to "derivative". After this the endo and exo optimised transition state structures were re-optimised as they were before but using the DFT/B3LYP/6-31G* level of theory instead with the additional keyword "Opt=NoEigen".

{| class="wikitable" border="1" align="center"
|+ '''Table 14: Comparison of the exo and endo transition states '''
! rowspan="2"|-!! colspan="2"|''exo'' !! colspan="2"|''endo''
|-
| Semi-empirical/AM1 || DFT/B3LYP/6-31G* || Semi-empirical/AM1 || DFT/B3LYP/6-31G*
|-
| align="center"|Structure ||align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>After_opt2_exodh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>Endo_TS2_afteropt2dh081.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| align="center"| Energy (a. u) ||align="center"| -0.05041975 ||align="center"| -612.67931095 ||align="center"| -0.05150470 ||align="center"| -612.68339684
|-
| C-C bond forming length (Å) ||align="center"| 2.17 ||align="center"| 2.29 ||align="center"| 2.16 ||align="center"| 2.27
|-align="center"
| Imaginary frequency (cm-1) ||align="center"| -812 ||align="center"| -448 ||align="center"| -806 ||align="center"| -447
|-align="center"
| Animation ||colspan="2"|[[Image:-812_SEdh08.gif]] || colspan="2"|[[Image:-806_cm-1dh08.gif]]
|}

Again the C-C bond forming length between the termini carbons is larger (by 0.11 and 0.12 Å) for the DFT/B3LYP/6-31G* level of theory and this is due the same reasons talked about above. However the C-C bond forming length between the termini carbons are very similar in both the endo and exo transition states (0.02 Å difference).

From looking at the animations for the imaginary frequency above it can also be seen that the formation of the two bonds between the carbons in this case as well is synchronous. Again this is expected of the Diels-Alder reaction since it is concerted.

{| align=center
|+ '''Strain diagrams for the endo and exo transition states'''
| [[Image:Exostraindh08.jpg|thumb|200x200px|'''Exo TS''']]
| [[Image:Strainendodh08.jpg|thumb|200x200px|'''Endo TS''']]
|}

From studying the energies in table 13 it can be seen that the endo transition state is more stable than the exo by 4.08589 x 10-3 a. u (10.73 KJ mol-1). However by looking at the strain of both transition states one would expect the exo TS to be more stable. This is because the distance between the cyclohexa-1,3-diene ring and the -(C=O)-O-(C=O)- maleic anhydride fragment is 2.99 Å in the endo TS and 3.03 Å in the exo TS. Thus the endo form would have more strain due to steric hindrance.

However within the exo TS the vicinal hydrogens on the CH2-CH2 (carbon numbers 18 and 21) fragment on the cyclohexa-1,3-diene ring are found below the plane and next to the maleic anhydride. This leads to steric hindrance is this is where the strain originates from within the exo TS. Therefore this disfavours the exo form, which makes the endo more stable.

Another factor making the endo TS more favourable is secondary orbital overlap. These orbitals contribute to the bonds forming and they interact with each other through space. Within the endo TS below there are 2 more orbitals interacting (secondary orbitals) than within the exo TS. This therefore makes the endo TS more stable and therefore more favoured.

[[Image:Secondary_orbital_shizzledh08.jpg|500x500px|center]]

{| class="wikitable" border="1" align="center"
|+ '''Table 15: Frontier molecular orbitals of the the transition state for the cyclohexa-1,3-diene reaction '''
! rowspan="2"|Molecular Orbital !! colspan="4"|''exo'' !! colspan="4"|''endo''
|- align="center"
| '''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''||'''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''
|- align="center"
| LUMO || [[Image:LUMO1exoh081.jpg|150x150px]]|| [[Image:LUMO2exodh081.jpg|150x150px]]|| Anti-symmetric || -0.078 || [[Image:LUMO1ENDOdh08.jpg|150x150px]]||[[Image:LUMO2ENDOdh08.jpg|150x150px]] || Anti-symmetric || -0.067
|- align="center"
| HOMO || [[Image:HOMO1exodh081.jpg|150x150px]]||[[Image:HOMO2exodh081.jpg|150x150px]] || Anti-symmetric|| -0.242 || [[Image:HOMO1ENDOdh08.jpg|150x150px]]|| [[Image:HOMO2ENDOdh08.jpg|150x150px]]||Anti-symmetric || -0.242
|}

Digital repositories:

exo: http://hdl.handle.net/10042/to-5596

endo: http://hdl.handle.net/10042/to-5597

From examining table 14 it is clear to see that both the HOMO and LUMO are anti-symmetric for both the transition states. The LUMO of the maleic anhydride fragment is the π* anti-bonding orbital of the C=C bond. Within the reaction then this has electrons transferred to it and the C=C bond breaks, which leaves just a C-C bond. The LUMOs for both transition states are both anti-bonding with respect to the two fragments. The HOMOs of both transitions state represent the σ-bonding that will be formed between the termini carbons on the products and reactants. This is most likely from the overlap of the cyclohexa-1,3-diene's HOMO and the maleic anhydride's LUMO.

=='''Conclusion'''==

='''References'''=
<references/>

Rep:Mod:hampluscheese

2010-11-11T11:58:12Z

Dh08: /* '''The Diels Alder Cycloaddition''' */

='''Module 3:Physical Computational Chemistry Lab'''=

Douglas Hunt

=='''Objectives'''==

The aim of this experiment is to

=='''The Cope Rearrangement'''==

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangement] is a [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement |3,3|-sigmatropic rearrangement] that takes place through a single [http://en.wikipedia.org/wiki/Pericyclic pericyclic], concerted [http://en.wikipedia.org/wiki/Transition_state transition state]. Within this rearrangement a σ bond is formed in concert with another being broken within a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] π system.

The Cope rearrangement we will be studying will be that of 1,5-hexadiene, which is shown below.

[[Image:Cope_rearrangement_schemedh08.jpg|300x300px|center]]

The objective is to locate the low-energy minima and transition structures on the C6H10 potential energy surface and to determine the preferred reaction mechanism by using [http://en.wikipedia.org/wiki/GAUSSIAN Gaussian], with the [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)] methods.

==='''Optimisation of 1,5-hexadiene Conformers'''===

Rough <jmol>
<jmolAppletButton>
<uploadedFileContents>Anti_2_drawndh08.mol</uploadedFileContents>
<text>anti</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche3_drawndh08.mol</uploadedFileContents>
<text>gauche</text>
<color>white</color>
</jmolAppletButton>
</jmol> conformers of 1,5-hexadiene were drawn on Gaussview and their structures were tidied by the clean function. They were then further optimised by the HF/3-21G method and were identified which conformer they were from their point group and appendix 1 on the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3 physical lab page].

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 1: Optimisation and comparison of anti and gauche 1,5-hexadiene conformers using HF/3-21G methods
| ''' Conformer''' || '''Optimised Structure''' || '''Energy (Atomic Units)''' || '''[http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1 Energies (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (Anti2 - Gauche3)''' || '''Point group'''
|-
| '''Anti2''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Anti_2_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692535 || -231.69254 ||-6.08 x 105 || 1.2606 x 10-4 a. u|| Ci
|-
| '''Gauche3''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Gauche3_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692661 || -231.69266 || -6.08 x 105 || 3.31 x 10-1 KJ mol-1 || C1
|}

From studying table 1 it can be seen that the gauche conformer is 0.33 KJ mol-1 more stable than the anti conformation. One would assume that the anti2 conformer would be more stable due to [http://en.wikipedia.org/wiki/Steric steric] factors/repulsion within the gauche3 conformation. However within the gauche3 conformer there is a favourable donation of [http://en.wikipedia.org/wiki/Electron_density electron density] from the C=C π orbital to the C-H σ* orbital from the adjacent proton, which stabilises the conformer<ref name="ref 1">B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446.[http://www.informaworld.com/smpp/content~db=all?content=10.1080/00268970110081412]</ref>.

 

The anti2 conformer was then optimised using the B3LYP 6-31G* (6-31G (d)) method.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 2: Comparison of the anti2 conformer optimisations carried out at different calculation methods
| ''' Method''' || '''Structure and bond distances (Å)''' || '''Dihedral angle (C1 to C4)''' || '''Energy (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (HF/3-21G - DFT/B3LYP/6-31G)'''
|-
| HF/3-21G || [[Image:Anti2_opt1dh08.jpg|300x300px]] || 114.6 o || -231.692535 || -6.08 x 105 || 2.919171 a. u
|-
| DFT/B3LYP/6-31G* || [[Image:Anti2_opt2DH08.jpg|300x300px]] || 118.8 o || -234.611716 || -6.16 x 105 || 7.66 x 103 KJ mol-1
|}

From examining table 2 at first it is possible to see that the bond lengths are very similar to each other and that of the literature bond lengths<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref> (C1=C2: 1.340 Å, C2-C3: 1.508 Å and C3-C4: 1.538 Å). It is also possible to see that the C=C bond length is more accurate for the DFT/B3LYP/6-31G* method by it being 0.01 Å closer to the literature value. Another main difference in terms of geometry is that the DFT/B3LYP/6-31G* method gives a larger [http://en.wikipedia.org/wiki/Dihedral_angle dihedral angle] (and most likely more accurate) of 118.8o compared to that of 114.6o.

In terms of energy the DFT/B3LYP/6-31G* method produces a conformer of a higher stability by it possessing a lower energy with a difference of 7.66 x 103 KJ mol-1.

==='''Frequency analysis of 1,5-hexadiene Conformers'''===

[[Image:Vibrations_anti2dh08.jpg|right|300x300px]]

For the frequency analysis the DFT/B3LYP/6-31G* method was used.

Frequency analysis is essential within this calculation due to it being the second derivative of the potential energy surface. This means that if all the frequencies are all positive the system is at a minimum, if one of them is negative the optimised system is at a transition state, and if more than one are negative then the calculation has failed to find a critical point, which means the optimisation has failed. This therefore gives us a method of seeing whether the calculation has worked.

From looking to the image on the right it can deduced that the optimisation has taken place successfully by there being no negative vibrations and therefore the conformation is at a minimum.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 3: Energies under the thermochemistry section at 298.15 K and 0 K within the frequency logfiles'''
| '''Energies''' || '''298.15 K (atomic units)''' ||| '''0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (KJ mol-1)''' || '''Discussion'''
|-
| Sum of electronic and zero-point energies || -234.469177 || -234.466673 || -2.504 x 10-3 || -6.57 || This is the potential energy at 0 K, which includes the zero-point vibrational energy (E = Eelec + ZPE).

|-
| Sum of electronic and thermal energies || -234.461835 || -234.461085 || -7.5 x 10-4|| -1.97 || This is the energy at 298.15 K (0 K for 0 K calculation) and 1 atm of pressure, which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans).

|-
| Sum of electronic and thermal enthalpies || -234.460891 || -234.460141 || -7.5 x 10-4 || -1.97 || This contains an additional correction for RT (H = E + RT), which is especially important when looking at dissociation reactions.
|-
| Sum of electronic and thermal free energies || -234.500739 || -234.495931 || 4.808 x 10-3 || -12.62 || This includes the entropic contribution to the free energy (G = H - TS).
|}

{|align=right
|+
| [[Image:Anti2_freq_IRdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 298.15 K''']]
| [[Image:Anti2_freq_0Kdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 0 K''']]
|}

From looking at table 3 above it can be deduced that the anti2 conformer is more stable at room temperature to that of 0 K with respect to all the types of energies. The spectra also seem to not change much between the two temperatures but the main change is the change in the sum of electronic and thermal free energies, which is heavily dependent of temperature by it including the entropic contribution to the free energy (G = H - '''T'''S)

 
 
 
 

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

Within the Cope Rearrangement there are two possible transition states, the chair (C2h) and the boat (C2v), which are shown within [https://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_2 appendix two ].

==='''Comparison of optimisation methods for the chair transition state'''===

An <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2hdh08.mol</uploadedFileContents>
<text>allyl fragment</text>
<color>white</color>
</jmolAppletButton>
</jmol>, CH2CHCH2 was drawn on Gaussview and was optimized through the HF/3-21G method. After this the original allyl fragment was copied and pasted into a new window and place exactly above the first fragment with a bond distance of 2.20 Å between the terminal end carbons. This therefore produced the <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2h_fulldh08.mol</uploadedFileContents>
<text>chair transition structure.</text>
<color>white</color>
</jmolAppletButton>
</jmol>

Gaussian was then set up for an optimisation for the transition state with the additional key words "Opt=NoEigen", the optimisation set to TS (Berny), the force constants set to "once" and using the HF/3-21G method.

After this the "chair" transition state was optimised an alternative way by freezing the coordinates of the two sets of terminal carbons (where the bonds form/break during the rearrangement) within the Redundant Coord Editor on Gauusview.
Following this a second calculation was run where the freeze coordinates option was changed to "Derivative" and the force constants were set to "never".

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 4: Comparison of chair transition state optimisation methods
| || '''Optimised through a TS (Berny)''' || '''Optimised through a frozen coordinate method'''
|-
| '''Optimised TS Structure''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>C2h_optfreq_TS_Bdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| '''Terminal C-C bond length (Å)''' || 2.02 || 2.02
|-
|'''Energy (atomic units)''' || -231.61932247 || -231.61932190
|-
| '''ΔE, TS (Berny) - frozen coordinate''' || 5.7 x 10-7 a.u || 1.50 x 10-3 KJ mol-1
|-
| '''Imaginary frequency (cm-1)''' || -818 || -818
|-
| '''Animation''' ||[[Image:818dh08TSB.gif|350x350px]] || [[Image:818cm-1Fdh08.gif]]
|}

The data in table 4 shows that both methods give the same results and they both give the same transition state. They both produced the same energy with a negligible energy difference between them and the bond lengths between the terminal carbons are both 2.02 Å. In addition to this the imaginary frequencies were both -818cm-1 and within the animations above it is possible to see that when one C-C bond is forming the other is breaking. This means it is a asynchronous bond formation, which is expected for the Cope Rearrangement since it is a pericyclic reaction.

The only difference between the two optimisations is the way in which the transition states were calculated. This is due to the TS (Berny) method calculating the force constant matrix within the first optimisation step and is then being recalculated as the optimisation carries on. When the frozen coordinate method freezes the reaction coordinate in order for the rest of the structure to be optimised around it, which does not require the whole hessian (force constant matrix) to be calculated but only that the reaction coordinate is differentiated (second step). The main advantage of this, is that you do not need to calculate the entire force constant matrix and just need to differentiate the reaction coordinate in order to obtain a good enough guess for the initial force constant matrix. The frozen coordinate method will therefore take less time to compute the transition state, which can be very expensive within certain systems. However to carry out this method one needs to have an idea of how far apart the terminal carbons are.

==='''Boat transition state'''===

===='''QST2 optimisation'''====

The optimisation for the boat transition state had a different method and this was the QST2 method. This calculation interpolates between two structures (product and reactant) and tries to find a transition state between them.

The reactants and products were numbered in the same way. Therefore both 1,5-hexadiene molecules had their numbering manually changed for the product molecule so it corresponded to the numbering of the reactant if it rearranged. This is illustrated in the reaction scheme below.

[[Image:Boat_rerrangmentdh08.jpg|300x300px|center]]

The chk file for the optimized anti2 conformer was opened and copied into a new window. A second window was opened within the new window and the same anti2 conformer was copied into the new window. The numbering was changed within the second molecule so the two looked like they do in the image below.

[[Image:TS_(QST2)_numbering_1dh08.jpg|400x400px|center]]

The job type was set as "Opt+Freq" and "TS (QST2)" was chosen to optimise the transition state. However this failed and produced
a structure a bit like a <jmol>
<jmolAppletButton>
<uploadedFileContents>Trialone_BPdh08.mol</uploadedFileContents>
<text>chair transition structure </text>
<color>white</color>
</jmolAppletButton>
</jmol> but more dissociated. This is because it interpolated between the two structures and translated the top allyl fragment and therefore did not consider the possibility of a rotation around the central bonds.

To get around this the C-C-C-C dihedral angle (C2-C3-C4-C5) was set to 0 o and the C-C-C angle (C2-C3-C4 and C3-C4-C5 for the product molecule) was set to 100 o for both molecules. This gave rise to the two structures below.

[[Image:TS_(QST2)_numbering_2dh08.jpg|400x400px|center]]

After this the QST2 calculation was run again and the results in table 5 were produced. The imaginary frequency at -839 cm-1 shows that while one bond is forming the other one is breaking, which is again an asynchronous bond formation.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5: Data for the boat transition state from the QST2 optimisation method
| Optimised TS Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.14
|-
|Energy (atomic units) || -231.60280151
|-
| Energy (KJ mol-1) || -6.08 x 105
|-
| Imaginary frequency (cm-1) || -839
|-
| Animation ||[[Image:-839dh08.gif]]
|}

===='''Intrinsic Reaction Coordinate'''====

It is not possible to predict which product the chair and boat transition state will lead to. To solve this problem within Gaussian one can use the IRC method, which allows one to follow the minimum energy path from a transition structure down to its local minimum on a potential energy surface. This creates a succession of points by taking small geometry steps in the direction to where the gradient of the energy surface is at its steepest until the minimum in energy is reached.

The chk file for the optimized chair transition structure was opened and the job type selected was "IRC", the reaction coordinate was computed in the forward direction (since molecule is symmetric) with 50 points along the IRC. This was done with two type of force constants being calculated, once and always.

Within the "once" method the output showed no bond formation process by the 26th step and therefore the calculation was incomplete, since a complete 1,5-hexadiene structure was supposed to be formed. From looking at the IRC pathway it can be seen that the minimum on the potential energy surface was not reached at that point, since the points on the pathway did not form an asymptote and the gradient did not go close enough to zero.

For the "always" method the computation of force constants were carried out at every step. This therefore lets the IRC calculation distinguish whether it is moving along the minimum energy pathway while going down the potential energy surface. While this is happening it can make changes to the pathway if necessary. Within the "always" case the bond forming arrives at 1,5-hexadiene and this is backed up by the IRC pathway reaching a minimum within its energy by the gradient going close to zero and the pathway reaches an asymptote.

The structure produced through the always method had an energy of -231.69199434 atomic units, which is very close to the energy of the gauche2 conformer in the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 appendix 1]. This suggests that the reaction takes place through the chair transition state, which later produces the gauche2 conformer of 1,5-hexadiene.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 6: Comparison of IRC calculations for the chair structure
| || '''Calculate force constants once''' || '''Calculate force constants always'''
|-
| '''Number of points along IRC''' || 26 || 47
|-
| '''Structure at the last point of IRC''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Once_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Always_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|'''Energy (atomic units)''' || -231.68906923 || -231.69199434
|-
| '''ΔE, once - always''' || 2.92511 x 10-3 a.u || 7.68 KJ mol-1
|-
| '''IRC pathway''' || [[Image:One_pathwaydh08.jpg|400x400px]] || [[Image:Always_pathwaydh08.jpg|400x400px]]
|-
| '''IRC gradient''' || [[Image:One_gradientdh08.jpg|400x400px]]|| [[Image:Always_gradientdh08.jpg|400x400px]]
|}

 
 
 

===='''Comparison of activation energies for the transition states'''====

The HF/3-21G optimised chair and boat transition structures were both reoptimised using the B3LYP/6-31G* (B3LYP/6-31G (d)) method. This gives a more accurate minimisation however this method takes longer to run. The values for the zero-point energies and thermal energies were found within the thermochemistry section of the log file.

The values for the HF/3-21G reactant (anti2) optimisation were calculated and the B3LYP/6-31G* values were taken from the "Frequency analysis of 1,5-hexadiene Conformers" earlier in this module.

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 7: Energy data for chair, boat and the anti2 conformation from the HF/3-21G and B3LYP/6-31G* methods of optimisation '''
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466677
| align="center" | -231.461320
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450936
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402342
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539524
| align="center" | -231.532562
| align="center" | -234.611716
| align="center" | -234.469177
| align="center" | -234.461835
|}
Digital repositories

Chair:
HF/3-21G: http://hdl.handle.net/10042/to-5569
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5568

Boat:
HF/3-21G: http://hdl.handle.net/10042/to-5570
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5571

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 8: Summary of activation energies for the chair and boat conformations (ΔE = chair TS or boat TS minus anti2)'''
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| align="center" | 1 a.u = 627.509 kcal mol-1
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE, chair (a.u)'''
| align="center" | 0.072847
| align="center" | 0.071242
| align="center" | 0.054248
| align="center" | 0.052827
| align="center" | -
|-
| '''ΔE, chair (kcal mol-1)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE, boat (a. u)'''
| align="center" | 0.088588
| align="center" | 0.087259
| align="center" | 0.066835
| align="center" | 0.065827
| align="center" | -
|-
| '''ΔE, boat (kcal mol-1)'''
| align="center" | 55.59
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.31
| align="center" | 44.7 ± 2.0
|}

By examining table 8 it is clear to see that the B3LYP/6-31G* method gives values which are much closer to the experimental ones. This is due to the B3LYP/6-31G* giving a higher level of optimisation than the HF/3-21G method. Table 8 also shows that the chair transition state has a lower activation energy to that of the boat transition state by 8.16 kcal mol-1. This suggests that the Cope rearrangement mechanism would most likely proceed through the chair transition state to that of the boat. Therefore the reaction would take place from the anti2 conformer to the chair transition state and then onto the gauche2 conformer within the Cope rearrangement.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9: Geometric parameters for the optimised chair and boat structures '''
| || ''' HF/3-21G chair TS''' || ''' B3LYP/6-31G* chair TS''' || '''HF/3-21G boat TS''' || '''B3LYP/6-31G* boat TS'''
|-
| Optimised Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Chair_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Boat_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.02 (C5-C10 or C3-C14)|| 1.97 (C5-C10 or C3-C14) || 2.14 (C3-C4 or C1-C6) || 2.21 (C3-C4 or C1-C6)
|-
| Fragment angle || 120 o (C3-C1-C5) || 120 o (C3-C1-C5) || 122 o (C4-C5-C6) || 122 o (C4-C5-C6)
|-
| Dihedral angle || 55 o (C1-C3-C14-C11)|| 54 o (C1-C3-C14-C11)|| 0 o (C2-C1-C6-C5) || 0 o (C2-C1-C6-C5)
|-
| Numbering of carbons || [[Image:Numbering_chairdh08.jpg|200x200px]]|| [[Image:Numbering_chairdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]]
|}

From studying table 9 it can be seen that the C-C bond breaking/forming length has changed between the two different types of optimisation methods. However the angles between the bonds remains unchanged.

=='''The Diels Alder Cycloaddition'''==

The Diels-Alder reaction is a cycloaddition between a conjugated diene and a dienophile (alkene system), which forms a cyclohexene system. It is often referred to as a [4s+2s] cycloaddition and proceeds through a single cyclic, concerted transition state where 2 σ bonds are formed in between the termini carbons of the 2 conjugated п systems. This is shown in the reaction scheme below where ethene is reacting with cis-butadiene.

[[Image:Reaction_scheme2dh08.jpg|300x300px|center]]

The π orbitals of the dienophile form new σ bonds with the π orbitals of the diene. The number of π electrons involved determine whether the reaction proceeds in a concerted stereospecific fashion (allowed) or not (forbidden). The HOMO/LUMO of one fragment interacts with the HOMO/LUMO of the other fragment to form two new bonding and anti-bonding MOs. The nodal properties allow one to predict how the reaction will occur.

Semi-Empirical/AM1 method and the B3LYP/6-31G* method of optimisation will be used on Gaussian to determine particular properites of the reactants to predict the way in which the reaction will proceed.

==='''cis-butadiene and ethylene: reagents'''===

<jmol>
<jmolAppletButton>
<uploadedFileContents>Cis_butadienedh08.mol</uploadedFileContents>
<text>Cis-butadiene</text>
<color>whi-te</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Ethenedh08.mol</uploadedFileContents>
<text>ethylene</text>
<color>white</color>
</jmolAppletButton>
</jmol> were optimised separately using the AM1 semi-empirical method on Gaussview.The cis-butadiene had an energy of 0.04879725 a. u (1.28 x 102 KJ mol-1) and ethylene had an energy of 0.02619027 a. u (68.76 KJ mol-1).

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10: Frontier molecular orbitals of cis-butadiene and ethylene '''
| '''Reagent''' || '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| cis-butadiene || LUMO || [[Image:LUMO1_cis_butadieneDH08.jpg|100x100px]] || [[Image:LUMO2_cis_butadienedh08.jpg|100x100px]]|| Symmetric || +0.017
|-
| cis-butadiene || HOMO || [[Image:HOMO1_cis_butadienedh08.jpg|100x100px]] || [[Image:HOMO2_cis_butadienedh08.jpg|100x100px]]|| Anti-symmetric || -0.343
|-
| ethylene || LUMO || [[Image:LUMO1_ethenedh08.jpg|100x100px]] || [[Image:LUMO2_ethenedh08.jpg|100x100px]]|| Anti-symmetric || +0.052
|-
| ethylene || HOMO || [[Image:HOMO1_ethenedh08.jpg|100x100px]] || [[Image:HOMO2_ethenedh08.jpg|100x100px]]|| Symmetric || -0.387
|}

 
The symmetries of the LUMO and HOMO orbitals for cis-butadiene are opposite of those for the ethylene. This therefore means that the only interactions that can take place are those between ethylene's LUMO and cis-butadiene's HOMO and in addition to this the interaction between cis-butadiene's LUMO and ethylene's HOMO. This is because orbital symmetry has to be conserved (the conservation of orbital symmetry states that the symmetry of an new orbital formed by the overlap of the 2 original orbitals must be the same as the symmetry of the initial orbitals).

==='''cis-butadiene and ethylene: transition state'''===

A <jmol>
<jmolAppletButton>
<uploadedFileContents>Bicyclic_startingdh08.mol</uploadedFileContents>
<text>bicyclo system</text>
<color>white</color>
</jmolAppletButton>
</jmol> was drawn and one of the -CH2-CH2- fragments were removed. Three C=C bonds were added and the previous one was taken away so a <jmol>
<jmolAppletButton>
<uploadedFileContents>Frozen_coorddh08.mol</uploadedFileContents>
<text>guess transition state</text>
<color>white</color>
</jmolAppletButton>
</jmol> was produced.

After this an optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while bonds between the termini of the fragments were frozen at 2.1Å. Following this the resulting file had two separate optimisation and frequency calculations run on it using the TS (Berny) with one using the semi-empirical/AM1 method and the other using the DFT/B3LYP/6-31G* level of theory. Both of these optimisations had the bonds unfrozen and the "derivative" option was chosen instead.

The DFT/B3LYP/6-31G* level of theory was used since it gave more accurate results in the calculations before and the energies it produces are consistent with the rest of the report (semi-empirical/AM1 method gives very different values).

The QST2 optimisation was not used since it required numbering the atoms in both the reactants and products, which took a lot of time to do.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11: Comparison of two methods of optimsations for the transition state '''
| || '''semi-empirical/AM1 method''' || '''DFT/B3LYP/6-31G* method'''
|-
| Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>SE_AM1_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>DFT_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Energy (a. u) || 0.11165608 || -234.54389573
|-
| Bond lengths || [[Image:DistancesSEdh08.jpg|150x150px]]|| [[Image:DistancesDFTdh08.jpg|150x150px]]
|-
|Imaginary frequency (cm-1) || -957 || -525
|-
|Animation || [[Image:-957dh08.gif]] || [[Image:-525dh08.gif]]
|-
| Lowest real frequency (cm-1) || 148 || 136
|-
|Animation || [[Image: 148dh08.gif]] || [[Image: 136dh08.gif]]
|}

The C-C bond forming length between the termini carbons is 0.15 Å larger for the DFT/B3LYP/6-31G* level of theory and this is because this method gives carries out the calculation to a greater degree of accuracy. This larger bond length leads a lower potential energy gradient to the transition state, which gives a lower total energy for the transition state. Therefore the given values below will be the DFT/B3LYP/6-31G* and not the semi-empirical/AM1 method.

A typical sp3 C-C bond length is 1.54 Å and a typical sp2 C=C bond length is 1.34 Å<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref>. Within the optimised transition molecule above the C-C bond lengths are 1.40 Å and the C=C bond length is 1.38 Å. This suggests that the π-bonds are breaking in order for the C-C bonds to form and additionally to this a π-bond is forming in between the two sp3 carbons.

The Van der Waals radius of a C-atom is 1.72 Å<ref name="ref 3">M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer, D. G. Truhlar, 1995, 346, pp. 63-69.[http://pubs.acs.org/doi/abs/10.1021/jp8111556]</ref>. The partly formed C-C length is 2.27 Å, which is shorter than two times the the van der Waals radius of a carbon atom. Therefore the carbon atoms are within a close enough distance to each other to share electron density and yet the bond has not formed at this point.

From looking at the animations for the imaginary frequency above it can be seen that the formation of the two bonds between the termini of the reactants is synchronous. This is expected of the Diels-Alder reaction since the mechanism is concerted. However the lowest real frequency is asynchronous with respect to the two bonds forming between the termini.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12: Frontier molecular orbitals of the transition state '''
| '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| LUMO || [[Image:LUMO1TSdh08.jpg|100x100px]] || [[Image:LUMO2TSdh08.jpg|100x100px]]|| Symmetric || +0.023
|-
| HOMO || [[Image:HOMO1TSdh08.jpg|100x100px]] || [[Image:HOMO2TSdh08.jpg|100x100px]]|| Anti-symmetric || -0.323
|}

From examining the images from table 12 above it can be deduced that the LUMO of the transition state originates from the cis-butadiene's LUMO and the ethylene's HOMO, which represents the σ*-antibonding orbitals. However the HOMO of the transition state originates from the cis-butadiene's HOMO and the ethylene's LUMO, which represents the σ-bonding orbitals within the two single C-C bonds that are formed within the reaction.

==='''cis-butadiene and ethylene: activation energy'''===

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 13: Calculation for the activation energy for the reaction'''
| '''molecule''' || '''DFT/B3LYP/6-31G* energy (a. u)''' || '''Calculations''' || '''Energy '''
|-
| cis-butadiene || -155.98595591 || transition state || -234.54389573 a. u
|-
| ethylene || -78.58745828 || ΔE, activation energy || 0.02951846 a. u
|-
| cis-butadiene and ethylene combined || -234.57341419 || ΔE, activation energy || 77.50 KJ mol-1
|}

==='''Cyclohexa-1,3-diene reaction with maleic anhydride'''===

The <jmol>
<jmolAppletButton>
<uploadedFileContents>FrozenTS22dh08.mol</uploadedFileContents>
<text>exo</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_frozenTS22dh08.mol</uploadedFileContents>
<text>endo</text>
<color>white</color>
</jmolAppletButton>
</jmol> transition states were drawn on Gaussview and then the optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while the distance of the forming bonds were frozen at 2.1Å between the respective carbons. After this the structures changed slightly (<jmol>
<jmolAppletButton>
<uploadedFileContents>TS2_afteropt1_exo1dh08.mol</uploadedFileContents>
<text>exo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_TS2_afteropt1dh08.mol</uploadedFileContents>
<text>endo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol>) and the resulting files had an optimisation and frequency calculation run on it using the TA (Berny) with using the semi-empirical/AM1 level of theory while the unfrozen option as changed to "derivative". After this the endo and exo optimised transition state structures were re-optimised as they were before but using the DFT/B3LYP/6-31G* level of theory instead with the additional keyword "Opt=NoEigen".

{| class="wikitable" border="1" align="center"
|+ '''Table 14: Comparison of the exo and endo transition states '''
! rowspan="2"|-!! colspan="2"|''exo'' !! colspan="2"|''endo''
|-
| Semi-empirical/AM1 || DFT/B3LYP/6-31G* || Semi-empirical/AM1 || DFT/B3LYP/6-31G*
|-
| align="center"|Structure ||align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>After_opt2_exodh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>Endo_TS2_afteropt2dh081.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| align="center"| Energy (a. u) ||align="center"| -0.05041975 ||align="center"| -612.67931095 ||align="center"| -0.05150470 ||align="center"| -612.68339684
|-
| C-C bond forming length (Å) ||align="center"| 2.17 ||align="center"| 2.29 ||align="center"| 2.16 ||align="center"| 2.27
|-align="center"
| Imaginary frequency (cm-1) ||align="center"| -812 ||align="center"| -448 ||align="center"| -806 ||align="center"| -447
|-align="center"
| Animation ||colspan="2"|[[Image:-812_SEdh08.gif]] || colspan="2"|[[Image:-806_cm-1dh08.gif]]
|}

Again the C-C bond forming length between the termini carbons is larger (by 0.11 and 0.12 Å) for the DFT/B3LYP/6-31G* level of theory and this is due the same reasons talked about above. However the C-C bond forming length between the termini carbons are very similar in both the endo and exo transition states (0.02 Å difference).

From looking at the animations for the imaginary frequency above it can also be seen that the formation of the two bonds between the carbons in this case as well is synchronous. Again this is expected of the Diels-Alder reaction since it is concerted.

{| align=center
|+ '''Strain diagrams for the endo and exo transition states'''
| [[Image:Exostraindh08.jpg|thumb|200x200px|'''Exo TS''']]
| [[Image:Strainendodh08.jpg|thumb|200x200px|'''Endo TS''']]
|}

From studying the energies in table 13 it can be seen that the endo transition state is more stable than the exo by 4.08589 x 10-3 a. u (10.73 KJ mol-1). However by looking at the strain of both transition states one would expect the exo TS to be more stable. This is because the distance between the cyclohexa-1,3-diene ring and the -(C=O)-O-(C=O)- maleic anhydride fragment is 2.99 Å in the endo TS and 3.03 Å in the exo TS. Thus the endo form would have more strain due to steric hindrance.

However within the exo TS the vicinal hydrogens on the CH2-CH2 (carbon numbers 18 and 21) fragment on the cyclohexa-1,3-diene ring are found below the plane and next to the maleic anhydride. This leads to steric hindrance is this is where the strain originates from within the exo TS. Therefore this disfavours the exo form, which makes the endo more stable.

Another factor making the endo TS more favourable is secondary orbital overlap. These orbitals contribute to the bonds forming and they interact with each other through space. Within the endo TS below there are 2 more orbitals interacting (secondary orbitals) than within the exo TS. This therefore makes the endo TS more stable and therefore more favoured.

[[Image:Secondary_orbital_shizzledh08.jpg|500x500px|center]]

{| class="wikitable" border="1" align="center"
|+ '''Table 15: Frontier molecular orbitals of the the transition state for the cyclohexa-1,3-diene reaction '''
! rowspan="2"|Molecular Orbital !! colspan="4"|''exo'' !! colspan="4"|''endo''
|- align="center"
| '''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''||'''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''
|- align="center"
| LUMO || [[Image:LUMO1exoh081.jpg|150x150px]]|| [[Image:LUMO2exodh081.jpg|150x150px]]|| Anti-symmetric || -0.078 || [[Image:LUMO1ENDOdh08.jpg|150x150px]]||[[Image:LUMO2ENDOdh08.jpg|150x150px]] || Anti-symmetric || -0.067
|- align="center"
| HOMO || [[Image:HOMO1exodh081.jpg|150x150px]]||[[Image:HOMO2exodh081.jpg|150x150px]] || Anti-symmetric|| -0.242 || [[Image:HOMO1ENDOdh08.jpg|150x150px]]|| [[Image:HOMO2ENDOdh08.jpg|150x150px]]||Anti-symmetric || -0.242
|}

Digital repositories:

exo: http://hdl.handle.net/10042/to-5596

endo: http://hdl.handle.net/10042/to-5597

From examining table 14 it is clear to see that both the HOMO and LUMO are anti-symmetric for both the transition states. The LUMO of the maleic anhydride fragment is the π* anti-bonding orbital of the C=C bond. Within the reaction then this has electrons transferred to it and the C=C bond breaks, which leaves just a C-C bond. The LUMOs for both transition states are both anti-bonding with respect to the two fragments. The HOMOs of both transitions state represent the σ-bonding that will be formed between the termini carbons on the products and reactants. This is most likely from the overlap of the cyclohexa-1,3-diene's HOMO and the maleic anhydride's LUMO.

=='''Conclusion'''==

='''References'''=
<references/>

Rep:Mod:hampluscheese

2010-11-10T23:11:07Z

Dh08: /* '''cis-butadiene and ethylene: transition state''' */

='''Module 3:Physical Computational Chemistry Lab'''=

Douglas Hunt

=='''Objectives'''==

The aim of this experiment is to

=='''The Cope Rearrangement'''==

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangement] is a [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement |3,3|-sigmatropic rearrangement] that takes place through a single [http://en.wikipedia.org/wiki/Pericyclic pericyclic], concerted [http://en.wikipedia.org/wiki/Transition_state transition state]. Within this rearrangement a σ bond is formed in concert with another being broken within a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] π system.

The Cope rearrangement we will be studying will be that of 1,5-hexadiene, which is shown below.

[[Image:Cope_rearrangement_schemedh08.jpg|300x300px|center]]

The objective is to locate the low-energy minima and transition structures on the C6H10 potential energy surface and to determine the preferred reaction mechanism by using [http://en.wikipedia.org/wiki/GAUSSIAN Gaussian], with the [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)] methods.

==='''Optimisation of 1,5-hexadiene Conformers'''===

Rough <jmol>
<jmolAppletButton>
<uploadedFileContents>Anti_2_drawndh08.mol</uploadedFileContents>
<text>anti</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche3_drawndh08.mol</uploadedFileContents>
<text>gauche</text>
<color>white</color>
</jmolAppletButton>
</jmol> conformers of 1,5-hexadiene were drawn on Gaussview and their structures were tidied by the clean function. They were then further optimised by the HF/3-21G method and were identified which conformer they were from their point group and appendix 1 on the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3 physical lab page].

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 1: Optimisation and comparison of anti and gauche 1,5-hexadiene conformers using HF/3-21G methods
| ''' Conformer''' || '''Optimised Structure''' || '''Energy (Atomic Units)''' || '''[http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1 Energies (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (Anti2 - Gauche3)''' || '''Point group'''
|-
| '''Anti2''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Anti_2_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692535 || -231.69254 ||-6.08 x 105 || 1.2606 x 10-4 a. u|| Ci
|-
| '''Gauche3''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Gauche3_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692661 || -231.69266 || -6.08 x 105 || 3.31 x 10-1 KJ mol-1 || C1
|}

From studying table 1 it can be seen that the gauche conformer is 0.33 KJ mol-1 more stable than the anti conformation. One would assume that the anti2 conformer would be more stable due to [http://en.wikipedia.org/wiki/Steric steric] factors/repulsion within the gauche3 conformation. However within the gauche3 conformer there is a favourable donation of [http://en.wikipedia.org/wiki/Electron_density electron density] from the C=C π orbital to the C-H σ* orbital from the adjacent proton, which stabilises the conformer<ref name="ref 1">B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446.[http://www.informaworld.com/smpp/content~db=all?content=10.1080/00268970110081412]</ref>.

 

The anti2 conformer was then optimised using the B3LYP 6-31G* (6-31G (d)) method.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 2: Comparison of the anti2 conformer optimisations carried out at different calculation methods
| ''' Method''' || '''Structure and bond distances (Å)''' || '''Dihedral angle (C1 to C4)''' || '''Energy (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (HF/3-21G - DFT/B3LYP/6-31G)'''
|-
| HF/3-21G || [[Image:Anti2_opt1dh08.jpg|300x300px]] || 114.6 o || -231.692535 || -6.08 x 105 || 2.919171 a. u
|-
| DFT/B3LYP/6-31G* || [[Image:Anti2_opt2DH08.jpg|300x300px]] || 118.8 o || -234.611716 || -6.16 x 105 || 7.66 x 103 KJ mol-1
|}

From examining table 2 at first it is possible to see that the bond lengths are very similar to each other and that of the literature bond lengths<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref> (C1=C2: 1.340 Å, C2-C3: 1.508 Å and C3-C4: 1.538 Å). It is also possible to see that the C=C bond length is more accurate for the DFT/B3LYP/6-31G* method by it being 0.01 Å closer to the literature value. Another main difference in terms of geometry is that the DFT/B3LYP/6-31G* method gives a larger [http://en.wikipedia.org/wiki/Dihedral_angle dihedral angle] (and most likely more accurate) of 118.8o compared to that of 114.6o.

In terms of energy the DFT/B3LYP/6-31G* method produces a conformer of a higher stability by it possessing a lower energy with a difference of 7.66 x 103 KJ mol-1.

==='''Frequency analysis of 1,5-hexadiene Conformers'''===

[[Image:Vibrations_anti2dh08.jpg|right|300x300px]]

For the frequency analysis the DFT/B3LYP/6-31G* method was used.

Frequency analysis is essential within this calculation due to it being the second derivative of the potential energy surface. This means that if all the frequencies are all positive the system is at a minimum, if one of them is negative the optimised system is at a transition state, and if more than one are negative then the calculation has failed to find a critical point, which means the optimisation has failed. This therefore gives us a method of seeing whether the calculation has worked.

From looking to the image on the right it can deduced that the optimisation has taken place successfully by there being no negative vibrations and therefore the conformation is at a minimum.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 3: Energies under the thermochemistry section at 298.15 K and 0 K within the frequency logfiles'''
| '''Energies''' || '''298.15 K (atomic units)''' ||| '''0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (KJ mol-1)''' || '''Discussion'''
|-
| Sum of electronic and zero-point energies || -234.469177 || -234.466673 || -2.504 x 10-3 || -6.57 || This is the potential energy at 0 K, which includes the zero-point vibrational energy (E = Eelec + ZPE).

|-
| Sum of electronic and thermal energies || -234.461835 || -234.461085 || -7.5 x 10-4|| -1.97 || This is the energy at 298.15 K (0 K for 0 K calculation) and 1 atm of pressure, which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans).

|-
| Sum of electronic and thermal enthalpies || -234.460891 || -234.460141 || -7.5 x 10-4 || -1.97 || This contains an additional correction for RT (H = E + RT), which is especially important when looking at dissociation reactions.
|-
| Sum of electronic and thermal free energies || -234.500739 || -234.495931 || 4.808 x 10-3 || -12.62 || This includes the entropic contribution to the free energy (G = H - TS).
|}

{|align=right
|+
| [[Image:Anti2_freq_IRdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 298.15 K''']]
| [[Image:Anti2_freq_0Kdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 0 K''']]
|}

From looking at table 3 above it can be deduced that the anti2 conformer is more stable at room temperature to that of 0 K with respect to all the types of energies. The spectra also seem to not change much between the two temperatures but the main change is the change in the sum of electronic and thermal free energies, which is heavily dependent of temperature by it including the entropic contribution to the free energy (G = H - '''T'''S)

 
 
 
 

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

Within the Cope Rearrangement there are two possible transition states, the chair (C2h) and the boat (C2v), which are shown within [https://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_2 appendix two ].

==='''Comparison of optimisation methods for the chair transition state'''===

An <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2hdh08.mol</uploadedFileContents>
<text>allyl fragment</text>
<color>white</color>
</jmolAppletButton>
</jmol>, CH2CHCH2 was drawn on Gaussview and was optimized through the HF/3-21G method. After this the original allyl fragment was copied and pasted into a new window and place exactly above the first fragment with a bond distance of 2.20 Å between the terminal end carbons. This therefore produced the <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2h_fulldh08.mol</uploadedFileContents>
<text>chair transition structure.</text>
<color>white</color>
</jmolAppletButton>
</jmol>

Gaussian was then set up for an optimisation for the transition state with the additional key words "Opt=NoEigen", the optimisation set to TS (Berny), the force constants set to "once" and using the HF/3-21G method.

After this the "chair" transition state was optimised an alternative way by freezing the coordinates of the two sets of terminal carbons (where the bonds form/break during the rearrangement) within the Redundant Coord Editor on Gauusview.
Following this a second calculation was run where the freeze coordinates option was changed to "Derivative" and the force constants were set to "never".

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 4: Comparison of chair transition state optimisation methods
| || '''Optimised through a TS (Berny)''' || '''Optimised through a frozen coordinate method'''
|-
| '''Optimised TS Structure''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>C2h_optfreq_TS_Bdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| '''Terminal C-C bond length (Å)''' || 2.02 || 2.02
|-
|'''Energy (atomic units)''' || -231.61932247 || -231.61932190
|-
| '''ΔE, TS (Berny) - frozen coordinate''' || 5.7 x 10-7 a.u || 1.50 x 10-3 KJ mol-1
|-
| '''Imaginary frequency (cm-1)''' || -818 || -818
|-
| '''Animation''' ||[[Image:818dh08TSB.gif|350x350px]] || [[Image:818cm-1Fdh08.gif]]
|}

The data in table 4 shows that both methods give the same results and they both give the same transition state. They both produced the same energy with a negligible energy difference between them and the bond lengths between the terminal carbons are both 2.02 Å. In addition to this the imaginary frequencies were both -818cm-1 and within the animations above it is possible to see that when one C-C bond is forming the other is breaking. This means it is a asynchronous bond formation, which is expected for the Cope Rearrangement since it is a pericyclic reaction.

The only difference between the two optimisations is the way in which the transition states were calculated. This is due to the TS (Berny) method calculating the force constant matrix within the first optimisation step and is then being recalculated as the optimisation carries on. When the frozen coordinate method freezes the reaction coordinate in order for the rest of the structure to be optimised around it, which does not require the whole hessian (force constant matrix) to be calculated but only that the reaction coordinate is differentiated (second step). The main advantage of this, is that you do not need to calculate the entire force constant matrix and just need to differentiate the reaction coordinate in order to obtain a good enough guess for the initial force constant matrix. The frozen coordinate method will therefore take less time to compute the transition state, which can be very expensive within certain systems. However to carry out this method one needs to have an idea of how far apart the terminal carbons are.

==='''Boat transition state'''===

===='''QST2 optimisation'''====

The optimisation for the boat transition state had a different method and this was the QST2 method. This calculation interpolates between two structures (product and reactant) and tries to find a transition state between them.

The reactants and products were numbered in the same way. Therefore both 1,5-hexadiene molecules had their numbering manually changed for the product molecule so it corresponded to the numbering of the reactant if it rearranged. This is illustrated in the reaction scheme below.

[[Image:Boat_rerrangmentdh08.jpg|300x300px|center]]

The chk file for the optimized anti2 conformer was opened and copied into a new window. A second window was opened within the new window and the same anti2 conformer was copied into the new window. The numbering was changed within the second molecule so the two looked like they do in the image below.

[[Image:TS_(QST2)_numbering_1dh08.jpg|400x400px|center]]

The job type was set as "Opt+Freq" and "TS (QST2)" was chosen to optimise the transition state. However this failed and produced
a structure a bit like a <jmol>
<jmolAppletButton>
<uploadedFileContents>Trialone_BPdh08.mol</uploadedFileContents>
<text>chair transition structure </text>
<color>white</color>
</jmolAppletButton>
</jmol> but more dissociated. This is because it interpolated between the two structures and translated the top allyl fragment and therefore did not consider the possibility of a rotation around the central bonds.

To get around this the C-C-C-C dihedral angle (C2-C3-C4-C5) was set to 0 o and the C-C-C angle (C2-C3-C4 and C3-C4-C5 for the product molecule) was set to 100 o for both molecules. This gave rise to the two structures below.

[[Image:TS_(QST2)_numbering_2dh08.jpg|400x400px|center]]

After this the QST2 calculation was run again and the results in table 5 were produced. The imaginary frequency at -839 cm-1 shows that while one bond is forming the other one is breaking, which is again an asynchronous bond formation.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5: Data for the boat transition state from the QST2 optimisation method
| Optimised TS Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.14
|-
|Energy (atomic units) || -231.60280151
|-
| Energy (KJ mol-1) || -6.08 x 105
|-
| Imaginary frequency (cm-1) || -839
|-
| Animation ||[[Image:-839dh08.gif]]
|}

===='''Intrinsic Reaction Coordinate'''====

It is not possible to predict which product the chair and boat transition state will lead to. To solve this problem within Gaussian one can use the IRC method, which allows one to follow the minimum energy path from a transition structure down to its local minimum on a potential energy surface. This creates a succession of points by taking small geometry steps in the direction to where the gradient of the energy surface is at its steepest until the minimum in energy is reached.

The chk file for the optimized chair transition structure was opened and the job type selected was "IRC", the reaction coordinate was computed in the forward direction (since molecule is symmetric) with 50 points along the IRC. This was done with two type of force constants being calculated, once and always.

Within the "once" method the output showed no bond formation process by the 26th step and therefore the calculation was incomplete, since a complete 1,5-hexadiene structure was supposed to be formed. From looking at the IRC pathway it can be seen that the minimum on the potential energy surface was not reached at that point, since the points on the pathway did not form an asymptote and the gradient did not go close enough to zero.

For the "always" method the computation of force constants were carried out at every step. This therefore lets the IRC calculation distinguish whether it is moving along the minimum energy pathway while going down the potential energy surface. While this is happening it can make changes to the pathway if necessary. Within the "always" case the bond forming arrives at 1,5-hexadiene and this is backed up by the IRC pathway reaching a minimum within its energy by the gradient going close to zero and the pathway reaches an asymptote.

The structure produced through the always method had an energy of -231.69199434 atomic units, which is very close to the energy of the gauche2 conformer in the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 appendix 1]. This suggests that the reaction takes place through the chair transition state, which later produces the gauche2 conformer of 1,5-hexadiene.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 6: Comparison of IRC calculations for the chair structure
| || '''Calculate force constants once''' || '''Calculate force constants always'''
|-
| '''Number of points along IRC''' || 26 || 47
|-
| '''Structure at the last point of IRC''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Once_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Always_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|'''Energy (atomic units)''' || -231.68906923 || -231.69199434
|-
| '''ΔE, once - always''' || 2.92511 x 10-3 a.u || 7.68 KJ mol-1
|-
| '''IRC pathway''' || [[Image:One_pathwaydh08.jpg|400x400px]] || [[Image:Always_pathwaydh08.jpg|400x400px]]
|-
| '''IRC gradient''' || [[Image:One_gradientdh08.jpg|400x400px]]|| [[Image:Always_gradientdh08.jpg|400x400px]]
|}

 
 
 

===='''Comparison of activation energies for the transition states'''====

The HF/3-21G optimised chair and boat transition structures were both reoptimised using the B3LYP/6-31G* (B3LYP/6-31G (d)) method. This gives a more accurate minimisation however this method takes longer to run. The values for the zero-point energies and thermal energies were found within the thermochemistry section of the log file.

The values for the HF/3-21G reactant (anti2) optimisation were calculated and the B3LYP/6-31G* values were taken from the "Frequency analysis of 1,5-hexadiene Conformers" earlier in this module.

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 7: Energy data for chair, boat and the anti2 conformation from the HF/3-21G and B3LYP/6-31G* methods of optimisation '''
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466677
| align="center" | -231.461320
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450936
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402342
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539524
| align="center" | -231.532562
| align="center" | -234.611716
| align="center" | -234.469177
| align="center" | -234.461835
|}
Digital repositories

Chair:
HF/3-21G: http://hdl.handle.net/10042/to-5569
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5568

Boat:
HF/3-21G: http://hdl.handle.net/10042/to-5570
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5571

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 8: Summary of activation energies for the chair and boat conformations (ΔE = chair TS or boat TS minus anti2)'''
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| align="center" | 1 a.u = 627.509 kcal mol-1
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE, chair (a.u)'''
| align="center" | 0.072847
| align="center" | 0.071242
| align="center" | 0.054248
| align="center" | 0.052827
| align="center" | -
|-
| '''ΔE, chair (kcal mol-1)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE, boat (a. u)'''
| align="center" | 0.088588
| align="center" | 0.087259
| align="center" | 0.066835
| align="center" | 0.065827
| align="center" | -
|-
| '''ΔE, boat (kcal mol-1)'''
| align="center" | 55.59
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.31
| align="center" | 44.7 ± 2.0
|}

By examining table 8 it is clear to see that the B3LYP/6-31G* method gives values which are much closer to the experimental ones. This is due to the B3LYP/6-31G* giving a higher level of optimisation than the HF/3-21G method. Table 8 also shows that the chair transition state has a lower activation energy to that of the boat transition state by 8.16 kcal mol-1. This suggests that the Cope rearrangement mechanism would most likely proceed through the chair transition state to that of the boat. Therefore the reaction would take place from the anti2 conformer to the chair transition state and then onto the gauche2 conformer within the Cope rearrangement.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9: Geometric parameters for the optimised chair and boat structures '''
| || ''' HF/3-21G chair TS''' || ''' B3LYP/6-31G* chair TS''' || '''HF/3-21G boat TS''' || '''B3LYP/6-31G* boat TS'''
|-
| Optimised Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Chair_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Boat_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.02 (C5-C10 or C3-C14)|| 1.97 (C5-C10 or C3-C14) || 2.14 (C3-C4 or C1-C6) || 2.21 (C3-C4 or C1-C6)
|-
| Fragment angle || 120 o (C3-C1-C5) || 120 o (C3-C1-C5) || 122 o (C4-C5-C6) || 122 o (C4-C5-C6)
|-
| Dihedral angle || 55 o (C1-C3-C14-C11)|| 54 o (C1-C3-C14-C11)|| 0 o (C2-C1-C6-C5) || 0 o (C2-C1-C6-C5)
|-
| Numbering of carbons || [[Image:Numbering_chairdh08.jpg|200x200px]]|| [[Image:Numbering_chairdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]]
|}

From studying table 9 it can be seen that the C-C bond breaking/forming length has changed between the two different types of optimisation methods. However the angles between the bonds remains unchanged.

=='''The Diels Alder Cycloaddition'''==

The Diels-Alder reaction is a cycloaddition between a conjugated diene and a dienophile (alkene system), which forms a cyclohexene system. It is often referred to as a [4s+2s] cycloaddition and proceeds through a single cyclic, concerted transition state where 2 σ bonds are formed in between the termini carbons of the 2 conjugated п systems. This is shown in the reaction scheme below where ethene is reacting with cis-butadiene.

[[Image:Reaction_scheme2dh08.jpg|300x300px|center]]

The π orbitals of the dienophile form new σ bonds with the π orbitals of the diene. The number of π electrons involved determine whether the reaction proceeds in a concerted stereospecific fashion (allowed) or not (forbidden). The HOMO/LUMO of one fragment interacts with the HOMO/LUMO of the other fragment to form two new bonding and anti-bonding MOs. The nodal properties allow one to predict how the reaction will occur.

Semi-Empirical/AM1 method and the B3LYP/6-31G* method of optimisation will be used on Gaussian to determine particular properites of the reactants to predict the way in which the reaction will proceed.

==='''cis-butadiene and ethylene: reagents'''===

<jmol>
<jmolAppletButton>
<uploadedFileContents>Cis_butadienedh08.mol</uploadedFileContents>
<text>Cis-butadiene</text>
<color>whi-te</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Ethenedh08.mol</uploadedFileContents>
<text>ethylene</text>
<color>white</color>
</jmolAppletButton>
</jmol> were optimised separately using the AM1 semi-empirical method on Gaussview.The cis-butadiene had an energy of 0.04879725 a. u (1.28 x 102 KJ mol-1) and ethylene had an energy of 0.02619027 a. u (68.76 KJ mol-1).

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10: Frontier molecular orbitals of cis-butadiene and ethylene '''
| '''Reagent''' || '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| cis-butadiene || LUMO || [[Image:LUMO1_cis_butadieneDH08.jpg|100x100px]] || [[Image:LUMO2_cis_butadienedh08.jpg|100x100px]]|| Symmetric || +0.017
|-
| cis-butadiene || HOMO || [[Image:HOMO1_cis_butadienedh08.jpg|100x100px]] || [[Image:HOMO2_cis_butadienedh08.jpg|100x100px]]|| Anti-symmetric || -0.343
|-
| ethylene || LUMO || [[Image:LUMO1_ethenedh08.jpg|100x100px]] || [[Image:LUMO2_ethenedh08.jpg|100x100px]]|| Anti-symmetric || +0.052
|-
| ethylene || HOMO || [[Image:HOMO1_ethenedh08.jpg|100x100px]] || [[Image:HOMO2_ethenedh08.jpg|100x100px]]|| Symmetric || -0.387
|}

 
The symmetries of the LUMO and HOMO orbitals for cis-butadiene are opposite of those for the ethylene. This therefore means that the only interactions that can take place are those between ethylene's LUMO and cis-butadiene's HOMO and in addition to this the interaction between cis-butadiene's LUMO and ethylene's HOMO. This is because orbital symmetry has to be conserved (the conservation of orbital symmetry states that the symmetry of an new orbital formed by the overlap of the 2 original orbitals must be the same as the symmetry of the initial orbitals).

==='''cis-butadiene and ethylene: transition state'''===

A <jmol>
<jmolAppletButton>
<uploadedFileContents>Bicyclic_startingdh08.mol</uploadedFileContents>
<text>bicyclo system</text>
<color>white</color>
</jmolAppletButton>
</jmol> was drawn and one of the -CH2-CH2- fragments were removed. Three C=C bonds were added and the previous one was taken away so a <jmol>
<jmolAppletButton>
<uploadedFileContents>Frozen_coorddh08.mol</uploadedFileContents>
<text>guess transition state</text>
<color>white</color>
</jmolAppletButton>
</jmol> was produced.

After this an optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while bonds between the termini of the fragments were frozen at 2.1Å. Following this the resulting file had two separate optimisation and frequency calculations run on it using the TS (Berny) with one using the semi-empirical/AM1 method and the other using the DFT/B3LYP/6-31G* level of theory. Both of these optimisations had the bonds unfrozen and the "derivative" option was chosen instead.

The DFT/B3LYP/6-31G* level of theory was used since it gave more accurate results in the calculations before and the energies it produces are consistent with the rest of the report (semi-empirical/AM1 method gives very different values).

The QST2 optimisation was not used since it required numbering the atoms in both the reactants and products, which took a lot of time to do.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11: Comparison of two methods of optimsations for the transition state '''
| || '''semi-empirical/AM1 method''' || '''DFT/B3LYP/6-31G* method'''
|-
| Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>SE_AM1_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>DFT_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Energy (a. u) || 0.11165608 || -234.54389573
|-
| Bond lengths || [[Image:DistancesSEdh08.jpg|150x150px]]|| [[Image:DistancesDFTdh08.jpg|150x150px]]
|-
|Imaginary frequency (cm-1) || -957 || -525
|-
|Animation || [[Image:-957dh08.gif]] || [[Image:-525dh08.gif]]
|-
| Lowest real frequency (cm-1) || 148 || 136
|-
|Animation || [[Image: 148dh08.gif]] || [[Image: 136dh08.gif]]

|}

The C-C bond forming length between the termini carbons is 0.15 Å larger for the DFT/B3LYP/6-31G* level of theory and this is because this method gives carries out the calculation to a greater degree of accuracy. This larger bond length leads a lower potential energy gradient to the transition state, which gives a lower total energy for the transition state. Therefore the given values below will be the DFT/B3LYP/6-31G* and not the semi-empirical/AM1 method.

A typical sp3 C-C bond length is 1.54 Å and a typical sp2 C=C bond length is 1.34 Å<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref>. Within the optimised transition molecule above the C-C bond lengths are 1.40 Å and the C=C bond length is 1.38 Å. This suggests that the π-bonds are breaking in order for the C-C bonds to form and additionally to this a π-bond is forming in between the two sp3 carbons.

The Van der Waals radius of a C-atom is 1.72 Å<ref name="ref 3">M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer, D. G. Truhlar, 1995, 346, pp. 63-69.[http://pubs.acs.org/doi/abs/10.1021/jp8111556]</ref>. The partly formed C-C length is 2.27 Å, which is shorter than two times the the van der Waals radius of a carbon atom. Therefore the carbon atoms are within a close enough distance to each other to share electron density and yet the bond has not formed at this point.

From looking at the animations for the imaginary frequency above it can be seen that the formation of the two bonds between the termini of the reactants is synchronous. This is expected of the Diels-Alder reaction since the mechanism is concerted. However the lowest real frequency is asynchronous with respect to the two bonds forming between the termini.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12: Frontier molecular orbitals of the transition state '''
| '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| LUMO || [[Image:LUMO1TSdh08.jpg|100x100px]] || [[Image:LUMO2TSdh08.jpg|100x100px]]|| Symmetric || +0.023
|-
| HOMO || [[Image:HOMO1TSdh08.jpg|100x100px]] || [[Image:HOMO2TSdh08.jpg|100x100px]]|| Anti-symmetric || -0.323
|}

From examining the images from table 12 above it can be deduced that the LUMO of the transition state originates from the cis-butadiene's LUMO and the ethylene's HOMO, which represents the σ*-antibonding orbitals. However the HOMO of the transition state originates from the cis-butadiene's HOMO and the ethylene's LUMO, which represents the σ-bonding orbitals within the two single C-C bonds that are formed within the reaction.

==='''Cyclohexa-1,3-diene reaction with maleic anhydride'''===

The <jmol>
<jmolAppletButton>
<uploadedFileContents>FrozenTS22dh08.mol</uploadedFileContents>
<text>exo</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_frozenTS22dh08.mol</uploadedFileContents>
<text>endo</text>
<color>white</color>
</jmolAppletButton>
</jmol> transition states were drawn on Gaussview and then the optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while the distance of the forming bonds were frozen at 2.1Å between the respective carbons. After this the structures changed slightly (<jmol>
<jmolAppletButton>
<uploadedFileContents>TS2_afteropt1_exo1dh08.mol</uploadedFileContents>
<text>exo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_TS2_afteropt1dh08.mol</uploadedFileContents>
<text>endo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol>) and the resulting files had an optimisation and frequency calculation run on it using the TA (Berny) with using the semi-empirical/AM1 level of theory while the unfrozen option as changed to "derivative". After this the endo and exo optimised transition state structures were re-optimised as they were before but using the DFT/B3LYP/6-31G* level of theory instead with the additional keyword "Opt=NoEigen".

{| class="wikitable" border="1" align="center"
|+ '''Table 13: Comparison of the exo and endo transition states '''
! rowspan="2"|-!! colspan="2"|''exo'' !! colspan="2"|''endo''
|-
| Semi-empirical/AM1 || DFT/B3LYP/6-31G* || Semi-empirical/AM1 || DFT/B3LYP/6-31G*
|-
| align="center"|Structure ||align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>After_opt2_exodh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>Endo_TS2_afteropt2dh081.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| align="center"| Energy (a. u) ||align="center"| -0.05041975 ||align="center"| -612.67931095 ||align="center"| -0.05150470 ||align="center"| -612.68339684
|-
| C-C bond forming length (Å) ||align="center"| 2.17 ||align="center"| 2.29 ||align="center"| 2.16 ||align="center"| 2.27
|-align="center"
| Imaginary frequency (cm-1) ||align="center"| -812 ||align="center"| -448 ||align="center"| -806 ||align="center"| -447
|-align="center"
| Animation ||colspan="2"|[[Image:-812_SEdh08.gif]] || colspan="2"|[[Image:-806_cm-1dh08.gif]]
|}

Again the C-C bond forming length between the termini carbons is larger (by 0.11 and 0.12 Å) for the DFT/B3LYP/6-31G* level of theory and this is due the same reasons talked about above. However the C-C bond forming length between the termini carbons are very similar in both the endo and exo transition states (0.02 Å difference).

From looking at the animations for the imaginary frequency above it can also be seen that the formation of the two bonds between the carbons in this case as well is synchronous. Again this is expected of the Diels-Alder reaction since it is concerted.

{| align=center
|+ '''Strain diagrams for the endo and exo transition states'''
| [[Image:Exostraindh08.jpg|thumb|200x200px|'''Exo TS''']]
| [[Image:Strainendodh08.jpg|thumb|200x200px|'''Endo TS''']]
|}

From studying the energies in table 13 it can be seen that the endo transition state is more stable than the exo by 4.08589 x 10-3 a. u (10.73 KJ mol-1). However by looking at the strain of both transition states one would expect the exo TS to be more stable. This is because the distance between the cyclohexa-1,3-diene ring and the -(C=O)-O-(C=O)- maleic anhydride fragment is 2.99 Å in the endo TS and 3.03 Å in the exo TS. Thus the endo form would have more strain due to steric hindrance.

However within the exo TS the vicinal hydrogens on the CH2-CH2 (carbon numbers 18 and 21) fragment on the cyclohexa-1,3-diene ring are found below the plane and next to the maleic anhydride. This leads to steric hindrance is this is where the strain originates from within the exo TS. Therefore this disfavours the exo form, which makes the endo more stable.

Another factor making the endo TS more favourable is secondary orbital overlap. These orbitals contribute to the bonds forming and they interact with each other through space. Within the endo TS below there are 2 more orbitals interacting (secondary orbitals) than within the exo TS. This therefore makes the endo TS more stable and therefore more favoured.

[[Image:Secondary_orbital_shizzledh08.jpg|500x500px|center]]

{| class="wikitable" border="1" align="center"
|+ '''Table 14: Frontier molecular orbitals of the the transition state for the cyclohexa-1,3-diene reaction '''
! rowspan="2"|Molecular Orbital !! colspan="4"|''exo'' !! colspan="4"|''endo''
|- align="center"
| '''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''||'''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''
|- align="center"
| LUMO || [[Image:LUMO1exoh081.jpg|150x150px]]|| [[Image:LUMO2exodh081.jpg|150x150px]]|| Anti-symmetric || -0.078 || [[Image:LUMO1ENDOdh08.jpg|150x150px]]||[[Image:LUMO2ENDOdh08.jpg|150x150px]] || Anti-symmetric || -0.067
|- align="center"
| HOMO || [[Image:HOMO1exodh081.jpg|150x150px]]||[[Image:HOMO2exodh081.jpg|150x150px]] || Anti-symmetric|| -0.242 || [[Image:HOMO1ENDOdh08.jpg|150x150px]]|| [[Image:HOMO2ENDOdh08.jpg|150x150px]]||Anti-symmetric || -0.242
|}

Digital repositories:

exo: http://hdl.handle.net/10042/to-5596

endo: http://hdl.handle.net/10042/to-5597

From examining table 14 it is clear to see that both the HOMO and LUMO are anti-symmetric for both the transition states. The LUMO of the maleic anhydride fragment is the π* anti-bonding orbital of the C=C bond. Within the reaction then this has electrons transferred to it and the C=C bond breaks, which leaves just a C-C bond. The LUMOs for both transition states are both anti-bonding with respect to the two fragments. The HOMOs of both transitions state represent the σ-bonding that will be formed between the termini carbons on the products and reactants. This is most likely from the overlap of the cyclohexa-1,3-diene's HOMO and the maleic anhydride's LUMO.

=='''Conclusion'''==

='''References'''=
<references/>

Rep:Mod:hampluscheese

2010-11-10T22:44:12Z

Dh08: /* '''Transition State''' */

='''Module 3:Physical Computational Chemistry Lab'''=

Douglas Hunt

=='''Objectives'''==

The aim of this experiment is to

=='''The Cope Rearrangement'''==

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangement] is a [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement |3,3|-sigmatropic rearrangement] that takes place through a single [http://en.wikipedia.org/wiki/Pericyclic pericyclic], concerted [http://en.wikipedia.org/wiki/Transition_state transition state]. Within this rearrangement a σ bond is formed in concert with another being broken within a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] π system.

The Cope rearrangement we will be studying will be that of 1,5-hexadiene, which is shown below.

[[Image:Cope_rearrangement_schemedh08.jpg|300x300px|center]]

The objective is to locate the low-energy minima and transition structures on the C6H10 potential energy surface and to determine the preferred reaction mechanism by using [http://en.wikipedia.org/wiki/GAUSSIAN Gaussian], with the [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)] methods.

==='''Optimisation of 1,5-hexadiene Conformers'''===

Rough <jmol>
<jmolAppletButton>
<uploadedFileContents>Anti_2_drawndh08.mol</uploadedFileContents>
<text>anti</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche3_drawndh08.mol</uploadedFileContents>
<text>gauche</text>
<color>white</color>
</jmolAppletButton>
</jmol> conformers of 1,5-hexadiene were drawn on Gaussview and their structures were tidied by the clean function. They were then further optimised by the HF/3-21G method and were identified which conformer they were from their point group and appendix 1 on the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3 physical lab page].

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 1: Optimisation and comparison of anti and gauche 1,5-hexadiene conformers using HF/3-21G methods
| ''' Conformer''' || '''Optimised Structure''' || '''Energy (Atomic Units)''' || '''[http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1 Energies (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (Anti2 - Gauche3)''' || '''Point group'''
|-
| '''Anti2''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Anti_2_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692535 || -231.69254 ||-6.08 x 105 || 1.2606 x 10-4 a. u|| Ci
|-
| '''Gauche3''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Gauche3_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692661 || -231.69266 || -6.08 x 105 || 3.31 x 10-1 KJ mol-1 || C1
|}

From studying table 1 it can be seen that the gauche conformer is 0.33 KJ mol-1 more stable than the anti conformation. One would assume that the anti2 conformer would be more stable due to [http://en.wikipedia.org/wiki/Steric steric] factors/repulsion within the gauche3 conformation. However within the gauche3 conformer there is a favourable donation of [http://en.wikipedia.org/wiki/Electron_density electron density] from the C=C π orbital to the C-H σ* orbital from the adjacent proton, which stabilises the conformer<ref name="ref 1">B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446.[http://www.informaworld.com/smpp/content~db=all?content=10.1080/00268970110081412]</ref>.

 

The anti2 conformer was then optimised using the B3LYP 6-31G* (6-31G (d)) method.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 2: Comparison of the anti2 conformer optimisations carried out at different calculation methods
| ''' Method''' || '''Structure and bond distances (Å)''' || '''Dihedral angle (C1 to C4)''' || '''Energy (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (HF/3-21G - DFT/B3LYP/6-31G)'''
|-
| HF/3-21G || [[Image:Anti2_opt1dh08.jpg|300x300px]] || 114.6 o || -231.692535 || -6.08 x 105 || 2.919171 a. u
|-
| DFT/B3LYP/6-31G* || [[Image:Anti2_opt2DH08.jpg|300x300px]] || 118.8 o || -234.611716 || -6.16 x 105 || 7.66 x 103 KJ mol-1
|}

From examining table 2 at first it is possible to see that the bond lengths are very similar to each other and that of the literature bond lengths<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref> (C1=C2: 1.340 Å, C2-C3: 1.508 Å and C3-C4: 1.538 Å). It is also possible to see that the C=C bond length is more accurate for the DFT/B3LYP/6-31G* method by it being 0.01 Å closer to the literature value. Another main difference in terms of geometry is that the DFT/B3LYP/6-31G* method gives a larger [http://en.wikipedia.org/wiki/Dihedral_angle dihedral angle] (and most likely more accurate) of 118.8o compared to that of 114.6o.

In terms of energy the DFT/B3LYP/6-31G* method produces a conformer of a higher stability by it possessing a lower energy with a difference of 7.66 x 103 KJ mol-1.

==='''Frequency analysis of 1,5-hexadiene Conformers'''===

[[Image:Vibrations_anti2dh08.jpg|right|300x300px]]

For the frequency analysis the DFT/B3LYP/6-31G* method was used.

Frequency analysis is essential within this calculation due to it being the second derivative of the potential energy surface. This means that if all the frequencies are all positive the system is at a minimum, if one of them is negative the optimised system is at a transition state, and if more than one are negative then the calculation has failed to find a critical point, which means the optimisation has failed. This therefore gives us a method of seeing whether the calculation has worked.

From looking to the image on the right it can deduced that the optimisation has taken place successfully by there being no negative vibrations and therefore the conformation is at a minimum.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 3: Energies under the thermochemistry section at 298.15 K and 0 K within the frequency logfiles'''
| '''Energies''' || '''298.15 K (atomic units)''' ||| '''0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (KJ mol-1)''' || '''Discussion'''
|-
| Sum of electronic and zero-point energies || -234.469177 || -234.466673 || -2.504 x 10-3 || -6.57 || This is the potential energy at 0 K, which includes the zero-point vibrational energy (E = Eelec + ZPE).

|-
| Sum of electronic and thermal energies || -234.461835 || -234.461085 || -7.5 x 10-4|| -1.97 || This is the energy at 298.15 K (0 K for 0 K calculation) and 1 atm of pressure, which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans).

|-
| Sum of electronic and thermal enthalpies || -234.460891 || -234.460141 || -7.5 x 10-4 || -1.97 || This contains an additional correction for RT (H = E + RT), which is especially important when looking at dissociation reactions.
|-
| Sum of electronic and thermal free energies || -234.500739 || -234.495931 || 4.808 x 10-3 || -12.62 || This includes the entropic contribution to the free energy (G = H - TS).
|}

{|align=right
|+
| [[Image:Anti2_freq_IRdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 298.15 K''']]
| [[Image:Anti2_freq_0Kdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 0 K''']]
|}

From looking at table 3 above it can be deduced that the anti2 conformer is more stable at room temperature to that of 0 K with respect to all the types of energies. The spectra also seem to not change much between the two temperatures but the main change is the change in the sum of electronic and thermal free energies, which is heavily dependent of temperature by it including the entropic contribution to the free energy (G = H - '''T'''S)

 
 
 
 

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

Within the Cope Rearrangement there are two possible transition states, the chair (C2h) and the boat (C2v), which are shown within [https://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_2 appendix two ].

==='''Comparison of optimisation methods for the chair transition state'''===

An <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2hdh08.mol</uploadedFileContents>
<text>allyl fragment</text>
<color>white</color>
</jmolAppletButton>
</jmol>, CH2CHCH2 was drawn on Gaussview and was optimized through the HF/3-21G method. After this the original allyl fragment was copied and pasted into a new window and place exactly above the first fragment with a bond distance of 2.20 Å between the terminal end carbons. This therefore produced the <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2h_fulldh08.mol</uploadedFileContents>
<text>chair transition structure.</text>
<color>white</color>
</jmolAppletButton>
</jmol>

Gaussian was then set up for an optimisation for the transition state with the additional key words "Opt=NoEigen", the optimisation set to TS (Berny), the force constants set to "once" and using the HF/3-21G method.

After this the "chair" transition state was optimised an alternative way by freezing the coordinates of the two sets of terminal carbons (where the bonds form/break during the rearrangement) within the Redundant Coord Editor on Gauusview.
Following this a second calculation was run where the freeze coordinates option was changed to "Derivative" and the force constants were set to "never".

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 4: Comparison of chair transition state optimisation methods
| || '''Optimised through a TS (Berny)''' || '''Optimised through a frozen coordinate method'''
|-
| '''Optimised TS Structure''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>C2h_optfreq_TS_Bdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| '''Terminal C-C bond length (Å)''' || 2.02 || 2.02
|-
|'''Energy (atomic units)''' || -231.61932247 || -231.61932190
|-
| '''ΔE, TS (Berny) - frozen coordinate''' || 5.7 x 10-7 a.u || 1.50 x 10-3 KJ mol-1
|-
| '''Imaginary frequency (cm-1)''' || -818 || -818
|-
| '''Animation''' ||[[Image:818dh08TSB.gif|350x350px]] || [[Image:818cm-1Fdh08.gif]]
|}

The data in table 4 shows that both methods give the same results and they both give the same transition state. They both produced the same energy with a negligible energy difference between them and the bond lengths between the terminal carbons are both 2.02 Å. In addition to this the imaginary frequencies were both -818cm-1 and within the animations above it is possible to see that when one C-C bond is forming the other is breaking. This means it is a asynchronous bond formation, which is expected for the Cope Rearrangement since it is a pericyclic reaction.

The only difference between the two optimisations is the way in which the transition states were calculated. This is due to the TS (Berny) method calculating the force constant matrix within the first optimisation step and is then being recalculated as the optimisation carries on. When the frozen coordinate method freezes the reaction coordinate in order for the rest of the structure to be optimised around it, which does not require the whole hessian (force constant matrix) to be calculated but only that the reaction coordinate is differentiated (second step). The main advantage of this, is that you do not need to calculate the entire force constant matrix and just need to differentiate the reaction coordinate in order to obtain a good enough guess for the initial force constant matrix. The frozen coordinate method will therefore take less time to compute the transition state, which can be very expensive within certain systems. However to carry out this method one needs to have an idea of how far apart the terminal carbons are.

==='''Boat transition state'''===

===='''QST2 optimisation'''====

The optimisation for the boat transition state had a different method and this was the QST2 method. This calculation interpolates between two structures (product and reactant) and tries to find a transition state between them.

The reactants and products were numbered in the same way. Therefore both 1,5-hexadiene molecules had their numbering manually changed for the product molecule so it corresponded to the numbering of the reactant if it rearranged. This is illustrated in the reaction scheme below.

[[Image:Boat_rerrangmentdh08.jpg|300x300px|center]]

The chk file for the optimized anti2 conformer was opened and copied into a new window. A second window was opened within the new window and the same anti2 conformer was copied into the new window. The numbering was changed within the second molecule so the two looked like they do in the image below.

[[Image:TS_(QST2)_numbering_1dh08.jpg|400x400px|center]]

The job type was set as "Opt+Freq" and "TS (QST2)" was chosen to optimise the transition state. However this failed and produced
a structure a bit like a <jmol>
<jmolAppletButton>
<uploadedFileContents>Trialone_BPdh08.mol</uploadedFileContents>
<text>chair transition structure </text>
<color>white</color>
</jmolAppletButton>
</jmol> but more dissociated. This is because it interpolated between the two structures and translated the top allyl fragment and therefore did not consider the possibility of a rotation around the central bonds.

To get around this the C-C-C-C dihedral angle (C2-C3-C4-C5) was set to 0 o and the C-C-C angle (C2-C3-C4 and C3-C4-C5 for the product molecule) was set to 100 o for both molecules. This gave rise to the two structures below.

[[Image:TS_(QST2)_numbering_2dh08.jpg|400x400px|center]]

After this the QST2 calculation was run again and the results in table 5 were produced. The imaginary frequency at -839 cm-1 shows that while one bond is forming the other one is breaking, which is again an asynchronous bond formation.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5: Data for the boat transition state from the QST2 optimisation method
| Optimised TS Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.14
|-
|Energy (atomic units) || -231.60280151
|-
| Energy (KJ mol-1) || -6.08 x 105
|-
| Imaginary frequency (cm-1) || -839
|-
| Animation ||[[Image:-839dh08.gif]]
|}

===='''Intrinsic Reaction Coordinate'''====

It is not possible to predict which product the chair and boat transition state will lead to. To solve this problem within Gaussian one can use the IRC method, which allows one to follow the minimum energy path from a transition structure down to its local minimum on a potential energy surface. This creates a succession of points by taking small geometry steps in the direction to where the gradient of the energy surface is at its steepest until the minimum in energy is reached.

The chk file for the optimized chair transition structure was opened and the job type selected was "IRC", the reaction coordinate was computed in the forward direction (since molecule is symmetric) with 50 points along the IRC. This was done with two type of force constants being calculated, once and always.

Within the "once" method the output showed no bond formation process by the 26th step and therefore the calculation was incomplete, since a complete 1,5-hexadiene structure was supposed to be formed. From looking at the IRC pathway it can be seen that the minimum on the potential energy surface was not reached at that point, since the points on the pathway did not form an asymptote and the gradient did not go close enough to zero.

For the "always" method the computation of force constants were carried out at every step. This therefore lets the IRC calculation distinguish whether it is moving along the minimum energy pathway while going down the potential energy surface. While this is happening it can make changes to the pathway if necessary. Within the "always" case the bond forming arrives at 1,5-hexadiene and this is backed up by the IRC pathway reaching a minimum within its energy by the gradient going close to zero and the pathway reaches an asymptote.

The structure produced through the always method had an energy of -231.69199434 atomic units, which is very close to the energy of the gauche2 conformer in the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 appendix 1]. This suggests that the reaction takes place through the chair transition state, which later produces the gauche2 conformer of 1,5-hexadiene.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 6: Comparison of IRC calculations for the chair structure
| || '''Calculate force constants once''' || '''Calculate force constants always'''
|-
| '''Number of points along IRC''' || 26 || 47
|-
| '''Structure at the last point of IRC''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Once_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Always_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|'''Energy (atomic units)''' || -231.68906923 || -231.69199434
|-
| '''ΔE, once - always''' || 2.92511 x 10-3 a.u || 7.68 KJ mol-1
|-
| '''IRC pathway''' || [[Image:One_pathwaydh08.jpg|400x400px]] || [[Image:Always_pathwaydh08.jpg|400x400px]]
|-
| '''IRC gradient''' || [[Image:One_gradientdh08.jpg|400x400px]]|| [[Image:Always_gradientdh08.jpg|400x400px]]
|}

 
 
 

===='''Comparison of activation energies for the transition states'''====

The HF/3-21G optimised chair and boat transition structures were both reoptimised using the B3LYP/6-31G* (B3LYP/6-31G (d)) method. This gives a more accurate minimisation however this method takes longer to run. The values for the zero-point energies and thermal energies were found within the thermochemistry section of the log file.

The values for the HF/3-21G reactant (anti2) optimisation were calculated and the B3LYP/6-31G* values were taken from the "Frequency analysis of 1,5-hexadiene Conformers" earlier in this module.

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 7: Energy data for chair, boat and the anti2 conformation from the HF/3-21G and B3LYP/6-31G* methods of optimisation '''
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466677
| align="center" | -231.461320
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450936
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402342
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539524
| align="center" | -231.532562
| align="center" | -234.611716
| align="center" | -234.469177
| align="center" | -234.461835
|}
Digital repositories

Chair:
HF/3-21G: http://hdl.handle.net/10042/to-5569
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5568

Boat:
HF/3-21G: http://hdl.handle.net/10042/to-5570
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5571

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 8: Summary of activation energies for the chair and boat conformations (ΔE = chair TS or boat TS minus anti2)'''
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| align="center" | 1 a.u = 627.509 kcal mol-1
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE, chair (a.u)'''
| align="center" | 0.072847
| align="center" | 0.071242
| align="center" | 0.054248
| align="center" | 0.052827
| align="center" | -
|-
| '''ΔE, chair (kcal mol-1)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE, boat (a. u)'''
| align="center" | 0.088588
| align="center" | 0.087259
| align="center" | 0.066835
| align="center" | 0.065827
| align="center" | -
|-
| '''ΔE, boat (kcal mol-1)'''
| align="center" | 55.59
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.31
| align="center" | 44.7 ± 2.0
|}

By examining table 8 it is clear to see that the B3LYP/6-31G* method gives values which are much closer to the experimental ones. This is due to the B3LYP/6-31G* giving a higher level of optimisation than the HF/3-21G method. Table 8 also shows that the chair transition state has a lower activation energy to that of the boat transition state by 8.16 kcal mol-1. This suggests that the Cope rearrangement mechanism would most likely proceed through the chair transition state to that of the boat. Therefore the reaction would take place from the anti2 conformer to the chair transition state and then onto the gauche2 conformer within the Cope rearrangement.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9: Geometric parameters for the optimised chair and boat structures '''
| || ''' HF/3-21G chair TS''' || ''' B3LYP/6-31G* chair TS''' || '''HF/3-21G boat TS''' || '''B3LYP/6-31G* boat TS'''
|-
| Optimised Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Chair_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Boat_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.02 (C5-C10 or C3-C14)|| 1.97 (C5-C10 or C3-C14) || 2.14 (C3-C4 or C1-C6) || 2.21 (C3-C4 or C1-C6)
|-
| Fragment angle || 120 o (C3-C1-C5) || 120 o (C3-C1-C5) || 122 o (C4-C5-C6) || 122 o (C4-C5-C6)
|-
| Dihedral angle || 55 o (C1-C3-C14-C11)|| 54 o (C1-C3-C14-C11)|| 0 o (C2-C1-C6-C5) || 0 o (C2-C1-C6-C5)
|-
| Numbering of carbons || [[Image:Numbering_chairdh08.jpg|200x200px]]|| [[Image:Numbering_chairdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]]
|}

From studying table 9 it can be seen that the C-C bond breaking/forming length has changed between the two different types of optimisation methods. However the angles between the bonds remains unchanged.

=='''The Diels Alder Cycloaddition'''==

The Diels-Alder reaction is a cycloaddition between a conjugated diene and a dienophile (alkene system), which forms a cyclohexene system. It is often referred to as a [4s+2s] cycloaddition and proceeds through a single cyclic, concerted transition state where 2 σ bonds are formed in between the termini carbons of the 2 conjugated п systems. This is shown in the reaction scheme below where ethene is reacting with cis-butadiene.

[[Image:Reaction_scheme2dh08.jpg|300x300px|center]]

The π orbitals of the dienophile form new σ bonds with the π orbitals of the diene. The number of π electrons involved determine whether the reaction proceeds in a concerted stereospecific fashion (allowed) or not (forbidden). The HOMO/LUMO of one fragment interacts with the HOMO/LUMO of the other fragment to form two new bonding and anti-bonding MOs. The nodal properties allow one to predict how the reaction will occur.

Semi-Empirical/AM1 method and the B3LYP/6-31G* method of optimisation will be used on Gaussian to determine particular properites of the reactants to predict the way in which the reaction will proceed.

==='''cis-butadiene and ethylene: reagents'''===

<jmol>
<jmolAppletButton>
<uploadedFileContents>Cis_butadienedh08.mol</uploadedFileContents>
<text>Cis-butadiene</text>
<color>whi-te</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Ethenedh08.mol</uploadedFileContents>
<text>ethylene</text>
<color>white</color>
</jmolAppletButton>
</jmol> were optimised separately using the AM1 semi-empirical method on Gaussview.The cis-butadiene had an energy of 0.04879725 a. u (1.28 x 102 KJ mol-1) and ethylene had an energy of 0.02619027 a. u (68.76 KJ mol-1).

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10: Frontier molecular orbitals of cis-butadiene and ethylene '''
| '''Reagent''' || '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| cis-butadiene || LUMO || [[Image:LUMO1_cis_butadieneDH08.jpg|100x100px]] || [[Image:LUMO2_cis_butadienedh08.jpg|100x100px]]|| Symmetric || +0.017
|-
| cis-butadiene || HOMO || [[Image:HOMO1_cis_butadienedh08.jpg|100x100px]] || [[Image:HOMO2_cis_butadienedh08.jpg|100x100px]]|| Anti-symmetric || -0.343
|-
| ethylene || LUMO || [[Image:LUMO1_ethenedh08.jpg|100x100px]] || [[Image:LUMO2_ethenedh08.jpg|100x100px]]|| Anti-symmetric || +0.052
|-
| ethylene || HOMO || [[Image:HOMO1_ethenedh08.jpg|100x100px]] || [[Image:HOMO2_ethenedh08.jpg|100x100px]]|| Symmetric || -0.387
|}

 
The symmetries of the LUMO and HOMO orbitals for cis-butadiene are opposite of those for the ethylene. This therefore means that the only interactions that can take place are those between ethylene's LUMO and cis-butadiene's HOMO and in addition to this the interaction between cis-butadiene's LUMO and ethylene's HOMO. This is because orbital symmetry has to be conserved (the conservation of orbital symmetry states that the symmetry of an new orbital formed by the overlap of the 2 original orbitals must be the same as the symmetry of the initial orbitals).

==='''cis-butadiene and ethylene: transition state'''===

A <jmol>
<jmolAppletButton>
<uploadedFileContents>Bicyclic_startingdh08.mol</uploadedFileContents>
<text>bicyclo system</text>
<color>white</color>
</jmolAppletButton>
</jmol> was drawn and one of the -CH2-CH2- fragments were removed. Three C=C bonds were added and the previous one was taken away so a <jmol>
<jmolAppletButton>
<uploadedFileContents>Frozen_coorddh08.mol</uploadedFileContents>
<text>guess transition state</text>
<color>white</color>
</jmolAppletButton>
</jmol> was produced.

After this an optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while bonds between the termini of the fragments were frozen at 2.1Å. Following this the resulting file had two separate optimisation and frequency calculations run on it using the TS (Berny) with one using the semi-empirical/AM1 method and the other using the DFT/B3LYP/6-31G* level of theory. Both of these optimisations had the bonds unfrozen and the "derivative" option was chosen instead.

The FT/B3LYP/6-31G* level of theory was used since it gave more accurate results in the calculations before and the energies it produces are consistent with the rest of the report (semi-empirical/AM1 method gives very different values).

The QST2 optimisation was not used since it required numbering the atoms in both the reactants and products, which took a lot of time to do.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11: Comparison of two methods of optimsations for the transition state '''
| || '''semi-empirical/AM1 method''' || '''DFT/B3LYP/6-31G* method'''
|-
| Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>SE_AM1_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>DFT_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Energy (a. u) || 0.11165608 || -234.54389573
|-
| Bond lengths || [[Image:DistancesSEdh08.jpg|150x150px]]|| [[Image:DistancesDFTdh08.jpg|150x150px]]
|-
|Imaginary frequency (cm-1) || -957 || -525
|-
|Animation || [[Image:-957dh08.gif]] || [[Image:-525dh08.gif]]
|-
| Lowest real frequency (cm-1) || 148 || 136
|-
|Animation || [[Image: 148dh08.gif]] || [[Image: 136dh08.gif]]

|}

The C-C bond forming length between the termini carbons is 0.15 Å larger for the DFT/B3LYP/6-31G* level of theory and this is because this method gives carries out the calculation to a greater degree of accuracy. This larger bond length leads a lower potential energy gradient to the transition state, which gives a lower total energy for the transition state. Therefore the given values below will be the DFT/B3LYP/6-31G* and not the semi-empirical/AM1 method.

A typical sp3 C-C bond length is 1.54 Å and a typical sp2 C=C bond length is 1.34 Å<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref>. Within the optimised transition molecule above the C-C bond lengths are 1.40 Å and the C=C bond length is 1.38 Å. This suggests that the π-bonds are breaking in order for the C-C bonds to form and additionally to this a π-bond is forming in between the two sp3 carbons.

The Van der Waals radius of a C-atom is 1.72 Å<ref name="ref 3">M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer, D. G. Truhlar, 1995, 346, pp. 63-69.[http://pubs.acs.org/doi/abs/10.1021/jp8111556]</ref>. The partly formed C-C length is 2.27 Å, which is shorter than two times the the van der Waals radius of a carbon atom. Therefore the carbon atoms are within a close enough distance to each other to share electron density and yet the bond has not formed at this point.

From looking at the animations for the imaginary frequency above it can be seen that the formation of the two bonds between the termini of the reactants is synchronous. This is expected of the Diels-Alder reaction since the mechanism is concerted. However the lowest real frequency is asynchronous with respect to the two bonds forming between the termini.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12: Frontier molecular orbitals of the transition state '''
| '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| LUMO || [[Image:LUMO1TSdh08.jpg|100x100px]] || [[Image:LUMO2TSdh08.jpg|100x100px]]|| Symmetric || +0.023
|-
| HOMO || [[Image:HOMO1TSdh08.jpg|100x100px]] || [[Image:HOMO2TSdh08.jpg|100x100px]]|| Anti-symmetric || -0.323
|}

From examining the images from table 12 above it can be deduced that the LUMO of the transition state originates from the cis-butadiene's LUMO and the ethylene's HOMO, which represents the σ*-antibonding orbitals. However the HOMO of the transition state originates from the cis-butadiene's HOMO and the ethylene's LUMO, which represents the σ-bonding orbitals within the two single C-C bonds that are formed within the reaction.

==='''Cyclohexa-1,3-diene reaction with maleic anhydride'''===

The <jmol>
<jmolAppletButton>
<uploadedFileContents>FrozenTS22dh08.mol</uploadedFileContents>
<text>exo</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_frozenTS22dh08.mol</uploadedFileContents>
<text>endo</text>
<color>white</color>
</jmolAppletButton>
</jmol> transition states were drawn on Gaussview and then the optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while the distance of the forming bonds were frozen at 2.1Å between the respective carbons. After this the structures changed slightly (<jmol>
<jmolAppletButton>
<uploadedFileContents>TS2_afteropt1_exo1dh08.mol</uploadedFileContents>
<text>exo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_TS2_afteropt1dh08.mol</uploadedFileContents>
<text>endo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol>) and the resulting files had an optimisation and frequency calculation run on it using the TA (Berny) with using the semi-empirical/AM1 level of theory while the unfrozen option as changed to "derivative". After this the endo and exo optimised transition state structures were re-optimised as they were before but using the DFT/B3LYP/6-31G* level of theory instead with the additional keyword "Opt=NoEigen".

{| class="wikitable" border="1" align="center"
|+ '''Table 13: Comparison of the exo and endo transition states '''
! rowspan="2"|-!! colspan="2"|''exo'' !! colspan="2"|''endo''
|-
| Semi-empirical/AM1 || DFT/B3LYP/6-31G* || Semi-empirical/AM1 || DFT/B3LYP/6-31G*
|-
| align="center"|Structure ||align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>After_opt2_exodh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>Endo_TS2_afteropt2dh081.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| align="center"| Energy (a. u) ||align="center"| -0.05041975 ||align="center"| -612.67931095 ||align="center"| -0.05150470 ||align="center"| -612.68339684
|-
| C-C bond forming length (Å) ||align="center"| 2.17 ||align="center"| 2.29 ||align="center"| 2.16 ||align="center"| 2.27
|-align="center"
| Imaginary frequency (cm-1) ||align="center"| -812 ||align="center"| -448 ||align="center"| -806 ||align="center"| -447
|-align="center"
| Animation ||colspan="2"|[[Image:-812_SEdh08.gif]] || colspan="2"|[[Image:-806_cm-1dh08.gif]]
|}

Again the C-C bond forming length between the termini carbons is larger (by 0.11 and 0.12 Å) for the DFT/B3LYP/6-31G* level of theory and this is due the same reasons talked about above. However the C-C bond forming length between the termini carbons are very similar in both the endo and exo transition states (0.02 Å difference).

From looking at the animations for the imaginary frequency above it can also be seen that the formation of the two bonds between the carbons in this case as well is synchronous. Again this is expected of the Diels-Alder reaction since it is concerted.

{| align=center
|+ '''Strain diagrams for the endo and exo transition states'''
| [[Image:Exostraindh08.jpg|thumb|200x200px|'''Exo TS''']]
| [[Image:Strainendodh08.jpg|thumb|200x200px|'''Endo TS''']]
|}

From studying the energies in table 13 it can be seen that the endo transition state is more stable than the exo by 4.08589 x 10-3 a. u (10.73 KJ mol-1). However by looking at the strain of both transition states one would expect the exo TS to be more stable. This is because the distance between the cyclohexa-1,3-diene ring and the -(C=O)-O-(C=O)- maleic anhydride fragment is 2.99 Å in the endo TS and 3.03 Å in the exo TS. Thus the endo form would have more strain due to steric hindrance.

However within the exo TS the vicinal hydrogens on the CH2-CH2 (carbon numbers 18 and 21) fragment on the cyclohexa-1,3-diene ring are found below the plane and next to the maleic anhydride. This leads to steric hindrance is this is where the strain originates from within the exo TS. Therefore this disfavours the exo form, which makes the endo more stable.

Another factor making the endo TS more favourable is secondary orbital overlap. These orbitals contribute to the bonds forming and they interact with each other through space. Within the endo TS below there are 2 more orbitals interacting (secondary orbitals) than within the exo TS. This therefore makes the endo TS more stable and therefore more favoured.

[[Image:Secondary_orbital_shizzledh08.jpg|500x500px|center]]

{| class="wikitable" border="1" align="center"
|+ '''Table 14: Frontier molecular orbitals of the the transition state for the cyclohexa-1,3-diene reaction '''
! rowspan="2"|Molecular Orbital !! colspan="4"|''exo'' !! colspan="4"|''endo''
|- align="center"
| '''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''||'''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''
|- align="center"
| LUMO || [[Image:LUMO1exoh081.jpg|150x150px]]|| [[Image:LUMO2exodh081.jpg|150x150px]]|| Anti-symmetric || -0.078 || [[Image:LUMO1ENDOdh08.jpg|150x150px]]||[[Image:LUMO2ENDOdh08.jpg|150x150px]] || Anti-symmetric || -0.067
|- align="center"
| HOMO || [[Image:HOMO1exodh081.jpg|150x150px]]||[[Image:HOMO2exodh081.jpg|150x150px]] || Anti-symmetric|| -0.242 || [[Image:HOMO1ENDOdh08.jpg|150x150px]]|| [[Image:HOMO2ENDOdh08.jpg|150x150px]]||Anti-symmetric || -0.242
|}

Digital repositories:

exo: http://hdl.handle.net/10042/to-5596

endo: http://hdl.handle.net/10042/to-5597

From examining table 14 it is clear to see that both the HOMO and LUMO are anti-symmetric for both the transition states. The LUMO of the maleic anhydride fragment is the π* anti-bonding orbital of the C=C bond. Within the reaction then this has electrons transferred to it and the C=C bond breaks, which leaves just a C-C bond. The LUMOs for both transition states are both anti-bonding with respect to the two fragments. The HOMOs of both transitions state represent the σ-bonding that will be formed between the termini carbons on the products and reactants. This is most likely from the overlap of the cyclohexa-1,3-diene's HOMO and the maleic anhydride's LUMO.

=='''Conclusion'''==

='''References'''=
<references/>

Rep:Mod:hampluscheese

2010-11-10T22:43:50Z

Dh08: /* '''Reagents''' */

='''Module 3:Physical Computational Chemistry Lab'''=

Douglas Hunt

=='''Objectives'''==

The aim of this experiment is to

=='''The Cope Rearrangement'''==

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangement] is a [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement |3,3|-sigmatropic rearrangement] that takes place through a single [http://en.wikipedia.org/wiki/Pericyclic pericyclic], concerted [http://en.wikipedia.org/wiki/Transition_state transition state]. Within this rearrangement a σ bond is formed in concert with another being broken within a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] π system.

The Cope rearrangement we will be studying will be that of 1,5-hexadiene, which is shown below.

[[Image:Cope_rearrangement_schemedh08.jpg|300x300px|center]]

The objective is to locate the low-energy minima and transition structures on the C6H10 potential energy surface and to determine the preferred reaction mechanism by using [http://en.wikipedia.org/wiki/GAUSSIAN Gaussian], with the [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)] methods.

==='''Optimisation of 1,5-hexadiene Conformers'''===

Rough <jmol>
<jmolAppletButton>
<uploadedFileContents>Anti_2_drawndh08.mol</uploadedFileContents>
<text>anti</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche3_drawndh08.mol</uploadedFileContents>
<text>gauche</text>
<color>white</color>
</jmolAppletButton>
</jmol> conformers of 1,5-hexadiene were drawn on Gaussview and their structures were tidied by the clean function. They were then further optimised by the HF/3-21G method and were identified which conformer they were from their point group and appendix 1 on the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3 physical lab page].

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 1: Optimisation and comparison of anti and gauche 1,5-hexadiene conformers using HF/3-21G methods
| ''' Conformer''' || '''Optimised Structure''' || '''Energy (Atomic Units)''' || '''[http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1 Energies (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (Anti2 - Gauche3)''' || '''Point group'''
|-
| '''Anti2''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Anti_2_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692535 || -231.69254 ||-6.08 x 105 || 1.2606 x 10-4 a. u|| Ci
|-
| '''Gauche3''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Gauche3_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692661 || -231.69266 || -6.08 x 105 || 3.31 x 10-1 KJ mol-1 || C1
|}

From studying table 1 it can be seen that the gauche conformer is 0.33 KJ mol-1 more stable than the anti conformation. One would assume that the anti2 conformer would be more stable due to [http://en.wikipedia.org/wiki/Steric steric] factors/repulsion within the gauche3 conformation. However within the gauche3 conformer there is a favourable donation of [http://en.wikipedia.org/wiki/Electron_density electron density] from the C=C π orbital to the C-H σ* orbital from the adjacent proton, which stabilises the conformer<ref name="ref 1">B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446.[http://www.informaworld.com/smpp/content~db=all?content=10.1080/00268970110081412]</ref>.

 

The anti2 conformer was then optimised using the B3LYP 6-31G* (6-31G (d)) method.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 2: Comparison of the anti2 conformer optimisations carried out at different calculation methods
| ''' Method''' || '''Structure and bond distances (Å)''' || '''Dihedral angle (C1 to C4)''' || '''Energy (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (HF/3-21G - DFT/B3LYP/6-31G)'''
|-
| HF/3-21G || [[Image:Anti2_opt1dh08.jpg|300x300px]] || 114.6 o || -231.692535 || -6.08 x 105 || 2.919171 a. u
|-
| DFT/B3LYP/6-31G* || [[Image:Anti2_opt2DH08.jpg|300x300px]] || 118.8 o || -234.611716 || -6.16 x 105 || 7.66 x 103 KJ mol-1
|}

From examining table 2 at first it is possible to see that the bond lengths are very similar to each other and that of the literature bond lengths<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref> (C1=C2: 1.340 Å, C2-C3: 1.508 Å and C3-C4: 1.538 Å). It is also possible to see that the C=C bond length is more accurate for the DFT/B3LYP/6-31G* method by it being 0.01 Å closer to the literature value. Another main difference in terms of geometry is that the DFT/B3LYP/6-31G* method gives a larger [http://en.wikipedia.org/wiki/Dihedral_angle dihedral angle] (and most likely more accurate) of 118.8o compared to that of 114.6o.

In terms of energy the DFT/B3LYP/6-31G* method produces a conformer of a higher stability by it possessing a lower energy with a difference of 7.66 x 103 KJ mol-1.

==='''Frequency analysis of 1,5-hexadiene Conformers'''===

[[Image:Vibrations_anti2dh08.jpg|right|300x300px]]

For the frequency analysis the DFT/B3LYP/6-31G* method was used.

Frequency analysis is essential within this calculation due to it being the second derivative of the potential energy surface. This means that if all the frequencies are all positive the system is at a minimum, if one of them is negative the optimised system is at a transition state, and if more than one are negative then the calculation has failed to find a critical point, which means the optimisation has failed. This therefore gives us a method of seeing whether the calculation has worked.

From looking to the image on the right it can deduced that the optimisation has taken place successfully by there being no negative vibrations and therefore the conformation is at a minimum.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 3: Energies under the thermochemistry section at 298.15 K and 0 K within the frequency logfiles'''
| '''Energies''' || '''298.15 K (atomic units)''' ||| '''0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (KJ mol-1)''' || '''Discussion'''
|-
| Sum of electronic and zero-point energies || -234.469177 || -234.466673 || -2.504 x 10-3 || -6.57 || This is the potential energy at 0 K, which includes the zero-point vibrational energy (E = Eelec + ZPE).

|-
| Sum of electronic and thermal energies || -234.461835 || -234.461085 || -7.5 x 10-4|| -1.97 || This is the energy at 298.15 K (0 K for 0 K calculation) and 1 atm of pressure, which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans).

|-
| Sum of electronic and thermal enthalpies || -234.460891 || -234.460141 || -7.5 x 10-4 || -1.97 || This contains an additional correction for RT (H = E + RT), which is especially important when looking at dissociation reactions.
|-
| Sum of electronic and thermal free energies || -234.500739 || -234.495931 || 4.808 x 10-3 || -12.62 || This includes the entropic contribution to the free energy (G = H - TS).
|}

{|align=right
|+
| [[Image:Anti2_freq_IRdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 298.15 K''']]
| [[Image:Anti2_freq_0Kdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 0 K''']]
|}

From looking at table 3 above it can be deduced that the anti2 conformer is more stable at room temperature to that of 0 K with respect to all the types of energies. The spectra also seem to not change much between the two temperatures but the main change is the change in the sum of electronic and thermal free energies, which is heavily dependent of temperature by it including the entropic contribution to the free energy (G = H - '''T'''S)

 
 
 
 

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

Within the Cope Rearrangement there are two possible transition states, the chair (C2h) and the boat (C2v), which are shown within [https://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_2 appendix two ].

==='''Comparison of optimisation methods for the chair transition state'''===

An <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2hdh08.mol</uploadedFileContents>
<text>allyl fragment</text>
<color>white</color>
</jmolAppletButton>
</jmol>, CH2CHCH2 was drawn on Gaussview and was optimized through the HF/3-21G method. After this the original allyl fragment was copied and pasted into a new window and place exactly above the first fragment with a bond distance of 2.20 Å between the terminal end carbons. This therefore produced the <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2h_fulldh08.mol</uploadedFileContents>
<text>chair transition structure.</text>
<color>white</color>
</jmolAppletButton>
</jmol>

Gaussian was then set up for an optimisation for the transition state with the additional key words "Opt=NoEigen", the optimisation set to TS (Berny), the force constants set to "once" and using the HF/3-21G method.

After this the "chair" transition state was optimised an alternative way by freezing the coordinates of the two sets of terminal carbons (where the bonds form/break during the rearrangement) within the Redundant Coord Editor on Gauusview.
Following this a second calculation was run where the freeze coordinates option was changed to "Derivative" and the force constants were set to "never".

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 4: Comparison of chair transition state optimisation methods
| || '''Optimised through a TS (Berny)''' || '''Optimised through a frozen coordinate method'''
|-
| '''Optimised TS Structure''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>C2h_optfreq_TS_Bdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| '''Terminal C-C bond length (Å)''' || 2.02 || 2.02
|-
|'''Energy (atomic units)''' || -231.61932247 || -231.61932190
|-
| '''ΔE, TS (Berny) - frozen coordinate''' || 5.7 x 10-7 a.u || 1.50 x 10-3 KJ mol-1
|-
| '''Imaginary frequency (cm-1)''' || -818 || -818
|-
| '''Animation''' ||[[Image:818dh08TSB.gif|350x350px]] || [[Image:818cm-1Fdh08.gif]]
|}

The data in table 4 shows that both methods give the same results and they both give the same transition state. They both produced the same energy with a negligible energy difference between them and the bond lengths between the terminal carbons are both 2.02 Å. In addition to this the imaginary frequencies were both -818cm-1 and within the animations above it is possible to see that when one C-C bond is forming the other is breaking. This means it is a asynchronous bond formation, which is expected for the Cope Rearrangement since it is a pericyclic reaction.

The only difference between the two optimisations is the way in which the transition states were calculated. This is due to the TS (Berny) method calculating the force constant matrix within the first optimisation step and is then being recalculated as the optimisation carries on. When the frozen coordinate method freezes the reaction coordinate in order for the rest of the structure to be optimised around it, which does not require the whole hessian (force constant matrix) to be calculated but only that the reaction coordinate is differentiated (second step). The main advantage of this, is that you do not need to calculate the entire force constant matrix and just need to differentiate the reaction coordinate in order to obtain a good enough guess for the initial force constant matrix. The frozen coordinate method will therefore take less time to compute the transition state, which can be very expensive within certain systems. However to carry out this method one needs to have an idea of how far apart the terminal carbons are.

==='''Boat transition state'''===

===='''QST2 optimisation'''====

The optimisation for the boat transition state had a different method and this was the QST2 method. This calculation interpolates between two structures (product and reactant) and tries to find a transition state between them.

The reactants and products were numbered in the same way. Therefore both 1,5-hexadiene molecules had their numbering manually changed for the product molecule so it corresponded to the numbering of the reactant if it rearranged. This is illustrated in the reaction scheme below.

[[Image:Boat_rerrangmentdh08.jpg|300x300px|center]]

The chk file for the optimized anti2 conformer was opened and copied into a new window. A second window was opened within the new window and the same anti2 conformer was copied into the new window. The numbering was changed within the second molecule so the two looked like they do in the image below.

[[Image:TS_(QST2)_numbering_1dh08.jpg|400x400px|center]]

The job type was set as "Opt+Freq" and "TS (QST2)" was chosen to optimise the transition state. However this failed and produced
a structure a bit like a <jmol>
<jmolAppletButton>
<uploadedFileContents>Trialone_BPdh08.mol</uploadedFileContents>
<text>chair transition structure </text>
<color>white</color>
</jmolAppletButton>
</jmol> but more dissociated. This is because it interpolated between the two structures and translated the top allyl fragment and therefore did not consider the possibility of a rotation around the central bonds.

To get around this the C-C-C-C dihedral angle (C2-C3-C4-C5) was set to 0 o and the C-C-C angle (C2-C3-C4 and C3-C4-C5 for the product molecule) was set to 100 o for both molecules. This gave rise to the two structures below.

[[Image:TS_(QST2)_numbering_2dh08.jpg|400x400px|center]]

After this the QST2 calculation was run again and the results in table 5 were produced. The imaginary frequency at -839 cm-1 shows that while one bond is forming the other one is breaking, which is again an asynchronous bond formation.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5: Data for the boat transition state from the QST2 optimisation method
| Optimised TS Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.14
|-
|Energy (atomic units) || -231.60280151
|-
| Energy (KJ mol-1) || -6.08 x 105
|-
| Imaginary frequency (cm-1) || -839
|-
| Animation ||[[Image:-839dh08.gif]]
|}

===='''Intrinsic Reaction Coordinate'''====

It is not possible to predict which product the chair and boat transition state will lead to. To solve this problem within Gaussian one can use the IRC method, which allows one to follow the minimum energy path from a transition structure down to its local minimum on a potential energy surface. This creates a succession of points by taking small geometry steps in the direction to where the gradient of the energy surface is at its steepest until the minimum in energy is reached.

The chk file for the optimized chair transition structure was opened and the job type selected was "IRC", the reaction coordinate was computed in the forward direction (since molecule is symmetric) with 50 points along the IRC. This was done with two type of force constants being calculated, once and always.

Within the "once" method the output showed no bond formation process by the 26th step and therefore the calculation was incomplete, since a complete 1,5-hexadiene structure was supposed to be formed. From looking at the IRC pathway it can be seen that the minimum on the potential energy surface was not reached at that point, since the points on the pathway did not form an asymptote and the gradient did not go close enough to zero.

For the "always" method the computation of force constants were carried out at every step. This therefore lets the IRC calculation distinguish whether it is moving along the minimum energy pathway while going down the potential energy surface. While this is happening it can make changes to the pathway if necessary. Within the "always" case the bond forming arrives at 1,5-hexadiene and this is backed up by the IRC pathway reaching a minimum within its energy by the gradient going close to zero and the pathway reaches an asymptote.

The structure produced through the always method had an energy of -231.69199434 atomic units, which is very close to the energy of the gauche2 conformer in the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 appendix 1]. This suggests that the reaction takes place through the chair transition state, which later produces the gauche2 conformer of 1,5-hexadiene.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 6: Comparison of IRC calculations for the chair structure
| || '''Calculate force constants once''' || '''Calculate force constants always'''
|-
| '''Number of points along IRC''' || 26 || 47
|-
| '''Structure at the last point of IRC''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Once_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Always_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|'''Energy (atomic units)''' || -231.68906923 || -231.69199434
|-
| '''ΔE, once - always''' || 2.92511 x 10-3 a.u || 7.68 KJ mol-1
|-
| '''IRC pathway''' || [[Image:One_pathwaydh08.jpg|400x400px]] || [[Image:Always_pathwaydh08.jpg|400x400px]]
|-
| '''IRC gradient''' || [[Image:One_gradientdh08.jpg|400x400px]]|| [[Image:Always_gradientdh08.jpg|400x400px]]
|}

 
 
 

===='''Comparison of activation energies for the transition states'''====

The HF/3-21G optimised chair and boat transition structures were both reoptimised using the B3LYP/6-31G* (B3LYP/6-31G (d)) method. This gives a more accurate minimisation however this method takes longer to run. The values for the zero-point energies and thermal energies were found within the thermochemistry section of the log file.

The values for the HF/3-21G reactant (anti2) optimisation were calculated and the B3LYP/6-31G* values were taken from the "Frequency analysis of 1,5-hexadiene Conformers" earlier in this module.

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 7: Energy data for chair, boat and the anti2 conformation from the HF/3-21G and B3LYP/6-31G* methods of optimisation '''
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466677
| align="center" | -231.461320
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450936
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402342
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539524
| align="center" | -231.532562
| align="center" | -234.611716
| align="center" | -234.469177
| align="center" | -234.461835
|}
Digital repositories

Chair:
HF/3-21G: http://hdl.handle.net/10042/to-5569
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5568

Boat:
HF/3-21G: http://hdl.handle.net/10042/to-5570
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5571

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 8: Summary of activation energies for the chair and boat conformations (ΔE = chair TS or boat TS minus anti2)'''
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| align="center" | 1 a.u = 627.509 kcal mol-1
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE, chair (a.u)'''
| align="center" | 0.072847
| align="center" | 0.071242
| align="center" | 0.054248
| align="center" | 0.052827
| align="center" | -
|-
| '''ΔE, chair (kcal mol-1)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE, boat (a. u)'''
| align="center" | 0.088588
| align="center" | 0.087259
| align="center" | 0.066835
| align="center" | 0.065827
| align="center" | -
|-
| '''ΔE, boat (kcal mol-1)'''
| align="center" | 55.59
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.31
| align="center" | 44.7 ± 2.0
|}

By examining table 8 it is clear to see that the B3LYP/6-31G* method gives values which are much closer to the experimental ones. This is due to the B3LYP/6-31G* giving a higher level of optimisation than the HF/3-21G method. Table 8 also shows that the chair transition state has a lower activation energy to that of the boat transition state by 8.16 kcal mol-1. This suggests that the Cope rearrangement mechanism would most likely proceed through the chair transition state to that of the boat. Therefore the reaction would take place from the anti2 conformer to the chair transition state and then onto the gauche2 conformer within the Cope rearrangement.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9: Geometric parameters for the optimised chair and boat structures '''
| || ''' HF/3-21G chair TS''' || ''' B3LYP/6-31G* chair TS''' || '''HF/3-21G boat TS''' || '''B3LYP/6-31G* boat TS'''
|-
| Optimised Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Chair_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Boat_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.02 (C5-C10 or C3-C14)|| 1.97 (C5-C10 or C3-C14) || 2.14 (C3-C4 or C1-C6) || 2.21 (C3-C4 or C1-C6)
|-
| Fragment angle || 120 o (C3-C1-C5) || 120 o (C3-C1-C5) || 122 o (C4-C5-C6) || 122 o (C4-C5-C6)
|-
| Dihedral angle || 55 o (C1-C3-C14-C11)|| 54 o (C1-C3-C14-C11)|| 0 o (C2-C1-C6-C5) || 0 o (C2-C1-C6-C5)
|-
| Numbering of carbons || [[Image:Numbering_chairdh08.jpg|200x200px]]|| [[Image:Numbering_chairdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]]
|}

From studying table 9 it can be seen that the C-C bond breaking/forming length has changed between the two different types of optimisation methods. However the angles between the bonds remains unchanged.

=='''The Diels Alder Cycloaddition'''==

The Diels-Alder reaction is a cycloaddition between a conjugated diene and a dienophile (alkene system), which forms a cyclohexene system. It is often referred to as a [4s+2s] cycloaddition and proceeds through a single cyclic, concerted transition state where 2 σ bonds are formed in between the termini carbons of the 2 conjugated п systems. This is shown in the reaction scheme below where ethene is reacting with cis-butadiene.

[[Image:Reaction_scheme2dh08.jpg|300x300px|center]]

The π orbitals of the dienophile form new σ bonds with the π orbitals of the diene. The number of π electrons involved determine whether the reaction proceeds in a concerted stereospecific fashion (allowed) or not (forbidden). The HOMO/LUMO of one fragment interacts with the HOMO/LUMO of the other fragment to form two new bonding and anti-bonding MOs. The nodal properties allow one to predict how the reaction will occur.

Semi-Empirical/AM1 method and the B3LYP/6-31G* method of optimisation will be used on Gaussian to determine particular properites of the reactants to predict the way in which the reaction will proceed.

==='''cis-butadiene and ethylene: reagents'''===

<jmol>
<jmolAppletButton>
<uploadedFileContents>Cis_butadienedh08.mol</uploadedFileContents>
<text>Cis-butadiene</text>
<color>whi-te</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Ethenedh08.mol</uploadedFileContents>
<text>ethylene</text>
<color>white</color>
</jmolAppletButton>
</jmol> were optimised separately using the AM1 semi-empirical method on Gaussview.The cis-butadiene had an energy of 0.04879725 a. u (1.28 x 102 KJ mol-1) and ethylene had an energy of 0.02619027 a. u (68.76 KJ mol-1).

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10: Frontier molecular orbitals of cis-butadiene and ethylene '''
| '''Reagent''' || '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| cis-butadiene || LUMO || [[Image:LUMO1_cis_butadieneDH08.jpg|100x100px]] || [[Image:LUMO2_cis_butadienedh08.jpg|100x100px]]|| Symmetric || +0.017
|-
| cis-butadiene || HOMO || [[Image:HOMO1_cis_butadienedh08.jpg|100x100px]] || [[Image:HOMO2_cis_butadienedh08.jpg|100x100px]]|| Anti-symmetric || -0.343
|-
| ethylene || LUMO || [[Image:LUMO1_ethenedh08.jpg|100x100px]] || [[Image:LUMO2_ethenedh08.jpg|100x100px]]|| Anti-symmetric || +0.052
|-
| ethylene || HOMO || [[Image:HOMO1_ethenedh08.jpg|100x100px]] || [[Image:HOMO2_ethenedh08.jpg|100x100px]]|| Symmetric || -0.387
|}

 
The symmetries of the LUMO and HOMO orbitals for cis-butadiene are opposite of those for the ethylene. This therefore means that the only interactions that can take place are those between ethylene's LUMO and cis-butadiene's HOMO and in addition to this the interaction between cis-butadiene's LUMO and ethylene's HOMO. This is because orbital symmetry has to be conserved (the conservation of orbital symmetry states that the symmetry of an new orbital formed by the overlap of the 2 original orbitals must be the same as the symmetry of the initial orbitals).

==='''Transition State'''===

A <jmol>
<jmolAppletButton>
<uploadedFileContents>Bicyclic_startingdh08.mol</uploadedFileContents>
<text>bicyclo system</text>
<color>white</color>
</jmolAppletButton>
</jmol> was drawn and one of the -CH2-CH2- fragments were removed. Three C=C bonds were added and the previous one was taken away so a <jmol>
<jmolAppletButton>
<uploadedFileContents>Frozen_coorddh08.mol</uploadedFileContents>
<text>guess transition state</text>
<color>white</color>
</jmolAppletButton>
</jmol> was produced.

After this an optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while bonds between the termini of the fragments were frozen at 2.1Å. Following this the resulting file had two separate optimisation and frequency calculations run on it using the TS (Berny) with one using the semi-empirical/AM1 method and the other using the DFT/B3LYP/6-31G* level of theory. Both of these optimisations had the bonds unfrozen and the "derivative" option was chosen instead.

The FT/B3LYP/6-31G* level of theory was used since it gave more accurate results in the calculations before and the energies it produces are consistent with the rest of the report (semi-empirical/AM1 method gives very different values).

The QST2 optimisation was not used since it required numbering the atoms in both the reactants and products, which took a lot of time to do.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11: Comparison of two methods of optimsations for the transition state '''
| || '''semi-empirical/AM1 method''' || '''DFT/B3LYP/6-31G* method'''
|-
| Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>SE_AM1_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>DFT_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Energy (a. u) || 0.11165608 || -234.54389573
|-
| Bond lengths || [[Image:DistancesSEdh08.jpg|150x150px]]|| [[Image:DistancesDFTdh08.jpg|150x150px]]
|-
|Imaginary frequency (cm-1) || -957 || -525
|-
|Animation || [[Image:-957dh08.gif]] || [[Image:-525dh08.gif]]
|-
| Lowest real frequency (cm-1) || 148 || 136
|-
|Animation || [[Image: 148dh08.gif]] || [[Image: 136dh08.gif]]

|}

The C-C bond forming length between the termini carbons is 0.15 Å larger for the DFT/B3LYP/6-31G* level of theory and this is because this method gives carries out the calculation to a greater degree of accuracy. This larger bond length leads a lower potential energy gradient to the transition state, which gives a lower total energy for the transition state. Therefore the given values below will be the DFT/B3LYP/6-31G* and not the semi-empirical/AM1 method.

A typical sp3 C-C bond length is 1.54 Å and a typical sp2 C=C bond length is 1.34 Å<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref>. Within the optimised transition molecule above the C-C bond lengths are 1.40 Å and the C=C bond length is 1.38 Å. This suggests that the π-bonds are breaking in order for the C-C bonds to form and additionally to this a π-bond is forming in between the two sp3 carbons.

The Van der Waals radius of a C-atom is 1.72 Å<ref name="ref 3">M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer, D. G. Truhlar, 1995, 346, pp. 63-69.[http://pubs.acs.org/doi/abs/10.1021/jp8111556]</ref>. The partly formed C-C length is 2.27 Å, which is shorter than two times the the van der Waals radius of a carbon atom. Therefore the carbon atoms are within a close enough distance to each other to share electron density and yet the bond has not formed at this point.

From looking at the animations for the imaginary frequency above it can be seen that the formation of the two bonds between the termini of the reactants is synchronous. This is expected of the Diels-Alder reaction since the mechanism is concerted. However the lowest real frequency is asynchronous with respect to the two bonds forming between the termini.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12: Frontier molecular orbitals of the transition state '''
| '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| LUMO || [[Image:LUMO1TSdh08.jpg|100x100px]] || [[Image:LUMO2TSdh08.jpg|100x100px]]|| Symmetric || +0.023
|-
| HOMO || [[Image:HOMO1TSdh08.jpg|100x100px]] || [[Image:HOMO2TSdh08.jpg|100x100px]]|| Anti-symmetric || -0.323
|}

From examining the images from table 12 above it can be deduced that the LUMO of the transition state originates from the cis-butadiene's LUMO and the ethylene's HOMO, which represents the σ*-antibonding orbitals. However the HOMO of the transition state originates from the cis-butadiene's HOMO and the ethylene's LUMO, which represents the σ-bonding orbitals within the two single C-C bonds that are formed within the reaction.

==='''Cyclohexa-1,3-diene reaction with maleic anhydride'''===

The <jmol>
<jmolAppletButton>
<uploadedFileContents>FrozenTS22dh08.mol</uploadedFileContents>
<text>exo</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_frozenTS22dh08.mol</uploadedFileContents>
<text>endo</text>
<color>white</color>
</jmolAppletButton>
</jmol> transition states were drawn on Gaussview and then the optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while the distance of the forming bonds were frozen at 2.1Å between the respective carbons. After this the structures changed slightly (<jmol>
<jmolAppletButton>
<uploadedFileContents>TS2_afteropt1_exo1dh08.mol</uploadedFileContents>
<text>exo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_TS2_afteropt1dh08.mol</uploadedFileContents>
<text>endo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol>) and the resulting files had an optimisation and frequency calculation run on it using the TA (Berny) with using the semi-empirical/AM1 level of theory while the unfrozen option as changed to "derivative". After this the endo and exo optimised transition state structures were re-optimised as they were before but using the DFT/B3LYP/6-31G* level of theory instead with the additional keyword "Opt=NoEigen".

{| class="wikitable" border="1" align="center"
|+ '''Table 13: Comparison of the exo and endo transition states '''
! rowspan="2"|-!! colspan="2"|''exo'' !! colspan="2"|''endo''
|-
| Semi-empirical/AM1 || DFT/B3LYP/6-31G* || Semi-empirical/AM1 || DFT/B3LYP/6-31G*
|-
| align="center"|Structure ||align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>After_opt2_exodh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>Endo_TS2_afteropt2dh081.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| align="center"| Energy (a. u) ||align="center"| -0.05041975 ||align="center"| -612.67931095 ||align="center"| -0.05150470 ||align="center"| -612.68339684
|-
| C-C bond forming length (Å) ||align="center"| 2.17 ||align="center"| 2.29 ||align="center"| 2.16 ||align="center"| 2.27
|-align="center"
| Imaginary frequency (cm-1) ||align="center"| -812 ||align="center"| -448 ||align="center"| -806 ||align="center"| -447
|-align="center"
| Animation ||colspan="2"|[[Image:-812_SEdh08.gif]] || colspan="2"|[[Image:-806_cm-1dh08.gif]]
|}

Again the C-C bond forming length between the termini carbons is larger (by 0.11 and 0.12 Å) for the DFT/B3LYP/6-31G* level of theory and this is due the same reasons talked about above. However the C-C bond forming length between the termini carbons are very similar in both the endo and exo transition states (0.02 Å difference).

From looking at the animations for the imaginary frequency above it can also be seen that the formation of the two bonds between the carbons in this case as well is synchronous. Again this is expected of the Diels-Alder reaction since it is concerted.

{| align=center
|+ '''Strain diagrams for the endo and exo transition states'''
| [[Image:Exostraindh08.jpg|thumb|200x200px|'''Exo TS''']]
| [[Image:Strainendodh08.jpg|thumb|200x200px|'''Endo TS''']]
|}

From studying the energies in table 13 it can be seen that the endo transition state is more stable than the exo by 4.08589 x 10-3 a. u (10.73 KJ mol-1). However by looking at the strain of both transition states one would expect the exo TS to be more stable. This is because the distance between the cyclohexa-1,3-diene ring and the -(C=O)-O-(C=O)- maleic anhydride fragment is 2.99 Å in the endo TS and 3.03 Å in the exo TS. Thus the endo form would have more strain due to steric hindrance.

However within the exo TS the vicinal hydrogens on the CH2-CH2 (carbon numbers 18 and 21) fragment on the cyclohexa-1,3-diene ring are found below the plane and next to the maleic anhydride. This leads to steric hindrance is this is where the strain originates from within the exo TS. Therefore this disfavours the exo form, which makes the endo more stable.

Another factor making the endo TS more favourable is secondary orbital overlap. These orbitals contribute to the bonds forming and they interact with each other through space. Within the endo TS below there are 2 more orbitals interacting (secondary orbitals) than within the exo TS. This therefore makes the endo TS more stable and therefore more favoured.

[[Image:Secondary_orbital_shizzledh08.jpg|500x500px|center]]

{| class="wikitable" border="1" align="center"
|+ '''Table 14: Frontier molecular orbitals of the the transition state for the cyclohexa-1,3-diene reaction '''
! rowspan="2"|Molecular Orbital !! colspan="4"|''exo'' !! colspan="4"|''endo''
|- align="center"
| '''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''||'''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''
|- align="center"
| LUMO || [[Image:LUMO1exoh081.jpg|150x150px]]|| [[Image:LUMO2exodh081.jpg|150x150px]]|| Anti-symmetric || -0.078 || [[Image:LUMO1ENDOdh08.jpg|150x150px]]||[[Image:LUMO2ENDOdh08.jpg|150x150px]] || Anti-symmetric || -0.067
|- align="center"
| HOMO || [[Image:HOMO1exodh081.jpg|150x150px]]||[[Image:HOMO2exodh081.jpg|150x150px]] || Anti-symmetric|| -0.242 || [[Image:HOMO1ENDOdh08.jpg|150x150px]]|| [[Image:HOMO2ENDOdh08.jpg|150x150px]]||Anti-symmetric || -0.242
|}

Digital repositories:

exo: http://hdl.handle.net/10042/to-5596

endo: http://hdl.handle.net/10042/to-5597

From examining table 14 it is clear to see that both the HOMO and LUMO are anti-symmetric for both the transition states. The LUMO of the maleic anhydride fragment is the π* anti-bonding orbital of the C=C bond. Within the reaction then this has electrons transferred to it and the C=C bond breaks, which leaves just a C-C bond. The LUMOs for both transition states are both anti-bonding with respect to the two fragments. The HOMOs of both transitions state represent the σ-bonding that will be formed between the termini carbons on the products and reactants. This is most likely from the overlap of the cyclohexa-1,3-diene's HOMO and the maleic anhydride's LUMO.

=='''Conclusion'''==

='''References'''=
<references/>

Rep:Mod:hampluscheese

2010-11-10T22:41:18Z

Dh08: /* '''Transition State''' */

='''Module 3:Physical Computational Chemistry Lab'''=

Douglas Hunt

=='''Objectives'''==

The aim of this experiment is to

=='''The Cope Rearrangement'''==

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangement] is a [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement |3,3|-sigmatropic rearrangement] that takes place through a single [http://en.wikipedia.org/wiki/Pericyclic pericyclic], concerted [http://en.wikipedia.org/wiki/Transition_state transition state]. Within this rearrangement a σ bond is formed in concert with another being broken within a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] π system.

The Cope rearrangement we will be studying will be that of 1,5-hexadiene, which is shown below.

[[Image:Cope_rearrangement_schemedh08.jpg|300x300px|center]]

The objective is to locate the low-energy minima and transition structures on the C6H10 potential energy surface and to determine the preferred reaction mechanism by using [http://en.wikipedia.org/wiki/GAUSSIAN Gaussian], with the [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)] methods.

==='''Optimisation of 1,5-hexadiene Conformers'''===

Rough <jmol>
<jmolAppletButton>
<uploadedFileContents>Anti_2_drawndh08.mol</uploadedFileContents>
<text>anti</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche3_drawndh08.mol</uploadedFileContents>
<text>gauche</text>
<color>white</color>
</jmolAppletButton>
</jmol> conformers of 1,5-hexadiene were drawn on Gaussview and their structures were tidied by the clean function. They were then further optimised by the HF/3-21G method and were identified which conformer they were from their point group and appendix 1 on the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3 physical lab page].

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 1: Optimisation and comparison of anti and gauche 1,5-hexadiene conformers using HF/3-21G methods
| ''' Conformer''' || '''Optimised Structure''' || '''Energy (Atomic Units)''' || '''[http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1 Energies (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (Anti2 - Gauche3)''' || '''Point group'''
|-
| '''Anti2''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Anti_2_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692535 || -231.69254 ||-6.08 x 105 || 1.2606 x 10-4 a. u|| Ci
|-
| '''Gauche3''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Gauche3_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692661 || -231.69266 || -6.08 x 105 || 3.31 x 10-1 KJ mol-1 || C1
|}

From studying table 1 it can be seen that the gauche conformer is 0.33 KJ mol-1 more stable than the anti conformation. One would assume that the anti2 conformer would be more stable due to [http://en.wikipedia.org/wiki/Steric steric] factors/repulsion within the gauche3 conformation. However within the gauche3 conformer there is a favourable donation of [http://en.wikipedia.org/wiki/Electron_density electron density] from the C=C π orbital to the C-H σ* orbital from the adjacent proton, which stabilises the conformer<ref name="ref 1">B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446.[http://www.informaworld.com/smpp/content~db=all?content=10.1080/00268970110081412]</ref>.

 

The anti2 conformer was then optimised using the B3LYP 6-31G* (6-31G (d)) method.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 2: Comparison of the anti2 conformer optimisations carried out at different calculation methods
| ''' Method''' || '''Structure and bond distances (Å)''' || '''Dihedral angle (C1 to C4)''' || '''Energy (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (HF/3-21G - DFT/B3LYP/6-31G)'''
|-
| HF/3-21G || [[Image:Anti2_opt1dh08.jpg|300x300px]] || 114.6 o || -231.692535 || -6.08 x 105 || 2.919171 a. u
|-
| DFT/B3LYP/6-31G* || [[Image:Anti2_opt2DH08.jpg|300x300px]] || 118.8 o || -234.611716 || -6.16 x 105 || 7.66 x 103 KJ mol-1
|}

From examining table 2 at first it is possible to see that the bond lengths are very similar to each other and that of the literature bond lengths<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref> (C1=C2: 1.340 Å, C2-C3: 1.508 Å and C3-C4: 1.538 Å). It is also possible to see that the C=C bond length is more accurate for the DFT/B3LYP/6-31G* method by it being 0.01 Å closer to the literature value. Another main difference in terms of geometry is that the DFT/B3LYP/6-31G* method gives a larger [http://en.wikipedia.org/wiki/Dihedral_angle dihedral angle] (and most likely more accurate) of 118.8o compared to that of 114.6o.

In terms of energy the DFT/B3LYP/6-31G* method produces a conformer of a higher stability by it possessing a lower energy with a difference of 7.66 x 103 KJ mol-1.

==='''Frequency analysis of 1,5-hexadiene Conformers'''===

[[Image:Vibrations_anti2dh08.jpg|right|300x300px]]

For the frequency analysis the DFT/B3LYP/6-31G* method was used.

Frequency analysis is essential within this calculation due to it being the second derivative of the potential energy surface. This means that if all the frequencies are all positive the system is at a minimum, if one of them is negative the optimised system is at a transition state, and if more than one are negative then the calculation has failed to find a critical point, which means the optimisation has failed. This therefore gives us a method of seeing whether the calculation has worked.

From looking to the image on the right it can deduced that the optimisation has taken place successfully by there being no negative vibrations and therefore the conformation is at a minimum.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 3: Energies under the thermochemistry section at 298.15 K and 0 K within the frequency logfiles'''
| '''Energies''' || '''298.15 K (atomic units)''' ||| '''0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (KJ mol-1)''' || '''Discussion'''
|-
| Sum of electronic and zero-point energies || -234.469177 || -234.466673 || -2.504 x 10-3 || -6.57 || This is the potential energy at 0 K, which includes the zero-point vibrational energy (E = Eelec + ZPE).

|-
| Sum of electronic and thermal energies || -234.461835 || -234.461085 || -7.5 x 10-4|| -1.97 || This is the energy at 298.15 K (0 K for 0 K calculation) and 1 atm of pressure, which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans).

|-
| Sum of electronic and thermal enthalpies || -234.460891 || -234.460141 || -7.5 x 10-4 || -1.97 || This contains an additional correction for RT (H = E + RT), which is especially important when looking at dissociation reactions.
|-
| Sum of electronic and thermal free energies || -234.500739 || -234.495931 || 4.808 x 10-3 || -12.62 || This includes the entropic contribution to the free energy (G = H - TS).
|}

{|align=right
|+
| [[Image:Anti2_freq_IRdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 298.15 K''']]
| [[Image:Anti2_freq_0Kdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 0 K''']]
|}

From looking at table 3 above it can be deduced that the anti2 conformer is more stable at room temperature to that of 0 K with respect to all the types of energies. The spectra also seem to not change much between the two temperatures but the main change is the change in the sum of electronic and thermal free energies, which is heavily dependent of temperature by it including the entropic contribution to the free energy (G = H - '''T'''S)

 
 
 
 

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

Within the Cope Rearrangement there are two possible transition states, the chair (C2h) and the boat (C2v), which are shown within [https://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_2 appendix two ].

==='''Comparison of optimisation methods for the chair transition state'''===

An <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2hdh08.mol</uploadedFileContents>
<text>allyl fragment</text>
<color>white</color>
</jmolAppletButton>
</jmol>, CH2CHCH2 was drawn on Gaussview and was optimized through the HF/3-21G method. After this the original allyl fragment was copied and pasted into a new window and place exactly above the first fragment with a bond distance of 2.20 Å between the terminal end carbons. This therefore produced the <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2h_fulldh08.mol</uploadedFileContents>
<text>chair transition structure.</text>
<color>white</color>
</jmolAppletButton>
</jmol>

Gaussian was then set up for an optimisation for the transition state with the additional key words "Opt=NoEigen", the optimisation set to TS (Berny), the force constants set to "once" and using the HF/3-21G method.

After this the "chair" transition state was optimised an alternative way by freezing the coordinates of the two sets of terminal carbons (where the bonds form/break during the rearrangement) within the Redundant Coord Editor on Gauusview.
Following this a second calculation was run where the freeze coordinates option was changed to "Derivative" and the force constants were set to "never".

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 4: Comparison of chair transition state optimisation methods
| || '''Optimised through a TS (Berny)''' || '''Optimised through a frozen coordinate method'''
|-
| '''Optimised TS Structure''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>C2h_optfreq_TS_Bdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| '''Terminal C-C bond length (Å)''' || 2.02 || 2.02
|-
|'''Energy (atomic units)''' || -231.61932247 || -231.61932190
|-
| '''ΔE, TS (Berny) - frozen coordinate''' || 5.7 x 10-7 a.u || 1.50 x 10-3 KJ mol-1
|-
| '''Imaginary frequency (cm-1)''' || -818 || -818
|-
| '''Animation''' ||[[Image:818dh08TSB.gif|350x350px]] || [[Image:818cm-1Fdh08.gif]]
|}

The data in table 4 shows that both methods give the same results and they both give the same transition state. They both produced the same energy with a negligible energy difference between them and the bond lengths between the terminal carbons are both 2.02 Å. In addition to this the imaginary frequencies were both -818cm-1 and within the animations above it is possible to see that when one C-C bond is forming the other is breaking. This means it is a asynchronous bond formation, which is expected for the Cope Rearrangement since it is a pericyclic reaction.

The only difference between the two optimisations is the way in which the transition states were calculated. This is due to the TS (Berny) method calculating the force constant matrix within the first optimisation step and is then being recalculated as the optimisation carries on. When the frozen coordinate method freezes the reaction coordinate in order for the rest of the structure to be optimised around it, which does not require the whole hessian (force constant matrix) to be calculated but only that the reaction coordinate is differentiated (second step). The main advantage of this, is that you do not need to calculate the entire force constant matrix and just need to differentiate the reaction coordinate in order to obtain a good enough guess for the initial force constant matrix. The frozen coordinate method will therefore take less time to compute the transition state, which can be very expensive within certain systems. However to carry out this method one needs to have an idea of how far apart the terminal carbons are.

==='''Boat transition state'''===

===='''QST2 optimisation'''====

The optimisation for the boat transition state had a different method and this was the QST2 method. This calculation interpolates between two structures (product and reactant) and tries to find a transition state between them.

The reactants and products were numbered in the same way. Therefore both 1,5-hexadiene molecules had their numbering manually changed for the product molecule so it corresponded to the numbering of the reactant if it rearranged. This is illustrated in the reaction scheme below.

[[Image:Boat_rerrangmentdh08.jpg|300x300px|center]]

The chk file for the optimized anti2 conformer was opened and copied into a new window. A second window was opened within the new window and the same anti2 conformer was copied into the new window. The numbering was changed within the second molecule so the two looked like they do in the image below.

[[Image:TS_(QST2)_numbering_1dh08.jpg|400x400px|center]]

The job type was set as "Opt+Freq" and "TS (QST2)" was chosen to optimise the transition state. However this failed and produced
a structure a bit like a <jmol>
<jmolAppletButton>
<uploadedFileContents>Trialone_BPdh08.mol</uploadedFileContents>
<text>chair transition structure </text>
<color>white</color>
</jmolAppletButton>
</jmol> but more dissociated. This is because it interpolated between the two structures and translated the top allyl fragment and therefore did not consider the possibility of a rotation around the central bonds.

To get around this the C-C-C-C dihedral angle (C2-C3-C4-C5) was set to 0 o and the C-C-C angle (C2-C3-C4 and C3-C4-C5 for the product molecule) was set to 100 o for both molecules. This gave rise to the two structures below.

[[Image:TS_(QST2)_numbering_2dh08.jpg|400x400px|center]]

After this the QST2 calculation was run again and the results in table 5 were produced. The imaginary frequency at -839 cm-1 shows that while one bond is forming the other one is breaking, which is again an asynchronous bond formation.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5: Data for the boat transition state from the QST2 optimisation method
| Optimised TS Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.14
|-
|Energy (atomic units) || -231.60280151
|-
| Energy (KJ mol-1) || -6.08 x 105
|-
| Imaginary frequency (cm-1) || -839
|-
| Animation ||[[Image:-839dh08.gif]]
|}

===='''Intrinsic Reaction Coordinate'''====

It is not possible to predict which product the chair and boat transition state will lead to. To solve this problem within Gaussian one can use the IRC method, which allows one to follow the minimum energy path from a transition structure down to its local minimum on a potential energy surface. This creates a succession of points by taking small geometry steps in the direction to where the gradient of the energy surface is at its steepest until the minimum in energy is reached.

The chk file for the optimized chair transition structure was opened and the job type selected was "IRC", the reaction coordinate was computed in the forward direction (since molecule is symmetric) with 50 points along the IRC. This was done with two type of force constants being calculated, once and always.

Within the "once" method the output showed no bond formation process by the 26th step and therefore the calculation was incomplete, since a complete 1,5-hexadiene structure was supposed to be formed. From looking at the IRC pathway it can be seen that the minimum on the potential energy surface was not reached at that point, since the points on the pathway did not form an asymptote and the gradient did not go close enough to zero.

For the "always" method the computation of force constants were carried out at every step. This therefore lets the IRC calculation distinguish whether it is moving along the minimum energy pathway while going down the potential energy surface. While this is happening it can make changes to the pathway if necessary. Within the "always" case the bond forming arrives at 1,5-hexadiene and this is backed up by the IRC pathway reaching a minimum within its energy by the gradient going close to zero and the pathway reaches an asymptote.

The structure produced through the always method had an energy of -231.69199434 atomic units, which is very close to the energy of the gauche2 conformer in the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 appendix 1]. This suggests that the reaction takes place through the chair transition state, which later produces the gauche2 conformer of 1,5-hexadiene.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 6: Comparison of IRC calculations for the chair structure
| || '''Calculate force constants once''' || '''Calculate force constants always'''
|-
| '''Number of points along IRC''' || 26 || 47
|-
| '''Structure at the last point of IRC''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Once_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Always_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|'''Energy (atomic units)''' || -231.68906923 || -231.69199434
|-
| '''ΔE, once - always''' || 2.92511 x 10-3 a.u || 7.68 KJ mol-1
|-
| '''IRC pathway''' || [[Image:One_pathwaydh08.jpg|400x400px]] || [[Image:Always_pathwaydh08.jpg|400x400px]]
|-
| '''IRC gradient''' || [[Image:One_gradientdh08.jpg|400x400px]]|| [[Image:Always_gradientdh08.jpg|400x400px]]
|}

 
 
 

===='''Comparison of activation energies for the transition states'''====

The HF/3-21G optimised chair and boat transition structures were both reoptimised using the B3LYP/6-31G* (B3LYP/6-31G (d)) method. This gives a more accurate minimisation however this method takes longer to run. The values for the zero-point energies and thermal energies were found within the thermochemistry section of the log file.

The values for the HF/3-21G reactant (anti2) optimisation were calculated and the B3LYP/6-31G* values were taken from the "Frequency analysis of 1,5-hexadiene Conformers" earlier in this module.

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 7: Energy data for chair, boat and the anti2 conformation from the HF/3-21G and B3LYP/6-31G* methods of optimisation '''
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466677
| align="center" | -231.461320
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450936
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402342
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539524
| align="center" | -231.532562
| align="center" | -234.611716
| align="center" | -234.469177
| align="center" | -234.461835
|}
Digital repositories

Chair:
HF/3-21G: http://hdl.handle.net/10042/to-5569
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5568

Boat:
HF/3-21G: http://hdl.handle.net/10042/to-5570
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5571

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 8: Summary of activation energies for the chair and boat conformations (ΔE = chair TS or boat TS minus anti2)'''
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| align="center" | 1 a.u = 627.509 kcal mol-1
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE, chair (a.u)'''
| align="center" | 0.072847
| align="center" | 0.071242
| align="center" | 0.054248
| align="center" | 0.052827
| align="center" | -
|-
| '''ΔE, chair (kcal mol-1)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE, boat (a. u)'''
| align="center" | 0.088588
| align="center" | 0.087259
| align="center" | 0.066835
| align="center" | 0.065827
| align="center" | -
|-
| '''ΔE, boat (kcal mol-1)'''
| align="center" | 55.59
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.31
| align="center" | 44.7 ± 2.0
|}

By examining table 8 it is clear to see that the B3LYP/6-31G* method gives values which are much closer to the experimental ones. This is due to the B3LYP/6-31G* giving a higher level of optimisation than the HF/3-21G method. Table 8 also shows that the chair transition state has a lower activation energy to that of the boat transition state by 8.16 kcal mol-1. This suggests that the Cope rearrangement mechanism would most likely proceed through the chair transition state to that of the boat. Therefore the reaction would take place from the anti2 conformer to the chair transition state and then onto the gauche2 conformer within the Cope rearrangement.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9: Geometric parameters for the optimised chair and boat structures '''
| || ''' HF/3-21G chair TS''' || ''' B3LYP/6-31G* chair TS''' || '''HF/3-21G boat TS''' || '''B3LYP/6-31G* boat TS'''
|-
| Optimised Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Chair_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Boat_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.02 (C5-C10 or C3-C14)|| 1.97 (C5-C10 or C3-C14) || 2.14 (C3-C4 or C1-C6) || 2.21 (C3-C4 or C1-C6)
|-
| Fragment angle || 120 o (C3-C1-C5) || 120 o (C3-C1-C5) || 122 o (C4-C5-C6) || 122 o (C4-C5-C6)
|-
| Dihedral angle || 55 o (C1-C3-C14-C11)|| 54 o (C1-C3-C14-C11)|| 0 o (C2-C1-C6-C5) || 0 o (C2-C1-C6-C5)
|-
| Numbering of carbons || [[Image:Numbering_chairdh08.jpg|200x200px]]|| [[Image:Numbering_chairdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]]
|}

From studying table 9 it can be seen that the C-C bond breaking/forming length has changed between the two different types of optimisation methods. However the angles between the bonds remains unchanged.

=='''The Diels Alder Cycloaddition'''==

The Diels-Alder reaction is a cycloaddition between a conjugated diene and a dienophile (alkene system), which forms a cyclohexene system. It is often referred to as a [4s+2s] cycloaddition and proceeds through a single cyclic, concerted transition state where 2 σ bonds are formed in between the termini carbons of the 2 conjugated п systems. This is shown in the reaction scheme below where ethene is reacting with cis-butadiene.

[[Image:Reaction_scheme2dh08.jpg|300x300px|center]]

The π orbitals of the dienophile form new σ bonds with the π orbitals of the diene. The number of π electrons involved determine whether the reaction proceeds in a concerted stereospecific fashion (allowed) or not (forbidden). The HOMO/LUMO of one fragment interacts with the HOMO/LUMO of the other fragment to form two new bonding and anti-bonding MOs. The nodal properties allow one to predict how the reaction will occur.

Semi-Empirical/AM1 method and the B3LYP/6-31G* method of optimisation will be used on Gaussian to determine particular properites of the reactants to predict the way in which the reaction will proceed.

==='''Reagents'''===

<jmol>
<jmolAppletButton>
<uploadedFileContents>Cis_butadienedh08.mol</uploadedFileContents>
<text>Cis-butadiene</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Ethenedh08.mol</uploadedFileContents>
<text>ethylene</text>
<color>white</color>
</jmolAppletButton>
</jmol> were optimised separately using the AM1 semi-empirical method on Gaussview.The cis-butadiene had an energy of 0.04879725 a. u (1.28 x 102 KJ mol-1) and ethylene had an energy of 0.02619027 a. u (68.76 KJ mol-1).

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10: Frontier molecular orbitals of cis-butadiene and ethylene '''
| '''Reagent''' || '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| cis-butadiene || LUMO || [[Image:LUMO1_cis_butadieneDH08.jpg|100x100px]] || [[Image:LUMO2_cis_butadienedh08.jpg|100x100px]]|| Symmetric || +0.017
|-
| cis-butadiene || HOMO || [[Image:HOMO1_cis_butadienedh08.jpg|100x100px]] || [[Image:HOMO2_cis_butadienedh08.jpg|100x100px]]|| Anti-symmetric || -0.343
|-
| ethylene || LUMO || [[Image:LUMO1_ethenedh08.jpg|100x100px]] || [[Image:LUMO2_ethenedh08.jpg|100x100px]]|| Anti-symmetric || +0.052
|-
| ethylene || HOMO || [[Image:HOMO1_ethenedh08.jpg|100x100px]] || [[Image:HOMO2_ethenedh08.jpg|100x100px]]|| Symmetric || -0.387
|}

 
The symmetries of the LUMO and HOMO orbitals for cis-butadiene are opposite of those for the ethylene. This therefore means that the only interactions that can take place are those between ethylene's LUMO and cis-butadiene's HOMO and in addition to this the interaction between cis-butadiene's LUMO and ethylene's HOMO. This is because orbital symmetry has to be conserved (the conservation of orbital symmetry states that the symmetry of an new orbital formed by the overlap of the 2 original orbitals must be the same as the symmetry of the initial orbitals).

==='''Transition State'''===

A <jmol>
<jmolAppletButton>
<uploadedFileContents>Bicyclic_startingdh08.mol</uploadedFileContents>
<text>bicyclo system</text>
<color>white</color>
</jmolAppletButton>
</jmol> was drawn and one of the -CH2-CH2- fragments were removed. Three C=C bonds were added and the previous one was taken away so a <jmol>
<jmolAppletButton>
<uploadedFileContents>Frozen_coorddh08.mol</uploadedFileContents>
<text>guess transition state</text>
<color>white</color>
</jmolAppletButton>
</jmol> was produced.

After this an optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while bonds between the termini of the fragments were frozen at 2.1Å. Following this the resulting file had two separate optimisation and frequency calculations run on it using the TS (Berny) with one using the semi-empirical/AM1 method and the other using the DFT/B3LYP/6-31G* level of theory. Both of these optimisations had the bonds unfrozen and the "derivative" option was chosen instead.

The FT/B3LYP/6-31G* level of theory was used since it gave more accurate results in the calculations before and the energies it produces are consistent with the rest of the report (semi-empirical/AM1 method gives very different values).

The QST2 optimisation was not used since it required numbering the atoms in both the reactants and products, which took a lot of time to do.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11: Comparison of two methods of optimsations for the transition state '''
| || '''semi-empirical/AM1 method''' || '''DFT/B3LYP/6-31G* method'''
|-
| Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>SE_AM1_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>DFT_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Energy (a. u) || 0.11165608 || -234.54389573
|-
| Bond lengths || [[Image:DistancesSEdh08.jpg|150x150px]]|| [[Image:DistancesDFTdh08.jpg|150x150px]]
|-
|Imaginary frequency (cm-1) || -957 || -525
|-
|Animation || [[Image:-957dh08.gif]] || [[Image:-525dh08.gif]]
|-
| Lowest real frequency (cm-1) || 148 || 136
|-
|Animation || [[Image: 148dh08.gif]] || [[Image: 136dh08.gif]]

|}

The C-C bond forming length between the termini carbons is 0.15 Å larger for the DFT/B3LYP/6-31G* level of theory and this is because this method gives carries out the calculation to a greater degree of accuracy. This larger bond length leads a lower potential energy gradient to the transition state, which gives a lower total energy for the transition state. Therefore the given values below will be the DFT/B3LYP/6-31G* and not the semi-empirical/AM1 method.

A typical sp3 C-C bond length is 1.54 Å and a typical sp2 C=C bond length is 1.34 Å<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref>. Within the optimised transition molecule above the C-C bond lengths are 1.40 Å and the C=C bond length is 1.38 Å. This suggests that the π-bonds are breaking in order for the C-C bonds to form and additionally to this a π-bond is forming in between the two sp3 carbons.

The Van der Waals radius of a C-atom is 1.72 Å<ref name="ref 3">M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer, D. G. Truhlar, 1995, 346, pp. 63-69.[http://pubs.acs.org/doi/abs/10.1021/jp8111556]</ref>. The partly formed C-C length is 2.27 Å, which is shorter than two times the the van der Waals radius of a carbon atom. Therefore the carbon atoms are within a close enough distance to each other to share electron density and yet the bond has not formed at this point.

From looking at the animations for the imaginary frequency above it can be seen that the formation of the two bonds between the termini of the reactants is synchronous. This is expected of the Diels-Alder reaction since the mechanism is concerted. However the lowest real frequency is asynchronous with respect to the two bonds forming between the termini.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12: Frontier molecular orbitals of the transition state '''
| '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| LUMO || [[Image:LUMO1TSdh08.jpg|100x100px]] || [[Image:LUMO2TSdh08.jpg|100x100px]]|| Symmetric || +0.023
|-
| HOMO || [[Image:HOMO1TSdh08.jpg|100x100px]] || [[Image:HOMO2TSdh08.jpg|100x100px]]|| Anti-symmetric || -0.323
|}

From examining the images from table 12 above it can be deduced that the LUMO of the transition state originates from the cis-butadiene's LUMO and the ethylene's HOMO, which represents the σ*-antibonding orbitals. However the HOMO of the transition state originates from the cis-butadiene's HOMO and the ethylene's LUMO, which represents the σ-bonding orbitals within the two single C-C bonds that are formed within the reaction.

==='''Cyclohexa-1,3-diene reaction with maleic anhydride'''===

The <jmol>
<jmolAppletButton>
<uploadedFileContents>FrozenTS22dh08.mol</uploadedFileContents>
<text>exo</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_frozenTS22dh08.mol</uploadedFileContents>
<text>endo</text>
<color>white</color>
</jmolAppletButton>
</jmol> transition states were drawn on Gaussview and then the optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while the distance of the forming bonds were frozen at 2.1Å between the respective carbons. After this the structures changed slightly (<jmol>
<jmolAppletButton>
<uploadedFileContents>TS2_afteropt1_exo1dh08.mol</uploadedFileContents>
<text>exo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_TS2_afteropt1dh08.mol</uploadedFileContents>
<text>endo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol>) and the resulting files had an optimisation and frequency calculation run on it using the TA (Berny) with using the semi-empirical/AM1 level of theory while the unfrozen option as changed to "derivative". After this the endo and exo optimised transition state structures were re-optimised as they were before but using the DFT/B3LYP/6-31G* level of theory instead with the additional keyword "Opt=NoEigen".

{| class="wikitable" border="1" align="center"
|+ '''Table 13: Comparison of the exo and endo transition states '''
! rowspan="2"|-!! colspan="2"|''exo'' !! colspan="2"|''endo''
|-
| Semi-empirical/AM1 || DFT/B3LYP/6-31G* || Semi-empirical/AM1 || DFT/B3LYP/6-31G*
|-
| align="center"|Structure ||align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>After_opt2_exodh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>Endo_TS2_afteropt2dh081.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| align="center"| Energy (a. u) ||align="center"| -0.05041975 ||align="center"| -612.67931095 ||align="center"| -0.05150470 ||align="center"| -612.68339684
|-
| C-C bond forming length (Å) ||align="center"| 2.17 ||align="center"| 2.29 ||align="center"| 2.16 ||align="center"| 2.27
|-align="center"
| Imaginary frequency (cm-1) ||align="center"| -812 ||align="center"| -448 ||align="center"| -806 ||align="center"| -447
|-align="center"
| Animation ||colspan="2"|[[Image:-812_SEdh08.gif]] || colspan="2"|[[Image:-806_cm-1dh08.gif]]
|}

Again the C-C bond forming length between the termini carbons is larger (by 0.11 and 0.12 Å) for the DFT/B3LYP/6-31G* level of theory and this is due the same reasons talked about above. However the C-C bond forming length between the termini carbons are very similar in both the endo and exo transition states (0.02 Å difference).

From looking at the animations for the imaginary frequency above it can also be seen that the formation of the two bonds between the carbons in this case as well is synchronous. Again this is expected of the Diels-Alder reaction since it is concerted.

{| align=center
|+ '''Strain diagrams for the endo and exo transition states'''
| [[Image:Exostraindh08.jpg|thumb|200x200px|'''Exo TS''']]
| [[Image:Strainendodh08.jpg|thumb|200x200px|'''Endo TS''']]
|}

From studying the energies in table 13 it can be seen that the endo transition state is more stable than the exo by 4.08589 x 10-3 a. u (10.73 KJ mol-1). However by looking at the strain of both transition states one would expect the exo TS to be more stable. This is because the distance between the cyclohexa-1,3-diene ring and the -(C=O)-O-(C=O)- maleic anhydride fragment is 2.99 Å in the endo TS and 3.03 Å in the exo TS. Thus the endo form would have more strain due to steric hindrance.

However within the exo TS the vicinal hydrogens on the CH2-CH2 (carbon numbers 18 and 21) fragment on the cyclohexa-1,3-diene ring are found below the plane and next to the maleic anhydride. This leads to steric hindrance is this is where the strain originates from within the exo TS. Therefore this disfavours the exo form, which makes the endo more stable.

Another factor making the endo TS more favourable is secondary orbital overlap. These orbitals contribute to the bonds forming and they interact with each other through space. Within the endo TS below there are 2 more orbitals interacting (secondary orbitals) than within the exo TS. This therefore makes the endo TS more stable and therefore more favoured.

[[Image:Secondary_orbital_shizzledh08.jpg|500x500px|center]]

{| class="wikitable" border="1" align="center"
|+ '''Table 14: Frontier molecular orbitals of the the transition state for the cyclohexa-1,3-diene reaction '''
! rowspan="2"|Molecular Orbital !! colspan="4"|''exo'' !! colspan="4"|''endo''
|- align="center"
| '''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''||'''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''
|- align="center"
| LUMO || [[Image:LUMO1exoh081.jpg|150x150px]]|| [[Image:LUMO2exodh081.jpg|150x150px]]|| Anti-symmetric || -0.078 || [[Image:LUMO1ENDOdh08.jpg|150x150px]]||[[Image:LUMO2ENDOdh08.jpg|150x150px]] || Anti-symmetric || -0.067
|- align="center"
| HOMO || [[Image:HOMO1exodh081.jpg|150x150px]]||[[Image:HOMO2exodh081.jpg|150x150px]] || Anti-symmetric|| -0.242 || [[Image:HOMO1ENDOdh08.jpg|150x150px]]|| [[Image:HOMO2ENDOdh08.jpg|150x150px]]||Anti-symmetric || -0.242
|}

Digital repositories:

exo: http://hdl.handle.net/10042/to-5596

endo: http://hdl.handle.net/10042/to-5597

From examining table 14 it is clear to see that both the HOMO and LUMO are anti-symmetric for both the transition states. The LUMO of the maleic anhydride fragment is the π* anti-bonding orbital of the C=C bond. Within the reaction then this has electrons transferred to it and the C=C bond breaks, which leaves just a C-C bond. The LUMOs for both transition states are both anti-bonding with respect to the two fragments. The HOMOs of both transitions state represent the σ-bonding that will be formed between the termini carbons on the products and reactants. This is most likely from the overlap of the cyclohexa-1,3-diene's HOMO and the maleic anhydride's LUMO.

=='''Conclusion'''==

='''References'''=
<references/>

Rep:Mod:hampluscheese

2010-11-10T22:36:59Z

Dh08: /* '''Transition State''' */

='''Module 3:Physical Computational Chemistry Lab'''=

Douglas Hunt

=='''Objectives'''==

The aim of this experiment is to

=='''The Cope Rearrangement'''==

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangement] is a [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement |3,3|-sigmatropic rearrangement] that takes place through a single [http://en.wikipedia.org/wiki/Pericyclic pericyclic], concerted [http://en.wikipedia.org/wiki/Transition_state transition state]. Within this rearrangement a σ bond is formed in concert with another being broken within a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] π system.

The Cope rearrangement we will be studying will be that of 1,5-hexadiene, which is shown below.

[[Image:Cope_rearrangement_schemedh08.jpg|300x300px|center]]

The objective is to locate the low-energy minima and transition structures on the C6H10 potential energy surface and to determine the preferred reaction mechanism by using [http://en.wikipedia.org/wiki/GAUSSIAN Gaussian], with the [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)] methods.

==='''Optimisation of 1,5-hexadiene Conformers'''===

Rough <jmol>
<jmolAppletButton>
<uploadedFileContents>Anti_2_drawndh08.mol</uploadedFileContents>
<text>anti</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche3_drawndh08.mol</uploadedFileContents>
<text>gauche</text>
<color>white</color>
</jmolAppletButton>
</jmol> conformers of 1,5-hexadiene were drawn on Gaussview and their structures were tidied by the clean function. They were then further optimised by the HF/3-21G method and were identified which conformer they were from their point group and appendix 1 on the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3 physical lab page].

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 1: Optimisation and comparison of anti and gauche 1,5-hexadiene conformers using HF/3-21G methods
| ''' Conformer''' || '''Optimised Structure''' || '''Energy (Atomic Units)''' || '''[http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1 Energies (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (Anti2 - Gauche3)''' || '''Point group'''
|-
| '''Anti2''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Anti_2_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692535 || -231.69254 ||-6.08 x 105 || 1.2606 x 10-4 a. u|| Ci
|-
| '''Gauche3''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Gauche3_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692661 || -231.69266 || -6.08 x 105 || 3.31 x 10-1 KJ mol-1 || C1
|}

From studying table 1 it can be seen that the gauche conformer is 0.33 KJ mol-1 more stable than the anti conformation. One would assume that the anti2 conformer would be more stable due to [http://en.wikipedia.org/wiki/Steric steric] factors/repulsion within the gauche3 conformation. However within the gauche3 conformer there is a favourable donation of [http://en.wikipedia.org/wiki/Electron_density electron density] from the C=C π orbital to the C-H σ* orbital from the adjacent proton, which stabilises the conformer<ref name="ref 1">B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446.[http://www.informaworld.com/smpp/content~db=all?content=10.1080/00268970110081412]</ref>.

 

The anti2 conformer was then optimised using the B3LYP 6-31G* (6-31G (d)) method.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 2: Comparison of the anti2 conformer optimisations carried out at different calculation methods
| ''' Method''' || '''Structure and bond distances (Å)''' || '''Dihedral angle (C1 to C4)''' || '''Energy (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (HF/3-21G - DFT/B3LYP/6-31G)'''
|-
| HF/3-21G || [[Image:Anti2_opt1dh08.jpg|300x300px]] || 114.6 o || -231.692535 || -6.08 x 105 || 2.919171 a. u
|-
| DFT/B3LYP/6-31G* || [[Image:Anti2_opt2DH08.jpg|300x300px]] || 118.8 o || -234.611716 || -6.16 x 105 || 7.66 x 103 KJ mol-1
|}

From examining table 2 at first it is possible to see that the bond lengths are very similar to each other and that of the literature bond lengths<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref> (C1=C2: 1.340 Å, C2-C3: 1.508 Å and C3-C4: 1.538 Å). It is also possible to see that the C=C bond length is more accurate for the DFT/B3LYP/6-31G* method by it being 0.01 Å closer to the literature value. Another main difference in terms of geometry is that the DFT/B3LYP/6-31G* method gives a larger [http://en.wikipedia.org/wiki/Dihedral_angle dihedral angle] (and most likely more accurate) of 118.8o compared to that of 114.6o.

In terms of energy the DFT/B3LYP/6-31G* method produces a conformer of a higher stability by it possessing a lower energy with a difference of 7.66 x 103 KJ mol-1.

==='''Frequency analysis of 1,5-hexadiene Conformers'''===

[[Image:Vibrations_anti2dh08.jpg|right|300x300px]]

For the frequency analysis the DFT/B3LYP/6-31G* method was used.

Frequency analysis is essential within this calculation due to it being the second derivative of the potential energy surface. This means that if all the frequencies are all positive the system is at a minimum, if one of them is negative the optimised system is at a transition state, and if more than one are negative then the calculation has failed to find a critical point, which means the optimisation has failed. This therefore gives us a method of seeing whether the calculation has worked.

From looking to the image on the right it can deduced that the optimisation has taken place successfully by there being no negative vibrations and therefore the conformation is at a minimum.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 3: Energies under the thermochemistry section at 298.15 K and 0 K within the frequency logfiles'''
| '''Energies''' || '''298.15 K (atomic units)''' ||| '''0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (KJ mol-1)''' || '''Discussion'''
|-
| Sum of electronic and zero-point energies || -234.469177 || -234.466673 || -2.504 x 10-3 || -6.57 || This is the potential energy at 0 K, which includes the zero-point vibrational energy (E = Eelec + ZPE).

|-
| Sum of electronic and thermal energies || -234.461835 || -234.461085 || -7.5 x 10-4|| -1.97 || This is the energy at 298.15 K (0 K for 0 K calculation) and 1 atm of pressure, which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans).

|-
| Sum of electronic and thermal enthalpies || -234.460891 || -234.460141 || -7.5 x 10-4 || -1.97 || This contains an additional correction for RT (H = E + RT), which is especially important when looking at dissociation reactions.
|-
| Sum of electronic and thermal free energies || -234.500739 || -234.495931 || 4.808 x 10-3 || -12.62 || This includes the entropic contribution to the free energy (G = H - TS).
|}

{|align=right
|+
| [[Image:Anti2_freq_IRdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 298.15 K''']]
| [[Image:Anti2_freq_0Kdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 0 K''']]
|}

From looking at table 3 above it can be deduced that the anti2 conformer is more stable at room temperature to that of 0 K with respect to all the types of energies. The spectra also seem to not change much between the two temperatures but the main change is the change in the sum of electronic and thermal free energies, which is heavily dependent of temperature by it including the entropic contribution to the free energy (G = H - '''T'''S)

 
 
 
 

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

Within the Cope Rearrangement there are two possible transition states, the chair (C2h) and the boat (C2v), which are shown within [https://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_2 appendix two ].

==='''Comparison of optimisation methods for the chair transition state'''===

An <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2hdh08.mol</uploadedFileContents>
<text>allyl fragment</text>
<color>white</color>
</jmolAppletButton>
</jmol>, CH2CHCH2 was drawn on Gaussview and was optimized through the HF/3-21G method. After this the original allyl fragment was copied and pasted into a new window and place exactly above the first fragment with a bond distance of 2.20 Å between the terminal end carbons. This therefore produced the <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2h_fulldh08.mol</uploadedFileContents>
<text>chair transition structure.</text>
<color>white</color>
</jmolAppletButton>
</jmol>

Gaussian was then set up for an optimisation for the transition state with the additional key words "Opt=NoEigen", the optimisation set to TS (Berny), the force constants set to "once" and using the HF/3-21G method.

After this the "chair" transition state was optimised an alternative way by freezing the coordinates of the two sets of terminal carbons (where the bonds form/break during the rearrangement) within the Redundant Coord Editor on Gauusview.
Following this a second calculation was run where the freeze coordinates option was changed to "Derivative" and the force constants were set to "never".

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 4: Comparison of chair transition state optimisation methods
| || '''Optimised through a TS (Berny)''' || '''Optimised through a frozen coordinate method'''
|-
| '''Optimised TS Structure''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>C2h_optfreq_TS_Bdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| '''Terminal C-C bond length (Å)''' || 2.02 || 2.02
|-
|'''Energy (atomic units)''' || -231.61932247 || -231.61932190
|-
| '''ΔE, TS (Berny) - frozen coordinate''' || 5.7 x 10-7 a.u || 1.50 x 10-3 KJ mol-1
|-
| '''Imaginary frequency (cm-1)''' || -818 || -818
|-
| '''Animation''' ||[[Image:818dh08TSB.gif|350x350px]] || [[Image:818cm-1Fdh08.gif]]
|}

The data in table 4 shows that both methods give the same results and they both give the same transition state. They both produced the same energy with a negligible energy difference between them and the bond lengths between the terminal carbons are both 2.02 Å. In addition to this the imaginary frequencies were both -818cm-1 and within the animations above it is possible to see that when one C-C bond is forming the other is breaking. This means it is a asynchronous bond formation, which is expected for the Cope Rearrangement since it is a pericyclic reaction.

The only difference between the two optimisations is the way in which the transition states were calculated. This is due to the TS (Berny) method calculating the force constant matrix within the first optimisation step and is then being recalculated as the optimisation carries on. When the frozen coordinate method freezes the reaction coordinate in order for the rest of the structure to be optimised around it, which does not require the whole hessian (force constant matrix) to be calculated but only that the reaction coordinate is differentiated (second step). The main advantage of this, is that you do not need to calculate the entire force constant matrix and just need to differentiate the reaction coordinate in order to obtain a good enough guess for the initial force constant matrix. The frozen coordinate method will therefore take less time to compute the transition state, which can be very expensive within certain systems. However to carry out this method one needs to have an idea of how far apart the terminal carbons are.

==='''Boat transition state'''===

===='''QST2 optimisation'''====

The optimisation for the boat transition state had a different method and this was the QST2 method. This calculation interpolates between two structures (product and reactant) and tries to find a transition state between them.

The reactants and products were numbered in the same way. Therefore both 1,5-hexadiene molecules had their numbering manually changed for the product molecule so it corresponded to the numbering of the reactant if it rearranged. This is illustrated in the reaction scheme below.

[[Image:Boat_rerrangmentdh08.jpg|300x300px|center]]

The chk file for the optimized anti2 conformer was opened and copied into a new window. A second window was opened within the new window and the same anti2 conformer was copied into the new window. The numbering was changed within the second molecule so the two looked like they do in the image below.

[[Image:TS_(QST2)_numbering_1dh08.jpg|400x400px|center]]

The job type was set as "Opt+Freq" and "TS (QST2)" was chosen to optimise the transition state. However this failed and produced
a structure a bit like a <jmol>
<jmolAppletButton>
<uploadedFileContents>Trialone_BPdh08.mol</uploadedFileContents>
<text>chair transition structure </text>
<color>white</color>
</jmolAppletButton>
</jmol> but more dissociated. This is because it interpolated between the two structures and translated the top allyl fragment and therefore did not consider the possibility of a rotation around the central bonds.

To get around this the C-C-C-C dihedral angle (C2-C3-C4-C5) was set to 0 o and the C-C-C angle (C2-C3-C4 and C3-C4-C5 for the product molecule) was set to 100 o for both molecules. This gave rise to the two structures below.

[[Image:TS_(QST2)_numbering_2dh08.jpg|400x400px|center]]

After this the QST2 calculation was run again and the results in table 5 were produced. The imaginary frequency at -839 cm-1 shows that while one bond is forming the other one is breaking, which is again an asynchronous bond formation.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5: Data for the boat transition state from the QST2 optimisation method
| Optimised TS Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.14
|-
|Energy (atomic units) || -231.60280151
|-
| Energy (KJ mol-1) || -6.08 x 105
|-
| Imaginary frequency (cm-1) || -839
|-
| Animation ||[[Image:-839dh08.gif]]
|}

===='''Intrinsic Reaction Coordinate'''====

It is not possible to predict which product the chair and boat transition state will lead to. To solve this problem within Gaussian one can use the IRC method, which allows one to follow the minimum energy path from a transition structure down to its local minimum on a potential energy surface. This creates a succession of points by taking small geometry steps in the direction to where the gradient of the energy surface is at its steepest until the minimum in energy is reached.

The chk file for the optimized chair transition structure was opened and the job type selected was "IRC", the reaction coordinate was computed in the forward direction (since molecule is symmetric) with 50 points along the IRC. This was done with two type of force constants being calculated, once and always.

Within the "once" method the output showed no bond formation process by the 26th step and therefore the calculation was incomplete, since a complete 1,5-hexadiene structure was supposed to be formed. From looking at the IRC pathway it can be seen that the minimum on the potential energy surface was not reached at that point, since the points on the pathway did not form an asymptote and the gradient did not go close enough to zero.

For the "always" method the computation of force constants were carried out at every step. This therefore lets the IRC calculation distinguish whether it is moving along the minimum energy pathway while going down the potential energy surface. While this is happening it can make changes to the pathway if necessary. Within the "always" case the bond forming arrives at 1,5-hexadiene and this is backed up by the IRC pathway reaching a minimum within its energy by the gradient going close to zero and the pathway reaches an asymptote.

The structure produced through the always method had an energy of -231.69199434 atomic units, which is very close to the energy of the gauche2 conformer in the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 appendix 1]. This suggests that the reaction takes place through the chair transition state, which later produces the gauche2 conformer of 1,5-hexadiene.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 6: Comparison of IRC calculations for the chair structure
| || '''Calculate force constants once''' || '''Calculate force constants always'''
|-
| '''Number of points along IRC''' || 26 || 47
|-
| '''Structure at the last point of IRC''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Once_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Always_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|'''Energy (atomic units)''' || -231.68906923 || -231.69199434
|-
| '''ΔE, once - always''' || 2.92511 x 10-3 a.u || 7.68 KJ mol-1
|-
| '''IRC pathway''' || [[Image:One_pathwaydh08.jpg|400x400px]] || [[Image:Always_pathwaydh08.jpg|400x400px]]
|-
| '''IRC gradient''' || [[Image:One_gradientdh08.jpg|400x400px]]|| [[Image:Always_gradientdh08.jpg|400x400px]]
|}

 
 
 

===='''Comparison of activation energies for the transition states'''====

The HF/3-21G optimised chair and boat transition structures were both reoptimised using the B3LYP/6-31G* (B3LYP/6-31G (d)) method. This gives a more accurate minimisation however this method takes longer to run. The values for the zero-point energies and thermal energies were found within the thermochemistry section of the log file.

The values for the HF/3-21G reactant (anti2) optimisation were calculated and the B3LYP/6-31G* values were taken from the "Frequency analysis of 1,5-hexadiene Conformers" earlier in this module.

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 7: Energy data for chair, boat and the anti2 conformation from the HF/3-21G and B3LYP/6-31G* methods of optimisation '''
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466677
| align="center" | -231.461320
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450936
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402342
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539524
| align="center" | -231.532562
| align="center" | -234.611716
| align="center" | -234.469177
| align="center" | -234.461835
|}
Digital repositories

Chair:
HF/3-21G: http://hdl.handle.net/10042/to-5569
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5568

Boat:
HF/3-21G: http://hdl.handle.net/10042/to-5570
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5571

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 8: Summary of activation energies for the chair and boat conformations (ΔE = chair TS or boat TS minus anti2)'''
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| align="center" | 1 a.u = 627.509 kcal mol-1
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE, chair (a.u)'''
| align="center" | 0.072847
| align="center" | 0.071242
| align="center" | 0.054248
| align="center" | 0.052827
| align="center" | -
|-
| '''ΔE, chair (kcal mol-1)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE, boat (a. u)'''
| align="center" | 0.088588
| align="center" | 0.087259
| align="center" | 0.066835
| align="center" | 0.065827
| align="center" | -
|-
| '''ΔE, boat (kcal mol-1)'''
| align="center" | 55.59
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.31
| align="center" | 44.7 ± 2.0
|}

By examining table 8 it is clear to see that the B3LYP/6-31G* method gives values which are much closer to the experimental ones. This is due to the B3LYP/6-31G* giving a higher level of optimisation than the HF/3-21G method. Table 8 also shows that the chair transition state has a lower activation energy to that of the boat transition state by 8.16 kcal mol-1. This suggests that the Cope rearrangement mechanism would most likely proceed through the chair transition state to that of the boat. Therefore the reaction would take place from the anti2 conformer to the chair transition state and then onto the gauche2 conformer within the Cope rearrangement.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9: Geometric parameters for the optimised chair and boat structures '''
| || ''' HF/3-21G chair TS''' || ''' B3LYP/6-31G* chair TS''' || '''HF/3-21G boat TS''' || '''B3LYP/6-31G* boat TS'''
|-
| Optimised Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Chair_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Boat_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.02 (C5-C10 or C3-C14)|| 1.97 (C5-C10 or C3-C14) || 2.14 (C3-C4 or C1-C6) || 2.21 (C3-C4 or C1-C6)
|-
| Fragment angle || 120 o (C3-C1-C5) || 120 o (C3-C1-C5) || 122 o (C4-C5-C6) || 122 o (C4-C5-C6)
|-
| Dihedral angle || 55 o (C1-C3-C14-C11)|| 54 o (C1-C3-C14-C11)|| 0 o (C2-C1-C6-C5) || 0 o (C2-C1-C6-C5)
|-
| Numbering of carbons || [[Image:Numbering_chairdh08.jpg|200x200px]]|| [[Image:Numbering_chairdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]]
|}

From studying table 9 it can be seen that the C-C bond breaking/forming length has changed between the two different types of optimisation methods. However the angles between the bonds remains unchanged.

=='''The Diels Alder Cycloaddition'''==

The Diels-Alder reaction is a cycloaddition between a conjugated diene and a dienophile (alkene system), which forms a cyclohexene system. It is often referred to as a [4s+2s] cycloaddition and proceeds through a single cyclic, concerted transition state where 2 σ bonds are formed in between the termini carbons of the 2 conjugated п systems. This is shown in the reaction scheme below where ethene is reacting with cis-butadiene.

[[Image:Reaction_scheme2dh08.jpg|300x300px|center]]

The π orbitals of the dienophile form new σ bonds with the π orbitals of the diene. The number of π electrons involved determine whether the reaction proceeds in a concerted stereospecific fashion (allowed) or not (forbidden). The HOMO/LUMO of one fragment interacts with the HOMO/LUMO of the other fragment to form two new bonding and anti-bonding MOs. The nodal properties allow one to predict how the reaction will occur.

Semi-Empirical/AM1 method and the B3LYP/6-31G* method of optimisation will be used on Gaussian to determine particular properites of the reactants to predict the way in which the reaction will proceed.

==='''Reagents'''===

<jmol>
<jmolAppletButton>
<uploadedFileContents>Cis_butadienedh08.mol</uploadedFileContents>
<text>Cis-butadiene</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Ethenedh08.mol</uploadedFileContents>
<text>ethylene</text>
<color>white</color>
</jmolAppletButton>
</jmol> were optimised separately using the AM1 semi-empirical method on Gaussview.The cis-butadiene had an energy of 0.04879725 a. u (1.28 x 102 KJ mol-1) and ethylene had an energy of 0.02619027 a. u (68.76 KJ mol-1).

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10: Frontier molecular orbitals of cis-butadiene and ethylene '''
| '''Reagent''' || '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| cis-butadiene || LUMO || [[Image:LUMO1_cis_butadieneDH08.jpg|100x100px]] || [[Image:LUMO2_cis_butadienedh08.jpg|100x100px]]|| Symmetric || +0.017
|-
| cis-butadiene || HOMO || [[Image:HOMO1_cis_butadienedh08.jpg|100x100px]] || [[Image:HOMO2_cis_butadienedh08.jpg|100x100px]]|| Anti-symmetric || -0.343
|-
| ethylene || LUMO || [[Image:LUMO1_ethenedh08.jpg|100x100px]] || [[Image:LUMO2_ethenedh08.jpg|100x100px]]|| Anti-symmetric || +0.052
|-
| ethylene || HOMO || [[Image:HOMO1_ethenedh08.jpg|100x100px]] || [[Image:HOMO2_ethenedh08.jpg|100x100px]]|| Symmetric || -0.387
|}

 
The symmetries of the LUMO and HOMO orbitals for cis-butadiene are opposite of those for the ethylene. This therefore means that the only interactions that can take place are those between ethylene's LUMO and cis-butadiene's HOMO and in addition to this the interaction between cis-butadiene's LUMO and ethylene's HOMO. This is because orbital symmetry has to be conserved (the conservation of orbital symmetry states that the symmetry of an new orbital formed by the overlap of the 2 original orbitals must be the same as the symmetry of the initial orbitals).

==='''Transition State'''===

A <jmol>
<jmolAppletButton>
<uploadedFileContents>Bicyclic_startingdh08.mol</uploadedFileContents>
<text>bicyclo system</text>
<color>white</color>
</jmolAppletButton>
</jmol> was drawn and one of the -CH2-CH2- fragments were removed. Three C=C bonds were added and the previous one was taken away so a <jmol>
<jmolAppletButton>
<uploadedFileContents>Frozen_coorddh08.mol</uploadedFileContents>
<text>guess transition state</text>
<color>white</color>
</jmolAppletButton>
</jmol> was produced.

After this an optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while bonds between the termini of the fragments were frozen at 2.1Å. Following this the resulting file had two separate optimisation and frequency calculations run on it using the TS (Berny) with one using the semi-empirical/AM1 method and the other using the DFT/B3LYP/6-31G* level of theory. Both of these optimisations had the bonds unfrozen and the "derivative" option was chosen instead.

The FT/B3LYP/6-31G* level of theory was used since it gave more accurate results in the calculations before and the energies it produces are consistent with the rest of the report (semi-empirical/AM1 method gives very different values).

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11: Comparison of two methods of optimsations for the transition state '''
| || '''semi-empirical/AM1 method''' || '''DFT/B3LYP/6-31G* method'''
|-
| Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>SE_AM1_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>DFT_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Energy (a. u) || 0.11165608 || -234.54389573
|-
| Bond lengths || [[Image:DistancesSEdh08.jpg|150x150px]]|| [[Image:DistancesDFTdh08.jpg|150x150px]]
|-
|Imaginary frequency (cm-1) || -957 || -525
|-
|Animation || [[Image:-957dh08.gif]] || [[Image:-525dh08.gif]]
|-
| Lowest real frequency (cm-1) || 148 || 136
|-
|Animation || [[Image: 148dh08.gif]] || [[Image: 136dh08.gif]]

|}

The C-C bond forming length between the termini carbons is 0.15 Å larger for the DFT/B3LYP/6-31G* level of theory and this is because this method gives carries out the calculation to a greater degree of accuracy. This larger bond length leads a lower potential energy gradient to the transition state, which gives a lower total energy for the transition state. Therefore the given values below will be the DFT/B3LYP/6-31G* and not the semi-empirical/AM1 method.

A typical sp3 C-C bond length is 1.54 Å and a typical sp2 C=C bond length is 1.34 Å<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref>. Within the optimised transition molecule above the C-C bond lengths are 1.40 Å and the C=C bond length is 1.38 Å. This suggests that the π-bonds are breaking in order for the C-C bonds to form and additionally to this a π-bond is forming in between the two sp3 carbons.

The Van der Waals radius of a C-atom is 1.72 Å<ref name="ref 3">M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer, D. G. Truhlar, 1995, 346, pp. 63-69.[http://pubs.acs.org/doi/abs/10.1021/jp8111556]</ref>. The partly formed C-C length is 2.27 Å, which is shorter than two times the the van der Waals radius of a carbon atom. Therefore the carbon atoms are within a close enough distance to each other to share electron density and yet the bond has not formed at this point.

From looking at the animations for the imaginary frequency above it can be seen that the formation of the two bonds between the termini of the reactants is synchronous. This is expected of the Diels-Alder reaction since the mechanism is concerted. However the lowest real frequency is asynchronous with respect to the two bonds forming between the termini.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12: Frontier molecular orbitals of the transition state '''
| '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| LUMO || [[Image:LUMO1TSdh08.jpg|100x100px]] || [[Image:LUMO2TSdh08.jpg|100x100px]]|| Symmetric || +0.023
|-
| HOMO || [[Image:HOMO1TSdh08.jpg|100x100px]] || [[Image:HOMO2TSdh08.jpg|100x100px]]|| Anti-symmetric || -0.323
|}

From examining the images from table 12 above it can be deduced that the LUMO of the transition state originates from the cis-butadiene's LUMO and the ethylene's HOMO, which represents the σ*-antibonding orbitals. However the HOMO of the transition state originates from the cis-butadiene's HOMO and the ethylene's LUMO, which represents the σ-bonding orbitals within the two single C-C bonds that are formed within the reaction.

==='''Cyclohexa-1,3-diene reaction with maleic anhydride'''===

The <jmol>
<jmolAppletButton>
<uploadedFileContents>FrozenTS22dh08.mol</uploadedFileContents>
<text>exo</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_frozenTS22dh08.mol</uploadedFileContents>
<text>endo</text>
<color>white</color>
</jmolAppletButton>
</jmol> transition states were drawn on Gaussview and then the optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while the distance of the forming bonds were frozen at 2.1Å between the respective carbons. After this the structures changed slightly (<jmol>
<jmolAppletButton>
<uploadedFileContents>TS2_afteropt1_exo1dh08.mol</uploadedFileContents>
<text>exo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_TS2_afteropt1dh08.mol</uploadedFileContents>
<text>endo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol>) and the resulting files had an optimisation and frequency calculation run on it using the TA (Berny) with using the semi-empirical/AM1 level of theory while the unfrozen option as changed to "derivative". After this the endo and exo optimised transition state structures were re-optimised as they were before but using the DFT/B3LYP/6-31G* level of theory instead with the additional keyword "Opt=NoEigen".

{| class="wikitable" border="1" align="center"
|+ '''Table 13: Comparison of the exo and endo transition states '''
! rowspan="2"|-!! colspan="2"|''exo'' !! colspan="2"|''endo''
|-
| Semi-empirical/AM1 || DFT/B3LYP/6-31G* || Semi-empirical/AM1 || DFT/B3LYP/6-31G*
|-
| align="center"|Structure ||align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>After_opt2_exodh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>Endo_TS2_afteropt2dh081.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| align="center"| Energy (a. u) ||align="center"| -0.05041975 ||align="center"| -612.67931095 ||align="center"| -0.05150470 ||align="center"| -612.68339684
|-
| C-C bond forming length (Å) ||align="center"| 2.17 ||align="center"| 2.29 ||align="center"| 2.16 ||align="center"| 2.27
|-align="center"
| Imaginary frequency (cm-1) ||align="center"| -812 ||align="center"| -448 ||align="center"| -806 ||align="center"| -447
|-align="center"
| Animation ||colspan="2"|[[Image:-812_SEdh08.gif]] || colspan="2"|[[Image:-806_cm-1dh08.gif]]
|}

Again the C-C bond forming length between the termini carbons is larger (by 0.11 and 0.12 Å) for the DFT/B3LYP/6-31G* level of theory and this is due the same reasons talked about above. However the C-C bond forming length between the termini carbons are very similar in both the endo and exo transition states (0.02 Å difference).

From looking at the animations for the imaginary frequency above it can also be seen that the formation of the two bonds between the carbons in this case as well is synchronous. Again this is expected of the Diels-Alder reaction since it is concerted.

{| align=center
|+ '''Strain diagrams for the endo and exo transition states'''
| [[Image:Exostraindh08.jpg|thumb|200x200px|'''Exo TS''']]
| [[Image:Strainendodh08.jpg|thumb|200x200px|'''Endo TS''']]
|}

From studying the energies in table 13 it can be seen that the endo transition state is more stable than the exo by 4.08589 x 10-3 a. u (10.73 KJ mol-1). However by looking at the strain of both transition states one would expect the exo TS to be more stable. This is because the distance between the cyclohexa-1,3-diene ring and the -(C=O)-O-(C=O)- maleic anhydride fragment is 2.99 Å in the endo TS and 3.03 Å in the exo TS. Thus the endo form would have more strain due to steric hindrance.

However within the exo TS the vicinal hydrogens on the CH2-CH2 (carbon numbers 18 and 21) fragment on the cyclohexa-1,3-diene ring are found below the plane and next to the maleic anhydride. This leads to steric hindrance is this is where the strain originates from within the exo TS. Therefore this disfavours the exo form, which makes the endo more stable.

Another factor making the endo TS more favourable is secondary orbital overlap. These orbitals contribute to the bonds forming and they interact with each other through space. Within the endo TS below there are 2 more orbitals interacting (secondary orbitals) than within the exo TS. This therefore makes the endo TS more stable and therefore more favoured.

[[Image:Secondary_orbital_shizzledh08.jpg|500x500px|center]]

{| class="wikitable" border="1" align="center"
|+ '''Table 14: Frontier molecular orbitals of the the transition state for the cyclohexa-1,3-diene reaction '''
! rowspan="2"|Molecular Orbital !! colspan="4"|''exo'' !! colspan="4"|''endo''
|- align="center"
| '''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''||'''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''
|- align="center"
| LUMO || [[Image:LUMO1exoh081.jpg|150x150px]]|| [[Image:LUMO2exodh081.jpg|150x150px]]|| Anti-symmetric || -0.078 || [[Image:LUMO1ENDOdh08.jpg|150x150px]]||[[Image:LUMO2ENDOdh08.jpg|150x150px]] || Anti-symmetric || -0.067
|- align="center"
| HOMO || [[Image:HOMO1exodh081.jpg|150x150px]]||[[Image:HOMO2exodh081.jpg|150x150px]] || Anti-symmetric|| -0.242 || [[Image:HOMO1ENDOdh08.jpg|150x150px]]|| [[Image:HOMO2ENDOdh08.jpg|150x150px]]||Anti-symmetric || -0.242
|}

Digital repositories:

exo: http://hdl.handle.net/10042/to-5596

endo: http://hdl.handle.net/10042/to-5597

From examining table 14 it is clear to see that both the HOMO and LUMO are anti-symmetric for both the transition states. The LUMO of the maleic anhydride fragment is the π* anti-bonding orbital of the C=C bond. Within the reaction then this has electrons transferred to it and the C=C bond breaks, which leaves just a C-C bond. The LUMOs for both transition states are both anti-bonding with respect to the two fragments. The HOMOs of both transitions state represent the σ-bonding that will be formed between the termini carbons on the products and reactants. This is most likely from the overlap of the cyclohexa-1,3-diene's HOMO and the maleic anhydride's LUMO.

=='''Conclusion'''==

='''References'''=
<references/>

Rep:Mod:hampluscheese

2010-11-10T22:33:17Z

Dh08: /* '''Cyclohexa-1,3-diene reaction with maleic anhydride''' */

='''Module 3:Physical Computational Chemistry Lab'''=

Douglas Hunt

=='''Objectives'''==

The aim of this experiment is to

=='''The Cope Rearrangement'''==

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangement] is a [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement |3,3|-sigmatropic rearrangement] that takes place through a single [http://en.wikipedia.org/wiki/Pericyclic pericyclic], concerted [http://en.wikipedia.org/wiki/Transition_state transition state]. Within this rearrangement a σ bond is formed in concert with another being broken within a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] π system.

The Cope rearrangement we will be studying will be that of 1,5-hexadiene, which is shown below.

[[Image:Cope_rearrangement_schemedh08.jpg|300x300px|center]]

The objective is to locate the low-energy minima and transition structures on the C6H10 potential energy surface and to determine the preferred reaction mechanism by using [http://en.wikipedia.org/wiki/GAUSSIAN Gaussian], with the [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)] methods.

==='''Optimisation of 1,5-hexadiene Conformers'''===

Rough <jmol>
<jmolAppletButton>
<uploadedFileContents>Anti_2_drawndh08.mol</uploadedFileContents>
<text>anti</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche3_drawndh08.mol</uploadedFileContents>
<text>gauche</text>
<color>white</color>
</jmolAppletButton>
</jmol> conformers of 1,5-hexadiene were drawn on Gaussview and their structures were tidied by the clean function. They were then further optimised by the HF/3-21G method and were identified which conformer they were from their point group and appendix 1 on the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3 physical lab page].

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 1: Optimisation and comparison of anti and gauche 1,5-hexadiene conformers using HF/3-21G methods
| ''' Conformer''' || '''Optimised Structure''' || '''Energy (Atomic Units)''' || '''[http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1 Energies (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (Anti2 - Gauche3)''' || '''Point group'''
|-
| '''Anti2''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Anti_2_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692535 || -231.69254 ||-6.08 x 105 || 1.2606 x 10-4 a. u|| Ci
|-
| '''Gauche3''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Gauche3_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692661 || -231.69266 || -6.08 x 105 || 3.31 x 10-1 KJ mol-1 || C1
|}

From studying table 1 it can be seen that the gauche conformer is 0.33 KJ mol-1 more stable than the anti conformation. One would assume that the anti2 conformer would be more stable due to [http://en.wikipedia.org/wiki/Steric steric] factors/repulsion within the gauche3 conformation. However within the gauche3 conformer there is a favourable donation of [http://en.wikipedia.org/wiki/Electron_density electron density] from the C=C π orbital to the C-H σ* orbital from the adjacent proton, which stabilises the conformer<ref name="ref 1">B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446.[http://www.informaworld.com/smpp/content~db=all?content=10.1080/00268970110081412]</ref>.

 

The anti2 conformer was then optimised using the B3LYP 6-31G* (6-31G (d)) method.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 2: Comparison of the anti2 conformer optimisations carried out at different calculation methods
| ''' Method''' || '''Structure and bond distances (Å)''' || '''Dihedral angle (C1 to C4)''' || '''Energy (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (HF/3-21G - DFT/B3LYP/6-31G)'''
|-
| HF/3-21G || [[Image:Anti2_opt1dh08.jpg|300x300px]] || 114.6 o || -231.692535 || -6.08 x 105 || 2.919171 a. u
|-
| DFT/B3LYP/6-31G* || [[Image:Anti2_opt2DH08.jpg|300x300px]] || 118.8 o || -234.611716 || -6.16 x 105 || 7.66 x 103 KJ mol-1
|}

From examining table 2 at first it is possible to see that the bond lengths are very similar to each other and that of the literature bond lengths<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref> (C1=C2: 1.340 Å, C2-C3: 1.508 Å and C3-C4: 1.538 Å). It is also possible to see that the C=C bond length is more accurate for the DFT/B3LYP/6-31G* method by it being 0.01 Å closer to the literature value. Another main difference in terms of geometry is that the DFT/B3LYP/6-31G* method gives a larger [http://en.wikipedia.org/wiki/Dihedral_angle dihedral angle] (and most likely more accurate) of 118.8o compared to that of 114.6o.

In terms of energy the DFT/B3LYP/6-31G* method produces a conformer of a higher stability by it possessing a lower energy with a difference of 7.66 x 103 KJ mol-1.

==='''Frequency analysis of 1,5-hexadiene Conformers'''===

[[Image:Vibrations_anti2dh08.jpg|right|300x300px]]

For the frequency analysis the DFT/B3LYP/6-31G* method was used.

Frequency analysis is essential within this calculation due to it being the second derivative of the potential energy surface. This means that if all the frequencies are all positive the system is at a minimum, if one of them is negative the optimised system is at a transition state, and if more than one are negative then the calculation has failed to find a critical point, which means the optimisation has failed. This therefore gives us a method of seeing whether the calculation has worked.

From looking to the image on the right it can deduced that the optimisation has taken place successfully by there being no negative vibrations and therefore the conformation is at a minimum.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 3: Energies under the thermochemistry section at 298.15 K and 0 K within the frequency logfiles'''
| '''Energies''' || '''298.15 K (atomic units)''' ||| '''0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (KJ mol-1)''' || '''Discussion'''
|-
| Sum of electronic and zero-point energies || -234.469177 || -234.466673 || -2.504 x 10-3 || -6.57 || This is the potential energy at 0 K, which includes the zero-point vibrational energy (E = Eelec + ZPE).

|-
| Sum of electronic and thermal energies || -234.461835 || -234.461085 || -7.5 x 10-4|| -1.97 || This is the energy at 298.15 K (0 K for 0 K calculation) and 1 atm of pressure, which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans).

|-
| Sum of electronic and thermal enthalpies || -234.460891 || -234.460141 || -7.5 x 10-4 || -1.97 || This contains an additional correction for RT (H = E + RT), which is especially important when looking at dissociation reactions.
|-
| Sum of electronic and thermal free energies || -234.500739 || -234.495931 || 4.808 x 10-3 || -12.62 || This includes the entropic contribution to the free energy (G = H - TS).
|}

{|align=right
|+
| [[Image:Anti2_freq_IRdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 298.15 K''']]
| [[Image:Anti2_freq_0Kdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 0 K''']]
|}

From looking at table 3 above it can be deduced that the anti2 conformer is more stable at room temperature to that of 0 K with respect to all the types of energies. The spectra also seem to not change much between the two temperatures but the main change is the change in the sum of electronic and thermal free energies, which is heavily dependent of temperature by it including the entropic contribution to the free energy (G = H - '''T'''S)

 
 
 
 

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

Within the Cope Rearrangement there are two possible transition states, the chair (C2h) and the boat (C2v), which are shown within [https://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_2 appendix two ].

==='''Comparison of optimisation methods for the chair transition state'''===

An <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2hdh08.mol</uploadedFileContents>
<text>allyl fragment</text>
<color>white</color>
</jmolAppletButton>
</jmol>, CH2CHCH2 was drawn on Gaussview and was optimized through the HF/3-21G method. After this the original allyl fragment was copied and pasted into a new window and place exactly above the first fragment with a bond distance of 2.20 Å between the terminal end carbons. This therefore produced the <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2h_fulldh08.mol</uploadedFileContents>
<text>chair transition structure.</text>
<color>white</color>
</jmolAppletButton>
</jmol>

Gaussian was then set up for an optimisation for the transition state with the additional key words "Opt=NoEigen", the optimisation set to TS (Berny), the force constants set to "once" and using the HF/3-21G method.

After this the "chair" transition state was optimised an alternative way by freezing the coordinates of the two sets of terminal carbons (where the bonds form/break during the rearrangement) within the Redundant Coord Editor on Gauusview.
Following this a second calculation was run where the freeze coordinates option was changed to "Derivative" and the force constants were set to "never".

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 4: Comparison of chair transition state optimisation methods
| || '''Optimised through a TS (Berny)''' || '''Optimised through a frozen coordinate method'''
|-
| '''Optimised TS Structure''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>C2h_optfreq_TS_Bdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| '''Terminal C-C bond length (Å)''' || 2.02 || 2.02
|-
|'''Energy (atomic units)''' || -231.61932247 || -231.61932190
|-
| '''ΔE, TS (Berny) - frozen coordinate''' || 5.7 x 10-7 a.u || 1.50 x 10-3 KJ mol-1
|-
| '''Imaginary frequency (cm-1)''' || -818 || -818
|-
| '''Animation''' ||[[Image:818dh08TSB.gif|350x350px]] || [[Image:818cm-1Fdh08.gif]]
|}

The data in table 4 shows that both methods give the same results and they both give the same transition state. They both produced the same energy with a negligible energy difference between them and the bond lengths between the terminal carbons are both 2.02 Å. In addition to this the imaginary frequencies were both -818cm-1 and within the animations above it is possible to see that when one C-C bond is forming the other is breaking. This means it is a asynchronous bond formation, which is expected for the Cope Rearrangement since it is a pericyclic reaction.

The only difference between the two optimisations is the way in which the transition states were calculated. This is due to the TS (Berny) method calculating the force constant matrix within the first optimisation step and is then being recalculated as the optimisation carries on. When the frozen coordinate method freezes the reaction coordinate in order for the rest of the structure to be optimised around it, which does not require the whole hessian (force constant matrix) to be calculated but only that the reaction coordinate is differentiated (second step). The main advantage of this, is that you do not need to calculate the entire force constant matrix and just need to differentiate the reaction coordinate in order to obtain a good enough guess for the initial force constant matrix. The frozen coordinate method will therefore take less time to compute the transition state, which can be very expensive within certain systems. However to carry out this method one needs to have an idea of how far apart the terminal carbons are.

==='''Boat transition state'''===

===='''QST2 optimisation'''====

The optimisation for the boat transition state had a different method and this was the QST2 method. This calculation interpolates between two structures (product and reactant) and tries to find a transition state between them.

The reactants and products were numbered in the same way. Therefore both 1,5-hexadiene molecules had their numbering manually changed for the product molecule so it corresponded to the numbering of the reactant if it rearranged. This is illustrated in the reaction scheme below.

[[Image:Boat_rerrangmentdh08.jpg|300x300px|center]]

The chk file for the optimized anti2 conformer was opened and copied into a new window. A second window was opened within the new window and the same anti2 conformer was copied into the new window. The numbering was changed within the second molecule so the two looked like they do in the image below.

[[Image:TS_(QST2)_numbering_1dh08.jpg|400x400px|center]]

The job type was set as "Opt+Freq" and "TS (QST2)" was chosen to optimise the transition state. However this failed and produced
a structure a bit like a <jmol>
<jmolAppletButton>
<uploadedFileContents>Trialone_BPdh08.mol</uploadedFileContents>
<text>chair transition structure </text>
<color>white</color>
</jmolAppletButton>
</jmol> but more dissociated. This is because it interpolated between the two structures and translated the top allyl fragment and therefore did not consider the possibility of a rotation around the central bonds.

To get around this the C-C-C-C dihedral angle (C2-C3-C4-C5) was set to 0 o and the C-C-C angle (C2-C3-C4 and C3-C4-C5 for the product molecule) was set to 100 o for both molecules. This gave rise to the two structures below.

[[Image:TS_(QST2)_numbering_2dh08.jpg|400x400px|center]]

After this the QST2 calculation was run again and the results in table 5 were produced. The imaginary frequency at -839 cm-1 shows that while one bond is forming the other one is breaking, which is again an asynchronous bond formation.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5: Data for the boat transition state from the QST2 optimisation method
| Optimised TS Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.14
|-
|Energy (atomic units) || -231.60280151
|-
| Energy (KJ mol-1) || -6.08 x 105
|-
| Imaginary frequency (cm-1) || -839
|-
| Animation ||[[Image:-839dh08.gif]]
|}

===='''Intrinsic Reaction Coordinate'''====

It is not possible to predict which product the chair and boat transition state will lead to. To solve this problem within Gaussian one can use the IRC method, which allows one to follow the minimum energy path from a transition structure down to its local minimum on a potential energy surface. This creates a succession of points by taking small geometry steps in the direction to where the gradient of the energy surface is at its steepest until the minimum in energy is reached.

The chk file for the optimized chair transition structure was opened and the job type selected was "IRC", the reaction coordinate was computed in the forward direction (since molecule is symmetric) with 50 points along the IRC. This was done with two type of force constants being calculated, once and always.

Within the "once" method the output showed no bond formation process by the 26th step and therefore the calculation was incomplete, since a complete 1,5-hexadiene structure was supposed to be formed. From looking at the IRC pathway it can be seen that the minimum on the potential energy surface was not reached at that point, since the points on the pathway did not form an asymptote and the gradient did not go close enough to zero.

For the "always" method the computation of force constants were carried out at every step. This therefore lets the IRC calculation distinguish whether it is moving along the minimum energy pathway while going down the potential energy surface. While this is happening it can make changes to the pathway if necessary. Within the "always" case the bond forming arrives at 1,5-hexadiene and this is backed up by the IRC pathway reaching a minimum within its energy by the gradient going close to zero and the pathway reaches an asymptote.

The structure produced through the always method had an energy of -231.69199434 atomic units, which is very close to the energy of the gauche2 conformer in the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 appendix 1]. This suggests that the reaction takes place through the chair transition state, which later produces the gauche2 conformer of 1,5-hexadiene.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 6: Comparison of IRC calculations for the chair structure
| || '''Calculate force constants once''' || '''Calculate force constants always'''
|-
| '''Number of points along IRC''' || 26 || 47
|-
| '''Structure at the last point of IRC''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Once_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Always_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|'''Energy (atomic units)''' || -231.68906923 || -231.69199434
|-
| '''ΔE, once - always''' || 2.92511 x 10-3 a.u || 7.68 KJ mol-1
|-
| '''IRC pathway''' || [[Image:One_pathwaydh08.jpg|400x400px]] || [[Image:Always_pathwaydh08.jpg|400x400px]]
|-
| '''IRC gradient''' || [[Image:One_gradientdh08.jpg|400x400px]]|| [[Image:Always_gradientdh08.jpg|400x400px]]
|}

 
 
 

===='''Comparison of activation energies for the transition states'''====

The HF/3-21G optimised chair and boat transition structures were both reoptimised using the B3LYP/6-31G* (B3LYP/6-31G (d)) method. This gives a more accurate minimisation however this method takes longer to run. The values for the zero-point energies and thermal energies were found within the thermochemistry section of the log file.

The values for the HF/3-21G reactant (anti2) optimisation were calculated and the B3LYP/6-31G* values were taken from the "Frequency analysis of 1,5-hexadiene Conformers" earlier in this module.

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 7: Energy data for chair, boat and the anti2 conformation from the HF/3-21G and B3LYP/6-31G* methods of optimisation '''
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466677
| align="center" | -231.461320
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450936
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402342
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539524
| align="center" | -231.532562
| align="center" | -234.611716
| align="center" | -234.469177
| align="center" | -234.461835
|}
Digital repositories

Chair:
HF/3-21G: http://hdl.handle.net/10042/to-5569
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5568

Boat:
HF/3-21G: http://hdl.handle.net/10042/to-5570
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5571

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 8: Summary of activation energies for the chair and boat conformations (ΔE = chair TS or boat TS minus anti2)'''
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| align="center" | 1 a.u = 627.509 kcal mol-1
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE, chair (a.u)'''
| align="center" | 0.072847
| align="center" | 0.071242
| align="center" | 0.054248
| align="center" | 0.052827
| align="center" | -
|-
| '''ΔE, chair (kcal mol-1)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE, boat (a. u)'''
| align="center" | 0.088588
| align="center" | 0.087259
| align="center" | 0.066835
| align="center" | 0.065827
| align="center" | -
|-
| '''ΔE, boat (kcal mol-1)'''
| align="center" | 55.59
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.31
| align="center" | 44.7 ± 2.0
|}

By examining table 8 it is clear to see that the B3LYP/6-31G* method gives values which are much closer to the experimental ones. This is due to the B3LYP/6-31G* giving a higher level of optimisation than the HF/3-21G method. Table 8 also shows that the chair transition state has a lower activation energy to that of the boat transition state by 8.16 kcal mol-1. This suggests that the Cope rearrangement mechanism would most likely proceed through the chair transition state to that of the boat. Therefore the reaction would take place from the anti2 conformer to the chair transition state and then onto the gauche2 conformer within the Cope rearrangement.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9: Geometric parameters for the optimised chair and boat structures '''
| || ''' HF/3-21G chair TS''' || ''' B3LYP/6-31G* chair TS''' || '''HF/3-21G boat TS''' || '''B3LYP/6-31G* boat TS'''
|-
| Optimised Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Chair_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Boat_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.02 (C5-C10 or C3-C14)|| 1.97 (C5-C10 or C3-C14) || 2.14 (C3-C4 or C1-C6) || 2.21 (C3-C4 or C1-C6)
|-
| Fragment angle || 120 o (C3-C1-C5) || 120 o (C3-C1-C5) || 122 o (C4-C5-C6) || 122 o (C4-C5-C6)
|-
| Dihedral angle || 55 o (C1-C3-C14-C11)|| 54 o (C1-C3-C14-C11)|| 0 o (C2-C1-C6-C5) || 0 o (C2-C1-C6-C5)
|-
| Numbering of carbons || [[Image:Numbering_chairdh08.jpg|200x200px]]|| [[Image:Numbering_chairdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]]
|}

From studying table 9 it can be seen that the C-C bond breaking/forming length has changed between the two different types of optimisation methods. However the angles between the bonds remains unchanged.

=='''The Diels Alder Cycloaddition'''==

The Diels-Alder reaction is a cycloaddition between a conjugated diene and a dienophile (alkene system), which forms a cyclohexene system. It is often referred to as a [4s+2s] cycloaddition and proceeds through a single cyclic, concerted transition state where 2 σ bonds are formed in between the termini carbons of the 2 conjugated п systems. This is shown in the reaction scheme below where ethene is reacting with cis-butadiene.

[[Image:Reaction_scheme2dh08.jpg|300x300px|center]]

The π orbitals of the dienophile form new σ bonds with the π orbitals of the diene. The number of π electrons involved determine whether the reaction proceeds in a concerted stereospecific fashion (allowed) or not (forbidden). The HOMO/LUMO of one fragment interacts with the HOMO/LUMO of the other fragment to form two new bonding and anti-bonding MOs. The nodal properties allow one to predict how the reaction will occur.

Semi-Empirical/AM1 method and the B3LYP/6-31G* method of optimisation will be used on Gaussian to determine particular properites of the reactants to predict the way in which the reaction will proceed.

==='''Reagents'''===

<jmol>
<jmolAppletButton>
<uploadedFileContents>Cis_butadienedh08.mol</uploadedFileContents>
<text>Cis-butadiene</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Ethenedh08.mol</uploadedFileContents>
<text>ethylene</text>
<color>white</color>
</jmolAppletButton>
</jmol> were optimised separately using the AM1 semi-empirical method on Gaussview.The cis-butadiene had an energy of 0.04879725 a. u (1.28 x 102 KJ mol-1) and ethylene had an energy of 0.02619027 a. u (68.76 KJ mol-1).

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10: Frontier molecular orbitals of cis-butadiene and ethylene '''
| '''Reagent''' || '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| cis-butadiene || LUMO || [[Image:LUMO1_cis_butadieneDH08.jpg|100x100px]] || [[Image:LUMO2_cis_butadienedh08.jpg|100x100px]]|| Symmetric || +0.017
|-
| cis-butadiene || HOMO || [[Image:HOMO1_cis_butadienedh08.jpg|100x100px]] || [[Image:HOMO2_cis_butadienedh08.jpg|100x100px]]|| Anti-symmetric || -0.343
|-
| ethylene || LUMO || [[Image:LUMO1_ethenedh08.jpg|100x100px]] || [[Image:LUMO2_ethenedh08.jpg|100x100px]]|| Anti-symmetric || +0.052
|-
| ethylene || HOMO || [[Image:HOMO1_ethenedh08.jpg|100x100px]] || [[Image:HOMO2_ethenedh08.jpg|100x100px]]|| Symmetric || -0.387
|}

 
The symmetries of the LUMO and HOMO orbitals for cis-butadiene are opposite of those for the ethylene. This therefore means that the only interactions that can take place are those between ethylene's LUMO and cis-butadiene's HOMO and in addition to this the interaction between cis-butadiene's LUMO and ethylene's HOMO. This is because orbital symmetry has to be conserved (the conservation of orbital symmetry states that the symmetry of an new orbital formed by the overlap of the 2 original orbitals must be the same as the symmetry of the initial orbitals).

==='''Transition State'''===

A <jmol>
<jmolAppletButton>
<uploadedFileContents>Bicyclic_startingdh08.mol</uploadedFileContents>
<text>bicyclo system</text>
<color>white</color>
</jmolAppletButton>
</jmol> was drawn and one of the -CH2-CH2- fragments were removed. Three C=C bonds were added and the previous one was taken away so a <jmol>
<jmolAppletButton>
<uploadedFileContents>Frozen_coorddh08.mol</uploadedFileContents>
<text>guess transition state</text>
<color>white</color>
</jmolAppletButton>
</jmol> was produced.

After this an optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while bonds between the termini of the fragments were frozen at 2.1Å. Following this the resulting file had two separate optimisation and frequency calculations run on it using the TS (Berny) with one using the semi-empirical/AM1 method and the other using the DFT/B3LYP/6-31G* level of theory. Both of these optimisations had the bonds unfrozen and the "derivative" option was chosen instead.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11: Comparison of two methods of optimsations for the transition state '''
| || '''semi-empirical/AM1 method''' || '''DFT/B3LYP/6-31G* method'''
|-
| Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>SE_AM1_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>DFT_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Energy (a. u) || 0.11165608 || -234.54389573
|-
| Bond lengths || [[Image:DistancesSEdh08.jpg|150x150px]]|| [[Image:DistancesDFTdh08.jpg|150x150px]]
|-
|Imaginary frequency (cm-1) || -957 || -525
|-
|Animation || [[Image:-957dh08.gif]] || [[Image:-525dh08.gif]]
|-
| Lowest real frequency (cm-1) || 148 || 136
|-
|Animation || [[Image: 148dh08.gif]] || [[Image: 136dh08.gif]]

|}

The C-C bond forming length between the termini carbons is 0.15 Å larger for the DFT/B3LYP/6-31G* level of theory and this is because this method gives carries out the calculation to a greater degree of accuracy. This larger bond length leads a lower potential energy gradient to the transition state, which gives a lower total energy for the transition state. Therefore the given values below will be the DFT/B3LYP/6-31G* and not the semi-empirical/AM1 method.

A typical sp3 C-C bond length is 1.54 Å and a typical sp2 C=C bond length is 1.34 Å<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref>. Within the optimised transition molecule above the C-C bond lengths are 1.40 Å and the C=C bond length is 1.38 Å. This suggests that the π-bonds are breaking in order for the C-C bonds to form and additionally to this a π-bond is forming in between the two sp3 carbons.

The Van der Waals radius of a C-atom is 1.72 Å<ref name="ref 3">M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer, D. G. Truhlar, 1995, 346, pp. 63-69.[http://pubs.acs.org/doi/abs/10.1021/jp8111556]</ref>. The partly formed C-C length is 2.27 Å, which is shorter than two times the the van der Waals radius of a carbon atom. Therefore the carbon atoms are within a close enough distance to each other to share electron density and yet the bond has not formed at this point.

From looking at the animations for the imaginary frequency above it can be seen that the formation of the two bonds between the termini of the reactants is synchronous. This is expected of the Diels-Alder reaction since the mechanism is concerted. However the lowest real frequency is asynchronous with respect to the two bonds forming between the termini.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12: Frontier molecular orbitals of the transition state '''
| '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| LUMO || [[Image:LUMO1TSdh08.jpg|100x100px]] || [[Image:LUMO2TSdh08.jpg|100x100px]]|| Symmetric || +0.023
|-
| HOMO || [[Image:HOMO1TSdh08.jpg|100x100px]] || [[Image:HOMO2TSdh08.jpg|100x100px]]|| Anti-symmetric || -0.323
|}

From examining the images from table 12 above it can be deduced that the LUMO of the transition state originates from the cis-butadiene's LUMO and the ethylene's HOMO, which represents the σ*-antibonding orbitals. However the HOMO of the transition state originates from the cis-butadiene's HOMO and the ethylene's LUMO, which represents the σ-bonding orbitals within the two single C-C bonds that are formed within the reaction.

==='''Cyclohexa-1,3-diene reaction with maleic anhydride'''===

The <jmol>
<jmolAppletButton>
<uploadedFileContents>FrozenTS22dh08.mol</uploadedFileContents>
<text>exo</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_frozenTS22dh08.mol</uploadedFileContents>
<text>endo</text>
<color>white</color>
</jmolAppletButton>
</jmol> transition states were drawn on Gaussview and then the optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while the distance of the forming bonds were frozen at 2.1Å between the respective carbons. After this the structures changed slightly (<jmol>
<jmolAppletButton>
<uploadedFileContents>TS2_afteropt1_exo1dh08.mol</uploadedFileContents>
<text>exo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_TS2_afteropt1dh08.mol</uploadedFileContents>
<text>endo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol>) and the resulting files had an optimisation and frequency calculation run on it using the TA (Berny) with using the semi-empirical/AM1 level of theory while the unfrozen option as changed to "derivative". After this the endo and exo optimised transition state structures were re-optimised as they were before but using the DFT/B3LYP/6-31G* level of theory instead with the additional keyword "Opt=NoEigen".

{| class="wikitable" border="1" align="center"
|+ '''Table 13: Comparison of the exo and endo transition states '''
! rowspan="2"|-!! colspan="2"|''exo'' !! colspan="2"|''endo''
|-
| Semi-empirical/AM1 || DFT/B3LYP/6-31G* || Semi-empirical/AM1 || DFT/B3LYP/6-31G*
|-
| align="center"|Structure ||align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>After_opt2_exodh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>Endo_TS2_afteropt2dh081.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| align="center"| Energy (a. u) ||align="center"| -0.05041975 ||align="center"| -612.67931095 ||align="center"| -0.05150470 ||align="center"| -612.68339684
|-
| C-C bond forming length (Å) ||align="center"| 2.17 ||align="center"| 2.29 ||align="center"| 2.16 ||align="center"| 2.27
|-align="center"
| Imaginary frequency (cm-1) ||align="center"| -812 ||align="center"| -448 ||align="center"| -806 ||align="center"| -447
|-align="center"
| Animation ||colspan="2"|[[Image:-812_SEdh08.gif]] || colspan="2"|[[Image:-806_cm-1dh08.gif]]
|}

Again the C-C bond forming length between the termini carbons is larger (by 0.11 and 0.12 Å) for the DFT/B3LYP/6-31G* level of theory and this is due the same reasons talked about above. However the C-C bond forming length between the termini carbons are very similar in both the endo and exo transition states (0.02 Å difference).

From looking at the animations for the imaginary frequency above it can also be seen that the formation of the two bonds between the carbons in this case as well is synchronous. Again this is expected of the Diels-Alder reaction since it is concerted.

{| align=center
|+ '''Strain diagrams for the endo and exo transition states'''
| [[Image:Exostraindh08.jpg|thumb|200x200px|'''Exo TS''']]
| [[Image:Strainendodh08.jpg|thumb|200x200px|'''Endo TS''']]
|}

From studying the energies in table 13 it can be seen that the endo transition state is more stable than the exo by 4.08589 x 10-3 a. u (10.73 KJ mol-1). However by looking at the strain of both transition states one would expect the exo TS to be more stable. This is because the distance between the cyclohexa-1,3-diene ring and the -(C=O)-O-(C=O)- maleic anhydride fragment is 2.99 Å in the endo TS and 3.03 Å in the exo TS. Thus the endo form would have more strain due to steric hindrance.

However within the exo TS the vicinal hydrogens on the CH2-CH2 (carbon numbers 18 and 21) fragment on the cyclohexa-1,3-diene ring are found below the plane and next to the maleic anhydride. This leads to steric hindrance is this is where the strain originates from within the exo TS. Therefore this disfavours the exo form, which makes the endo more stable.

Another factor making the endo TS more favourable is secondary orbital overlap. These orbitals contribute to the bonds forming and they interact with each other through space. Within the endo TS below there are 2 more orbitals interacting (secondary orbitals) than within the exo TS. This therefore makes the endo TS more stable and therefore more favoured.

[[Image:Secondary_orbital_shizzledh08.jpg|500x500px|center]]

{| class="wikitable" border="1" align="center"
|+ '''Table 14: Frontier molecular orbitals of the the transition state for the cyclohexa-1,3-diene reaction '''
! rowspan="2"|Molecular Orbital !! colspan="4"|''exo'' !! colspan="4"|''endo''
|- align="center"
| '''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''||'''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''
|- align="center"
| LUMO || [[Image:LUMO1exoh081.jpg|150x150px]]|| [[Image:LUMO2exodh081.jpg|150x150px]]|| Anti-symmetric || -0.078 || [[Image:LUMO1ENDOdh08.jpg|150x150px]]||[[Image:LUMO2ENDOdh08.jpg|150x150px]] || Anti-symmetric || -0.067
|- align="center"
| HOMO || [[Image:HOMO1exodh081.jpg|150x150px]]||[[Image:HOMO2exodh081.jpg|150x150px]] || Anti-symmetric|| -0.242 || [[Image:HOMO1ENDOdh08.jpg|150x150px]]|| [[Image:HOMO2ENDOdh08.jpg|150x150px]]||Anti-symmetric || -0.242
|}

Digital repositories:

exo: http://hdl.handle.net/10042/to-5596

endo: http://hdl.handle.net/10042/to-5597

From examining table 14 it is clear to see that both the HOMO and LUMO are anti-symmetric for both the transition states. The LUMO of the maleic anhydride fragment is the π* anti-bonding orbital of the C=C bond. Within the reaction then this has electrons transferred to it and the C=C bond breaks, which leaves just a C-C bond. The LUMOs for both transition states are both anti-bonding with respect to the two fragments. The HOMOs of both transitions state represent the σ-bonding that will be formed between the termini carbons on the products and reactants. This is most likely from the overlap of the cyclohexa-1,3-diene's HOMO and the maleic anhydride's LUMO.

=='''Conclusion'''==

='''References'''=
<references/>

Rep:Mod:hampluscheese

2010-11-10T22:29:07Z

Dh08: /* '''Cyclohexa-1,3-diene reaction with maleic anhydride''' */

='''Module 3:Physical Computational Chemistry Lab'''=

Douglas Hunt

=='''Objectives'''==

The aim of this experiment is to

=='''The Cope Rearrangement'''==

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangement] is a [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement |3,3|-sigmatropic rearrangement] that takes place through a single [http://en.wikipedia.org/wiki/Pericyclic pericyclic], concerted [http://en.wikipedia.org/wiki/Transition_state transition state]. Within this rearrangement a σ bond is formed in concert with another being broken within a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] π system.

The Cope rearrangement we will be studying will be that of 1,5-hexadiene, which is shown below.

[[Image:Cope_rearrangement_schemedh08.jpg|300x300px|center]]

The objective is to locate the low-energy minima and transition structures on the C6H10 potential energy surface and to determine the preferred reaction mechanism by using [http://en.wikipedia.org/wiki/GAUSSIAN Gaussian], with the [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)] methods.

==='''Optimisation of 1,5-hexadiene Conformers'''===

Rough <jmol>
<jmolAppletButton>
<uploadedFileContents>Anti_2_drawndh08.mol</uploadedFileContents>
<text>anti</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche3_drawndh08.mol</uploadedFileContents>
<text>gauche</text>
<color>white</color>
</jmolAppletButton>
</jmol> conformers of 1,5-hexadiene were drawn on Gaussview and their structures were tidied by the clean function. They were then further optimised by the HF/3-21G method and were identified which conformer they were from their point group and appendix 1 on the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3 physical lab page].

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 1: Optimisation and comparison of anti and gauche 1,5-hexadiene conformers using HF/3-21G methods
| ''' Conformer''' || '''Optimised Structure''' || '''Energy (Atomic Units)''' || '''[http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1 Energies (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (Anti2 - Gauche3)''' || '''Point group'''
|-
| '''Anti2''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Anti_2_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692535 || -231.69254 ||-6.08 x 105 || 1.2606 x 10-4 a. u|| Ci
|-
| '''Gauche3''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Gauche3_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692661 || -231.69266 || -6.08 x 105 || 3.31 x 10-1 KJ mol-1 || C1
|}

From studying table 1 it can be seen that the gauche conformer is 0.33 KJ mol-1 more stable than the anti conformation. One would assume that the anti2 conformer would be more stable due to [http://en.wikipedia.org/wiki/Steric steric] factors/repulsion within the gauche3 conformation. However within the gauche3 conformer there is a favourable donation of [http://en.wikipedia.org/wiki/Electron_density electron density] from the C=C π orbital to the C-H σ* orbital from the adjacent proton, which stabilises the conformer<ref name="ref 1">B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446.[http://www.informaworld.com/smpp/content~db=all?content=10.1080/00268970110081412]</ref>.

 

The anti2 conformer was then optimised using the B3LYP 6-31G* (6-31G (d)) method.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 2: Comparison of the anti2 conformer optimisations carried out at different calculation methods
| ''' Method''' || '''Structure and bond distances (Å)''' || '''Dihedral angle (C1 to C4)''' || '''Energy (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (HF/3-21G - DFT/B3LYP/6-31G)'''
|-
| HF/3-21G || [[Image:Anti2_opt1dh08.jpg|300x300px]] || 114.6 o || -231.692535 || -6.08 x 105 || 2.919171 a. u
|-
| DFT/B3LYP/6-31G* || [[Image:Anti2_opt2DH08.jpg|300x300px]] || 118.8 o || -234.611716 || -6.16 x 105 || 7.66 x 103 KJ mol-1
|}

From examining table 2 at first it is possible to see that the bond lengths are very similar to each other and that of the literature bond lengths<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref> (C1=C2: 1.340 Å, C2-C3: 1.508 Å and C3-C4: 1.538 Å). It is also possible to see that the C=C bond length is more accurate for the DFT/B3LYP/6-31G* method by it being 0.01 Å closer to the literature value. Another main difference in terms of geometry is that the DFT/B3LYP/6-31G* method gives a larger [http://en.wikipedia.org/wiki/Dihedral_angle dihedral angle] (and most likely more accurate) of 118.8o compared to that of 114.6o.

In terms of energy the DFT/B3LYP/6-31G* method produces a conformer of a higher stability by it possessing a lower energy with a difference of 7.66 x 103 KJ mol-1.

==='''Frequency analysis of 1,5-hexadiene Conformers'''===

[[Image:Vibrations_anti2dh08.jpg|right|300x300px]]

For the frequency analysis the DFT/B3LYP/6-31G* method was used.

Frequency analysis is essential within this calculation due to it being the second derivative of the potential energy surface. This means that if all the frequencies are all positive the system is at a minimum, if one of them is negative the optimised system is at a transition state, and if more than one are negative then the calculation has failed to find a critical point, which means the optimisation has failed. This therefore gives us a method of seeing whether the calculation has worked.

From looking to the image on the right it can deduced that the optimisation has taken place successfully by there being no negative vibrations and therefore the conformation is at a minimum.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 3: Energies under the thermochemistry section at 298.15 K and 0 K within the frequency logfiles'''
| '''Energies''' || '''298.15 K (atomic units)''' ||| '''0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (KJ mol-1)''' || '''Discussion'''
|-
| Sum of electronic and zero-point energies || -234.469177 || -234.466673 || -2.504 x 10-3 || -6.57 || This is the potential energy at 0 K, which includes the zero-point vibrational energy (E = Eelec + ZPE).

|-
| Sum of electronic and thermal energies || -234.461835 || -234.461085 || -7.5 x 10-4|| -1.97 || This is the energy at 298.15 K (0 K for 0 K calculation) and 1 atm of pressure, which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans).

|-
| Sum of electronic and thermal enthalpies || -234.460891 || -234.460141 || -7.5 x 10-4 || -1.97 || This contains an additional correction for RT (H = E + RT), which is especially important when looking at dissociation reactions.
|-
| Sum of electronic and thermal free energies || -234.500739 || -234.495931 || 4.808 x 10-3 || -12.62 || This includes the entropic contribution to the free energy (G = H - TS).
|}

{|align=right
|+
| [[Image:Anti2_freq_IRdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 298.15 K''']]
| [[Image:Anti2_freq_0Kdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 0 K''']]
|}

From looking at table 3 above it can be deduced that the anti2 conformer is more stable at room temperature to that of 0 K with respect to all the types of energies. The spectra also seem to not change much between the two temperatures but the main change is the change in the sum of electronic and thermal free energies, which is heavily dependent of temperature by it including the entropic contribution to the free energy (G = H - '''T'''S)

 
 
 
 

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

Within the Cope Rearrangement there are two possible transition states, the chair (C2h) and the boat (C2v), which are shown within [https://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_2 appendix two ].

==='''Comparison of optimisation methods for the chair transition state'''===

An <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2hdh08.mol</uploadedFileContents>
<text>allyl fragment</text>
<color>white</color>
</jmolAppletButton>
</jmol>, CH2CHCH2 was drawn on Gaussview and was optimized through the HF/3-21G method. After this the original allyl fragment was copied and pasted into a new window and place exactly above the first fragment with a bond distance of 2.20 Å between the terminal end carbons. This therefore produced the <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2h_fulldh08.mol</uploadedFileContents>
<text>chair transition structure.</text>
<color>white</color>
</jmolAppletButton>
</jmol>

Gaussian was then set up for an optimisation for the transition state with the additional key words "Opt=NoEigen", the optimisation set to TS (Berny), the force constants set to "once" and using the HF/3-21G method.

After this the "chair" transition state was optimised an alternative way by freezing the coordinates of the two sets of terminal carbons (where the bonds form/break during the rearrangement) within the Redundant Coord Editor on Gauusview.
Following this a second calculation was run where the freeze coordinates option was changed to "Derivative" and the force constants were set to "never".

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 4: Comparison of chair transition state optimisation methods
| || '''Optimised through a TS (Berny)''' || '''Optimised through a frozen coordinate method'''
|-
| '''Optimised TS Structure''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>C2h_optfreq_TS_Bdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| '''Terminal C-C bond length (Å)''' || 2.02 || 2.02
|-
|'''Energy (atomic units)''' || -231.61932247 || -231.61932190
|-
| '''ΔE, TS (Berny) - frozen coordinate''' || 5.7 x 10-7 a.u || 1.50 x 10-3 KJ mol-1
|-
| '''Imaginary frequency (cm-1)''' || -818 || -818
|-
| '''Animation''' ||[[Image:818dh08TSB.gif|350x350px]] || [[Image:818cm-1Fdh08.gif]]
|}

The data in table 4 shows that both methods give the same results and they both give the same transition state. They both produced the same energy with a negligible energy difference between them and the bond lengths between the terminal carbons are both 2.02 Å. In addition to this the imaginary frequencies were both -818cm-1 and within the animations above it is possible to see that when one C-C bond is forming the other is breaking. This means it is a asynchronous bond formation, which is expected for the Cope Rearrangement since it is a pericyclic reaction.

The only difference between the two optimisations is the way in which the transition states were calculated. This is due to the TS (Berny) method calculating the force constant matrix within the first optimisation step and is then being recalculated as the optimisation carries on. When the frozen coordinate method freezes the reaction coordinate in order for the rest of the structure to be optimised around it, which does not require the whole hessian (force constant matrix) to be calculated but only that the reaction coordinate is differentiated (second step). The main advantage of this, is that you do not need to calculate the entire force constant matrix and just need to differentiate the reaction coordinate in order to obtain a good enough guess for the initial force constant matrix. The frozen coordinate method will therefore take less time to compute the transition state, which can be very expensive within certain systems. However to carry out this method one needs to have an idea of how far apart the terminal carbons are.

==='''Boat transition state'''===

===='''QST2 optimisation'''====

The optimisation for the boat transition state had a different method and this was the QST2 method. This calculation interpolates between two structures (product and reactant) and tries to find a transition state between them.

The reactants and products were numbered in the same way. Therefore both 1,5-hexadiene molecules had their numbering manually changed for the product molecule so it corresponded to the numbering of the reactant if it rearranged. This is illustrated in the reaction scheme below.

[[Image:Boat_rerrangmentdh08.jpg|300x300px|center]]

The chk file for the optimized anti2 conformer was opened and copied into a new window. A second window was opened within the new window and the same anti2 conformer was copied into the new window. The numbering was changed within the second molecule so the two looked like they do in the image below.

[[Image:TS_(QST2)_numbering_1dh08.jpg|400x400px|center]]

The job type was set as "Opt+Freq" and "TS (QST2)" was chosen to optimise the transition state. However this failed and produced
a structure a bit like a <jmol>
<jmolAppletButton>
<uploadedFileContents>Trialone_BPdh08.mol</uploadedFileContents>
<text>chair transition structure </text>
<color>white</color>
</jmolAppletButton>
</jmol> but more dissociated. This is because it interpolated between the two structures and translated the top allyl fragment and therefore did not consider the possibility of a rotation around the central bonds.

To get around this the C-C-C-C dihedral angle (C2-C3-C4-C5) was set to 0 o and the C-C-C angle (C2-C3-C4 and C3-C4-C5 for the product molecule) was set to 100 o for both molecules. This gave rise to the two structures below.

[[Image:TS_(QST2)_numbering_2dh08.jpg|400x400px|center]]

After this the QST2 calculation was run again and the results in table 5 were produced. The imaginary frequency at -839 cm-1 shows that while one bond is forming the other one is breaking, which is again an asynchronous bond formation.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5: Data for the boat transition state from the QST2 optimisation method
| Optimised TS Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.14
|-
|Energy (atomic units) || -231.60280151
|-
| Energy (KJ mol-1) || -6.08 x 105
|-
| Imaginary frequency (cm-1) || -839
|-
| Animation ||[[Image:-839dh08.gif]]
|}

===='''Intrinsic Reaction Coordinate'''====

It is not possible to predict which product the chair and boat transition state will lead to. To solve this problem within Gaussian one can use the IRC method, which allows one to follow the minimum energy path from a transition structure down to its local minimum on a potential energy surface. This creates a succession of points by taking small geometry steps in the direction to where the gradient of the energy surface is at its steepest until the minimum in energy is reached.

The chk file for the optimized chair transition structure was opened and the job type selected was "IRC", the reaction coordinate was computed in the forward direction (since molecule is symmetric) with 50 points along the IRC. This was done with two type of force constants being calculated, once and always.

Within the "once" method the output showed no bond formation process by the 26th step and therefore the calculation was incomplete, since a complete 1,5-hexadiene structure was supposed to be formed. From looking at the IRC pathway it can be seen that the minimum on the potential energy surface was not reached at that point, since the points on the pathway did not form an asymptote and the gradient did not go close enough to zero.

For the "always" method the computation of force constants were carried out at every step. This therefore lets the IRC calculation distinguish whether it is moving along the minimum energy pathway while going down the potential energy surface. While this is happening it can make changes to the pathway if necessary. Within the "always" case the bond forming arrives at 1,5-hexadiene and this is backed up by the IRC pathway reaching a minimum within its energy by the gradient going close to zero and the pathway reaches an asymptote.

The structure produced through the always method had an energy of -231.69199434 atomic units, which is very close to the energy of the gauche2 conformer in the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 appendix 1]. This suggests that the reaction takes place through the chair transition state, which later produces the gauche2 conformer of 1,5-hexadiene.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 6: Comparison of IRC calculations for the chair structure
| || '''Calculate force constants once''' || '''Calculate force constants always'''
|-
| '''Number of points along IRC''' || 26 || 47
|-
| '''Structure at the last point of IRC''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Once_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Always_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|'''Energy (atomic units)''' || -231.68906923 || -231.69199434
|-
| '''ΔE, once - always''' || 2.92511 x 10-3 a.u || 7.68 KJ mol-1
|-
| '''IRC pathway''' || [[Image:One_pathwaydh08.jpg|400x400px]] || [[Image:Always_pathwaydh08.jpg|400x400px]]
|-
| '''IRC gradient''' || [[Image:One_gradientdh08.jpg|400x400px]]|| [[Image:Always_gradientdh08.jpg|400x400px]]
|}

 
 
 

===='''Comparison of activation energies for the transition states'''====

The HF/3-21G optimised chair and boat transition structures were both reoptimised using the B3LYP/6-31G* (B3LYP/6-31G (d)) method. This gives a more accurate minimisation however this method takes longer to run. The values for the zero-point energies and thermal energies were found within the thermochemistry section of the log file.

The values for the HF/3-21G reactant (anti2) optimisation were calculated and the B3LYP/6-31G* values were taken from the "Frequency analysis of 1,5-hexadiene Conformers" earlier in this module.

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 7: Energy data for chair, boat and the anti2 conformation from the HF/3-21G and B3LYP/6-31G* methods of optimisation '''
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466677
| align="center" | -231.461320
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450936
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402342
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539524
| align="center" | -231.532562
| align="center" | -234.611716
| align="center" | -234.469177
| align="center" | -234.461835
|}
Digital repositories

Chair:
HF/3-21G: http://hdl.handle.net/10042/to-5569
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5568

Boat:
HF/3-21G: http://hdl.handle.net/10042/to-5570
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5571

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 8: Summary of activation energies for the chair and boat conformations (ΔE = chair TS or boat TS minus anti2)'''
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| align="center" | 1 a.u = 627.509 kcal mol-1
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE, chair (a.u)'''
| align="center" | 0.072847
| align="center" | 0.071242
| align="center" | 0.054248
| align="center" | 0.052827
| align="center" | -
|-
| '''ΔE, chair (kcal mol-1)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE, boat (a. u)'''
| align="center" | 0.088588
| align="center" | 0.087259
| align="center" | 0.066835
| align="center" | 0.065827
| align="center" | -
|-
| '''ΔE, boat (kcal mol-1)'''
| align="center" | 55.59
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.31
| align="center" | 44.7 ± 2.0
|}

By examining table 8 it is clear to see that the B3LYP/6-31G* method gives values which are much closer to the experimental ones. This is due to the B3LYP/6-31G* giving a higher level of optimisation than the HF/3-21G method. Table 8 also shows that the chair transition state has a lower activation energy to that of the boat transition state by 8.16 kcal mol-1. This suggests that the Cope rearrangement mechanism would most likely proceed through the chair transition state to that of the boat. Therefore the reaction would take place from the anti2 conformer to the chair transition state and then onto the gauche2 conformer within the Cope rearrangement.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9: Geometric parameters for the optimised chair and boat structures '''
| || ''' HF/3-21G chair TS''' || ''' B3LYP/6-31G* chair TS''' || '''HF/3-21G boat TS''' || '''B3LYP/6-31G* boat TS'''
|-
| Optimised Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Chair_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Boat_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.02 (C5-C10 or C3-C14)|| 1.97 (C5-C10 or C3-C14) || 2.14 (C3-C4 or C1-C6) || 2.21 (C3-C4 or C1-C6)
|-
| Fragment angle || 120 o (C3-C1-C5) || 120 o (C3-C1-C5) || 122 o (C4-C5-C6) || 122 o (C4-C5-C6)
|-
| Dihedral angle || 55 o (C1-C3-C14-C11)|| 54 o (C1-C3-C14-C11)|| 0 o (C2-C1-C6-C5) || 0 o (C2-C1-C6-C5)
|-
| Numbering of carbons || [[Image:Numbering_chairdh08.jpg|200x200px]]|| [[Image:Numbering_chairdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]]
|}

From studying table 9 it can be seen that the C-C bond breaking/forming length has changed between the two different types of optimisation methods. However the angles between the bonds remains unchanged.

=='''The Diels Alder Cycloaddition'''==

The Diels-Alder reaction is a cycloaddition between a conjugated diene and a dienophile (alkene system), which forms a cyclohexene system. It is often referred to as a [4s+2s] cycloaddition and proceeds through a single cyclic, concerted transition state where 2 σ bonds are formed in between the termini carbons of the 2 conjugated п systems. This is shown in the reaction scheme below where ethene is reacting with cis-butadiene.

[[Image:Reaction_scheme2dh08.jpg|300x300px|center]]

The π orbitals of the dienophile form new σ bonds with the π orbitals of the diene. The number of π electrons involved determine whether the reaction proceeds in a concerted stereospecific fashion (allowed) or not (forbidden). The HOMO/LUMO of one fragment interacts with the HOMO/LUMO of the other fragment to form two new bonding and anti-bonding MOs. The nodal properties allow one to predict how the reaction will occur.

Semi-Empirical/AM1 method and the B3LYP/6-31G* method of optimisation will be used on Gaussian to determine particular properites of the reactants to predict the way in which the reaction will proceed.

==='''Reagents'''===

<jmol>
<jmolAppletButton>
<uploadedFileContents>Cis_butadienedh08.mol</uploadedFileContents>
<text>Cis-butadiene</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Ethenedh08.mol</uploadedFileContents>
<text>ethylene</text>
<color>white</color>
</jmolAppletButton>
</jmol> were optimised separately using the AM1 semi-empirical method on Gaussview.The cis-butadiene had an energy of 0.04879725 a. u (1.28 x 102 KJ mol-1) and ethylene had an energy of 0.02619027 a. u (68.76 KJ mol-1).

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10: Frontier molecular orbitals of cis-butadiene and ethylene '''
| '''Reagent''' || '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| cis-butadiene || LUMO || [[Image:LUMO1_cis_butadieneDH08.jpg|100x100px]] || [[Image:LUMO2_cis_butadienedh08.jpg|100x100px]]|| Symmetric || +0.017
|-
| cis-butadiene || HOMO || [[Image:HOMO1_cis_butadienedh08.jpg|100x100px]] || [[Image:HOMO2_cis_butadienedh08.jpg|100x100px]]|| Anti-symmetric || -0.343
|-
| ethylene || LUMO || [[Image:LUMO1_ethenedh08.jpg|100x100px]] || [[Image:LUMO2_ethenedh08.jpg|100x100px]]|| Anti-symmetric || +0.052
|-
| ethylene || HOMO || [[Image:HOMO1_ethenedh08.jpg|100x100px]] || [[Image:HOMO2_ethenedh08.jpg|100x100px]]|| Symmetric || -0.387
|}

 
The symmetries of the LUMO and HOMO orbitals for cis-butadiene are opposite of those for the ethylene. This therefore means that the only interactions that can take place are those between ethylene's LUMO and cis-butadiene's HOMO and in addition to this the interaction between cis-butadiene's LUMO and ethylene's HOMO. This is because orbital symmetry has to be conserved (the conservation of orbital symmetry states that the symmetry of an new orbital formed by the overlap of the 2 original orbitals must be the same as the symmetry of the initial orbitals).

==='''Transition State'''===

A <jmol>
<jmolAppletButton>
<uploadedFileContents>Bicyclic_startingdh08.mol</uploadedFileContents>
<text>bicyclo system</text>
<color>white</color>
</jmolAppletButton>
</jmol> was drawn and one of the -CH2-CH2- fragments were removed. Three C=C bonds were added and the previous one was taken away so a <jmol>
<jmolAppletButton>
<uploadedFileContents>Frozen_coorddh08.mol</uploadedFileContents>
<text>guess transition state</text>
<color>white</color>
</jmolAppletButton>
</jmol> was produced.

After this an optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while bonds between the termini of the fragments were frozen at 2.1Å. Following this the resulting file had two separate optimisation and frequency calculations run on it using the TS (Berny) with one using the semi-empirical/AM1 method and the other using the DFT/B3LYP/6-31G* level of theory. Both of these optimisations had the bonds unfrozen and the "derivative" option was chosen instead.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11: Comparison of two methods of optimsations for the transition state '''
| || '''semi-empirical/AM1 method''' || '''DFT/B3LYP/6-31G* method'''
|-
| Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>SE_AM1_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>DFT_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Energy (a. u) || 0.11165608 || -234.54389573
|-
| Bond lengths || [[Image:DistancesSEdh08.jpg|150x150px]]|| [[Image:DistancesDFTdh08.jpg|150x150px]]
|-
|Imaginary frequency (cm-1) || -957 || -525
|-
|Animation || [[Image:-957dh08.gif]] || [[Image:-525dh08.gif]]
|-
| Lowest real frequency (cm-1) || 148 || 136
|-
|Animation || [[Image: 148dh08.gif]] || [[Image: 136dh08.gif]]

|}

The C-C bond forming length between the termini carbons is 0.15 Å larger for the DFT/B3LYP/6-31G* level of theory and this is because this method gives carries out the calculation to a greater degree of accuracy. This larger bond length leads a lower potential energy gradient to the transition state, which gives a lower total energy for the transition state. Therefore the given values below will be the DFT/B3LYP/6-31G* and not the semi-empirical/AM1 method.

A typical sp3 C-C bond length is 1.54 Å and a typical sp2 C=C bond length is 1.34 Å<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref>. Within the optimised transition molecule above the C-C bond lengths are 1.40 Å and the C=C bond length is 1.38 Å. This suggests that the π-bonds are breaking in order for the C-C bonds to form and additionally to this a π-bond is forming in between the two sp3 carbons.

The Van der Waals radius of a C-atom is 1.72 Å<ref name="ref 3">M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer, D. G. Truhlar, 1995, 346, pp. 63-69.[http://pubs.acs.org/doi/abs/10.1021/jp8111556]</ref>. The partly formed C-C length is 2.27 Å, which is shorter than two times the the van der Waals radius of a carbon atom. Therefore the carbon atoms are within a close enough distance to each other to share electron density and yet the bond has not formed at this point.

From looking at the animations for the imaginary frequency above it can be seen that the formation of the two bonds between the termini of the reactants is synchronous. This is expected of the Diels-Alder reaction since the mechanism is concerted. However the lowest real frequency is asynchronous with respect to the two bonds forming between the termini.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12: Frontier molecular orbitals of the transition state '''
| '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| LUMO || [[Image:LUMO1TSdh08.jpg|100x100px]] || [[Image:LUMO2TSdh08.jpg|100x100px]]|| Symmetric || +0.023
|-
| HOMO || [[Image:HOMO1TSdh08.jpg|100x100px]] || [[Image:HOMO2TSdh08.jpg|100x100px]]|| Anti-symmetric || -0.323
|}

From examining the images from table 12 above it can be deduced that the LUMO of the transition state originates from the cis-butadiene's LUMO and the ethylene's HOMO, which represents the σ*-antibonding orbitals. However the HOMO of the transition state originates from the cis-butadiene's HOMO and the ethylene's LUMO, which represents the σ-bonding orbitals within the two single C-C bonds that are formed within the reaction.

==='''Cyclohexa-1,3-diene reaction with maleic anhydride'''===

The <jmol>
<jmolAppletButton>
<uploadedFileContents>FrozenTS22dh08.mol</uploadedFileContents>
<text>exo</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_frozenTS22dh08.mol</uploadedFileContents>
<text>endo</text>
<color>white</color>
</jmolAppletButton>
</jmol> transition states were drawn on Gaussview and then the optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while the distance of the forming bonds were frozen at 2.1Å between the respective carbons. After this the structures changed slightly (<jmol>
<jmolAppletButton>
<uploadedFileContents>TS2_afteropt1_exo1dh08.mol</uploadedFileContents>
<text>exo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_TS2_afteropt1dh08.mol</uploadedFileContents>
<text>endo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol>) and the resulting files had an optimisation and frequency calculation run on it using the TA (Berny) with using the semi-empirical/AM1 level of theory while the unfrozen option as changed to "derivative". After this the endo and exo optimised transition state structures were re-optimised as they were before but using the DFT/B3LYP/6-31G* level of theory instead with the additional keyword "Opt=NoEigen".

{| class="wikitable" border="1" align="center"
|+ '''Table 13: Comparison of the exo and endo transition states '''
! rowspan="2"|-!! colspan="2"|''exo'' !! colspan="2"|''endo''
|-
| Semi-empirical/AM1 || DFT/B3LYP/6-31G* || Semi-empirical/AM1 || DFT/B3LYP/6-31G*
|-
| align="center"|Structure ||align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>After_opt2_exodh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>Endo_TS2_afteropt2dh081.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| align="center"| Energy (a. u) ||align="center"| -0.05041975 ||align="center"| -612.67931095 ||align="center"| -0.05150470 ||align="center"| -612.68339684
|-
| C-C bond forming length (Å) ||align="center"| 2.17 ||align="center"| 2.29 ||align="center"| 2.16 ||align="center"| 2.27
|-align="center"
| Imaginary frequency (cm-1) ||align="center"| -812 ||align="center"| -448 ||align="center"| -806 ||align="center"| -447
|-align="center"
| Animation ||colspan="2"|[[Image:-812_SEdh08.gif]] || colspan="2"|[[Image:-806_cm-1dh08.gif]]
|}

Again the C-C bond forming length between the termini carbons is larger (by 0.11 and 0.12 Å) for the DFT/B3LYP/6-31G* level of theory and this is due the same reasons talked about above. However the C-C bond forming length between the termini carbons are very similar in both the endo and exo transition states (0.02 Å difference).

From looking at the animations for the imaginary frequency above it can also be seen that the formation of the two bonds between the carbons in this case as well is synchronous. Again this is expected of the Diels-Alder reaction since it is concerted.

{| align=center
|+ '''Strain diagrams for the endo and exo transition states'''
| [[Image:Exostraindh08.jpg|thumb|200x200px|'''Exo TS''']]
| [[Image:Strainendodh08.jpg|thumb|200x200px|'''Endo TS''']]
|}

From studying the energies in table 13 it can be seen that the endo transition state is more stable than the exo by 4.08589 x 10-3 a. u (10.73 KJ mol-1). However by looking at the strain of both transition states one would expect the exo TS to be more stable. This is because the distance between the cyclohexa-1,3-diene ring and the -(C=O)-O-(C=O)- maleic anhydride fragment is 2.99 Å in the endo TS and 3.03 Å in the exo TS. Thus the endo form would have more strain due to steric hindrance.

However within the exo TS the vicinal hydrogens on the CH2-CH2 (carbon numbers 18 and 21) fragment on the cyclohexa-1,3-diene ring are found below the plane and next to the maleic anhydride. This leads to steric hindrance is this is where the strain originates from within the exo TS. Therefore this disfavours the exo form, which makes the endo more stable.

Another factor making the endo TS more favourable is secondary orbital overlap. These orbitals contribute to the bonds forming and they interact with each other through space. Within the endo TS below there are 2 more orbitals interacting (secondary orbitals) than within the exo TS. This therefore makes the endo TS more stable and therefore more favoured.

[[Image:Secondary_orbital_shizzledh08.jpg|500x500px|center]]

{| class="wikitable" border="1" align="center"
|+ '''Table 14: Frontier molecular orbitals of the the transition state for the cyclohexa-1,3-diene reaction '''
! rowspan="2"|Molecular Orbital !! colspan="4"|''exo'' !! colspan="4"|''endo''
|- align="center"
| '''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''||'''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''
|- align="center"
| LUMO || [[Image:LUMO1exoh081.jpg|150x150px]]|| [[Image:LUMO2exodh081.jpg|150x150px]]|| Anti-symmetric || -0.078 || [[Image:LUMO1ENDOdh08.jpg|150x150px]]||[[Image:LUMO2ENDOdh08.jpg|150x150px]] || Anti-symmetric || -0.067
|- align="center"
| HOMO || [[Image:HOMO1exodh081.jpg|150x150px]]||[[Image:HOMO2exodh081.jpg|150x150px]] || Anti-symmetric|| -0.242 || [[Image:HOMO1ENDOdh08.jpg|150x150px]]|| [[Image:HOMO2ENDOdh08.jpg|150x150px]]||Anti-symmetric || -0.242
|}

Digital repositories:

exo: http://hdl.handle.net/10042/to-5596

endo: http://hdl.handle.net/10042/to-5597

From examining table 14 it is clear to see that both the HOMO and LUMO are anti-symmetric for both the transition states. The LUMO of the maleic anhydride fragment is the π* anti-bonding orbital of the C=C bond. Within the reaction then this has electrons transferred to it and the C=C bond breaks, which leaves a C-C bond. The LUMOs for both transition states are both anti-bonding with respect to the two fragments.

=='''Conclusion'''==

='''References'''=
<references/>

Rep:Mod:hampluscheese

2010-11-10T22:20:30Z

Dh08: /* '''Cyclohexa-1,3-diene reaction with maleic anhydride''' */

='''Module 3:Physical Computational Chemistry Lab'''=

Douglas Hunt

=='''Objectives'''==

The aim of this experiment is to

=='''The Cope Rearrangement'''==

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangement] is a [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement |3,3|-sigmatropic rearrangement] that takes place through a single [http://en.wikipedia.org/wiki/Pericyclic pericyclic], concerted [http://en.wikipedia.org/wiki/Transition_state transition state]. Within this rearrangement a σ bond is formed in concert with another being broken within a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] π system.

The Cope rearrangement we will be studying will be that of 1,5-hexadiene, which is shown below.

[[Image:Cope_rearrangement_schemedh08.jpg|300x300px|center]]

The objective is to locate the low-energy minima and transition structures on the C6H10 potential energy surface and to determine the preferred reaction mechanism by using [http://en.wikipedia.org/wiki/GAUSSIAN Gaussian], with the [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)] methods.

==='''Optimisation of 1,5-hexadiene Conformers'''===

Rough <jmol>
<jmolAppletButton>
<uploadedFileContents>Anti_2_drawndh08.mol</uploadedFileContents>
<text>anti</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche3_drawndh08.mol</uploadedFileContents>
<text>gauche</text>
<color>white</color>
</jmolAppletButton>
</jmol> conformers of 1,5-hexadiene were drawn on Gaussview and their structures were tidied by the clean function. They were then further optimised by the HF/3-21G method and were identified which conformer they were from their point group and appendix 1 on the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3 physical lab page].

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 1: Optimisation and comparison of anti and gauche 1,5-hexadiene conformers using HF/3-21G methods
| ''' Conformer''' || '''Optimised Structure''' || '''Energy (Atomic Units)''' || '''[http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1 Energies (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (Anti2 - Gauche3)''' || '''Point group'''
|-
| '''Anti2''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Anti_2_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692535 || -231.69254 ||-6.08 x 105 || 1.2606 x 10-4 a. u|| Ci
|-
| '''Gauche3''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Gauche3_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692661 || -231.69266 || -6.08 x 105 || 3.31 x 10-1 KJ mol-1 || C1
|}

From studying table 1 it can be seen that the gauche conformer is 0.33 KJ mol-1 more stable than the anti conformation. One would assume that the anti2 conformer would be more stable due to [http://en.wikipedia.org/wiki/Steric steric] factors/repulsion within the gauche3 conformation. However within the gauche3 conformer there is a favourable donation of [http://en.wikipedia.org/wiki/Electron_density electron density] from the C=C π orbital to the C-H σ* orbital from the adjacent proton, which stabilises the conformer<ref name="ref 1">B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446.[http://www.informaworld.com/smpp/content~db=all?content=10.1080/00268970110081412]</ref>.

 

The anti2 conformer was then optimised using the B3LYP 6-31G* (6-31G (d)) method.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 2: Comparison of the anti2 conformer optimisations carried out at different calculation methods
| ''' Method''' || '''Structure and bond distances (Å)''' || '''Dihedral angle (C1 to C4)''' || '''Energy (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (HF/3-21G - DFT/B3LYP/6-31G)'''
|-
| HF/3-21G || [[Image:Anti2_opt1dh08.jpg|300x300px]] || 114.6 o || -231.692535 || -6.08 x 105 || 2.919171 a. u
|-
| DFT/B3LYP/6-31G* || [[Image:Anti2_opt2DH08.jpg|300x300px]] || 118.8 o || -234.611716 || -6.16 x 105 || 7.66 x 103 KJ mol-1
|}

From examining table 2 at first it is possible to see that the bond lengths are very similar to each other and that of the literature bond lengths<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref> (C1=C2: 1.340 Å, C2-C3: 1.508 Å and C3-C4: 1.538 Å). It is also possible to see that the C=C bond length is more accurate for the DFT/B3LYP/6-31G* method by it being 0.01 Å closer to the literature value. Another main difference in terms of geometry is that the DFT/B3LYP/6-31G* method gives a larger [http://en.wikipedia.org/wiki/Dihedral_angle dihedral angle] (and most likely more accurate) of 118.8o compared to that of 114.6o.

In terms of energy the DFT/B3LYP/6-31G* method produces a conformer of a higher stability by it possessing a lower energy with a difference of 7.66 x 103 KJ mol-1.

==='''Frequency analysis of 1,5-hexadiene Conformers'''===

[[Image:Vibrations_anti2dh08.jpg|right|300x300px]]

For the frequency analysis the DFT/B3LYP/6-31G* method was used.

Frequency analysis is essential within this calculation due to it being the second derivative of the potential energy surface. This means that if all the frequencies are all positive the system is at a minimum, if one of them is negative the optimised system is at a transition state, and if more than one are negative then the calculation has failed to find a critical point, which means the optimisation has failed. This therefore gives us a method of seeing whether the calculation has worked.

From looking to the image on the right it can deduced that the optimisation has taken place successfully by there being no negative vibrations and therefore the conformation is at a minimum.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 3: Energies under the thermochemistry section at 298.15 K and 0 K within the frequency logfiles'''
| '''Energies''' || '''298.15 K (atomic units)''' ||| '''0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (KJ mol-1)''' || '''Discussion'''
|-
| Sum of electronic and zero-point energies || -234.469177 || -234.466673 || -2.504 x 10-3 || -6.57 || This is the potential energy at 0 K, which includes the zero-point vibrational energy (E = Eelec + ZPE).

|-
| Sum of electronic and thermal energies || -234.461835 || -234.461085 || -7.5 x 10-4|| -1.97 || This is the energy at 298.15 K (0 K for 0 K calculation) and 1 atm of pressure, which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans).

|-
| Sum of electronic and thermal enthalpies || -234.460891 || -234.460141 || -7.5 x 10-4 || -1.97 || This contains an additional correction for RT (H = E + RT), which is especially important when looking at dissociation reactions.
|-
| Sum of electronic and thermal free energies || -234.500739 || -234.495931 || 4.808 x 10-3 || -12.62 || This includes the entropic contribution to the free energy (G = H - TS).
|}

{|align=right
|+
| [[Image:Anti2_freq_IRdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 298.15 K''']]
| [[Image:Anti2_freq_0Kdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 0 K''']]
|}

From looking at table 3 above it can be deduced that the anti2 conformer is more stable at room temperature to that of 0 K with respect to all the types of energies. The spectra also seem to not change much between the two temperatures but the main change is the change in the sum of electronic and thermal free energies, which is heavily dependent of temperature by it including the entropic contribution to the free energy (G = H - '''T'''S)

 
 
 
 

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

Within the Cope Rearrangement there are two possible transition states, the chair (C2h) and the boat (C2v), which are shown within [https://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_2 appendix two ].

==='''Comparison of optimisation methods for the chair transition state'''===

An <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2hdh08.mol</uploadedFileContents>
<text>allyl fragment</text>
<color>white</color>
</jmolAppletButton>
</jmol>, CH2CHCH2 was drawn on Gaussview and was optimized through the HF/3-21G method. After this the original allyl fragment was copied and pasted into a new window and place exactly above the first fragment with a bond distance of 2.20 Å between the terminal end carbons. This therefore produced the <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2h_fulldh08.mol</uploadedFileContents>
<text>chair transition structure.</text>
<color>white</color>
</jmolAppletButton>
</jmol>

Gaussian was then set up for an optimisation for the transition state with the additional key words "Opt=NoEigen", the optimisation set to TS (Berny), the force constants set to "once" and using the HF/3-21G method.

After this the "chair" transition state was optimised an alternative way by freezing the coordinates of the two sets of terminal carbons (where the bonds form/break during the rearrangement) within the Redundant Coord Editor on Gauusview.
Following this a second calculation was run where the freeze coordinates option was changed to "Derivative" and the force constants were set to "never".

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 4: Comparison of chair transition state optimisation methods
| || '''Optimised through a TS (Berny)''' || '''Optimised through a frozen coordinate method'''
|-
| '''Optimised TS Structure''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>C2h_optfreq_TS_Bdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| '''Terminal C-C bond length (Å)''' || 2.02 || 2.02
|-
|'''Energy (atomic units)''' || -231.61932247 || -231.61932190
|-
| '''ΔE, TS (Berny) - frozen coordinate''' || 5.7 x 10-7 a.u || 1.50 x 10-3 KJ mol-1
|-
| '''Imaginary frequency (cm-1)''' || -818 || -818
|-
| '''Animation''' ||[[Image:818dh08TSB.gif|350x350px]] || [[Image:818cm-1Fdh08.gif]]
|}

The data in table 4 shows that both methods give the same results and they both give the same transition state. They both produced the same energy with a negligible energy difference between them and the bond lengths between the terminal carbons are both 2.02 Å. In addition to this the imaginary frequencies were both -818cm-1 and within the animations above it is possible to see that when one C-C bond is forming the other is breaking. This means it is a asynchronous bond formation, which is expected for the Cope Rearrangement since it is a pericyclic reaction.

The only difference between the two optimisations is the way in which the transition states were calculated. This is due to the TS (Berny) method calculating the force constant matrix within the first optimisation step and is then being recalculated as the optimisation carries on. When the frozen coordinate method freezes the reaction coordinate in order for the rest of the structure to be optimised around it, which does not require the whole hessian (force constant matrix) to be calculated but only that the reaction coordinate is differentiated (second step). The main advantage of this, is that you do not need to calculate the entire force constant matrix and just need to differentiate the reaction coordinate in order to obtain a good enough guess for the initial force constant matrix. The frozen coordinate method will therefore take less time to compute the transition state, which can be very expensive within certain systems. However to carry out this method one needs to have an idea of how far apart the terminal carbons are.

==='''Boat transition state'''===

===='''QST2 optimisation'''====

The optimisation for the boat transition state had a different method and this was the QST2 method. This calculation interpolates between two structures (product and reactant) and tries to find a transition state between them.

The reactants and products were numbered in the same way. Therefore both 1,5-hexadiene molecules had their numbering manually changed for the product molecule so it corresponded to the numbering of the reactant if it rearranged. This is illustrated in the reaction scheme below.

[[Image:Boat_rerrangmentdh08.jpg|300x300px|center]]

The chk file for the optimized anti2 conformer was opened and copied into a new window. A second window was opened within the new window and the same anti2 conformer was copied into the new window. The numbering was changed within the second molecule so the two looked like they do in the image below.

[[Image:TS_(QST2)_numbering_1dh08.jpg|400x400px|center]]

The job type was set as "Opt+Freq" and "TS (QST2)" was chosen to optimise the transition state. However this failed and produced
a structure a bit like a <jmol>
<jmolAppletButton>
<uploadedFileContents>Trialone_BPdh08.mol</uploadedFileContents>
<text>chair transition structure </text>
<color>white</color>
</jmolAppletButton>
</jmol> but more dissociated. This is because it interpolated between the two structures and translated the top allyl fragment and therefore did not consider the possibility of a rotation around the central bonds.

To get around this the C-C-C-C dihedral angle (C2-C3-C4-C5) was set to 0 o and the C-C-C angle (C2-C3-C4 and C3-C4-C5 for the product molecule) was set to 100 o for both molecules. This gave rise to the two structures below.

[[Image:TS_(QST2)_numbering_2dh08.jpg|400x400px|center]]

After this the QST2 calculation was run again and the results in table 5 were produced. The imaginary frequency at -839 cm-1 shows that while one bond is forming the other one is breaking, which is again an asynchronous bond formation.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5: Data for the boat transition state from the QST2 optimisation method
| Optimised TS Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.14
|-
|Energy (atomic units) || -231.60280151
|-
| Energy (KJ mol-1) || -6.08 x 105
|-
| Imaginary frequency (cm-1) || -839
|-
| Animation ||[[Image:-839dh08.gif]]
|}

===='''Intrinsic Reaction Coordinate'''====

It is not possible to predict which product the chair and boat transition state will lead to. To solve this problem within Gaussian one can use the IRC method, which allows one to follow the minimum energy path from a transition structure down to its local minimum on a potential energy surface. This creates a succession of points by taking small geometry steps in the direction to where the gradient of the energy surface is at its steepest until the minimum in energy is reached.

The chk file for the optimized chair transition structure was opened and the job type selected was "IRC", the reaction coordinate was computed in the forward direction (since molecule is symmetric) with 50 points along the IRC. This was done with two type of force constants being calculated, once and always.

Within the "once" method the output showed no bond formation process by the 26th step and therefore the calculation was incomplete, since a complete 1,5-hexadiene structure was supposed to be formed. From looking at the IRC pathway it can be seen that the minimum on the potential energy surface was not reached at that point, since the points on the pathway did not form an asymptote and the gradient did not go close enough to zero.

For the "always" method the computation of force constants were carried out at every step. This therefore lets the IRC calculation distinguish whether it is moving along the minimum energy pathway while going down the potential energy surface. While this is happening it can make changes to the pathway if necessary. Within the "always" case the bond forming arrives at 1,5-hexadiene and this is backed up by the IRC pathway reaching a minimum within its energy by the gradient going close to zero and the pathway reaches an asymptote.

The structure produced through the always method had an energy of -231.69199434 atomic units, which is very close to the energy of the gauche2 conformer in the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 appendix 1]. This suggests that the reaction takes place through the chair transition state, which later produces the gauche2 conformer of 1,5-hexadiene.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 6: Comparison of IRC calculations for the chair structure
| || '''Calculate force constants once''' || '''Calculate force constants always'''
|-
| '''Number of points along IRC''' || 26 || 47
|-
| '''Structure at the last point of IRC''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Once_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Always_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|'''Energy (atomic units)''' || -231.68906923 || -231.69199434
|-
| '''ΔE, once - always''' || 2.92511 x 10-3 a.u || 7.68 KJ mol-1
|-
| '''IRC pathway''' || [[Image:One_pathwaydh08.jpg|400x400px]] || [[Image:Always_pathwaydh08.jpg|400x400px]]
|-
| '''IRC gradient''' || [[Image:One_gradientdh08.jpg|400x400px]]|| [[Image:Always_gradientdh08.jpg|400x400px]]
|}

 
 
 

===='''Comparison of activation energies for the transition states'''====

The HF/3-21G optimised chair and boat transition structures were both reoptimised using the B3LYP/6-31G* (B3LYP/6-31G (d)) method. This gives a more accurate minimisation however this method takes longer to run. The values for the zero-point energies and thermal energies were found within the thermochemistry section of the log file.

The values for the HF/3-21G reactant (anti2) optimisation were calculated and the B3LYP/6-31G* values were taken from the "Frequency analysis of 1,5-hexadiene Conformers" earlier in this module.

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 7: Energy data for chair, boat and the anti2 conformation from the HF/3-21G and B3LYP/6-31G* methods of optimisation '''
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466677
| align="center" | -231.461320
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450936
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402342
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539524
| align="center" | -231.532562
| align="center" | -234.611716
| align="center" | -234.469177
| align="center" | -234.461835
|}
Digital repositories

Chair:
HF/3-21G: http://hdl.handle.net/10042/to-5569
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5568

Boat:
HF/3-21G: http://hdl.handle.net/10042/to-5570
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5571

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 8: Summary of activation energies for the chair and boat conformations (ΔE = chair TS or boat TS minus anti2)'''
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| align="center" | 1 a.u = 627.509 kcal mol-1
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE, chair (a.u)'''
| align="center" | 0.072847
| align="center" | 0.071242
| align="center" | 0.054248
| align="center" | 0.052827
| align="center" | -
|-
| '''ΔE, chair (kcal mol-1)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE, boat (a. u)'''
| align="center" | 0.088588
| align="center" | 0.087259
| align="center" | 0.066835
| align="center" | 0.065827
| align="center" | -
|-
| '''ΔE, boat (kcal mol-1)'''
| align="center" | 55.59
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.31
| align="center" | 44.7 ± 2.0
|}

By examining table 8 it is clear to see that the B3LYP/6-31G* method gives values which are much closer to the experimental ones. This is due to the B3LYP/6-31G* giving a higher level of optimisation than the HF/3-21G method. Table 8 also shows that the chair transition state has a lower activation energy to that of the boat transition state by 8.16 kcal mol-1. This suggests that the Cope rearrangement mechanism would most likely proceed through the chair transition state to that of the boat. Therefore the reaction would take place from the anti2 conformer to the chair transition state and then onto the gauche2 conformer within the Cope rearrangement.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9: Geometric parameters for the optimised chair and boat structures '''
| || ''' HF/3-21G chair TS''' || ''' B3LYP/6-31G* chair TS''' || '''HF/3-21G boat TS''' || '''B3LYP/6-31G* boat TS'''
|-
| Optimised Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Chair_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Boat_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.02 (C5-C10 or C3-C14)|| 1.97 (C5-C10 or C3-C14) || 2.14 (C3-C4 or C1-C6) || 2.21 (C3-C4 or C1-C6)
|-
| Fragment angle || 120 o (C3-C1-C5) || 120 o (C3-C1-C5) || 122 o (C4-C5-C6) || 122 o (C4-C5-C6)
|-
| Dihedral angle || 55 o (C1-C3-C14-C11)|| 54 o (C1-C3-C14-C11)|| 0 o (C2-C1-C6-C5) || 0 o (C2-C1-C6-C5)
|-
| Numbering of carbons || [[Image:Numbering_chairdh08.jpg|200x200px]]|| [[Image:Numbering_chairdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]]
|}

From studying table 9 it can be seen that the C-C bond breaking/forming length has changed between the two different types of optimisation methods. However the angles between the bonds remains unchanged.

=='''The Diels Alder Cycloaddition'''==

The Diels-Alder reaction is a cycloaddition between a conjugated diene and a dienophile (alkene system), which forms a cyclohexene system. It is often referred to as a [4s+2s] cycloaddition and proceeds through a single cyclic, concerted transition state where 2 σ bonds are formed in between the termini carbons of the 2 conjugated п systems. This is shown in the reaction scheme below where ethene is reacting with cis-butadiene.

[[Image:Reaction_scheme2dh08.jpg|300x300px|center]]

The π orbitals of the dienophile form new σ bonds with the π orbitals of the diene. The number of π electrons involved determine whether the reaction proceeds in a concerted stereospecific fashion (allowed) or not (forbidden). The HOMO/LUMO of one fragment interacts with the HOMO/LUMO of the other fragment to form two new bonding and anti-bonding MOs. The nodal properties allow one to predict how the reaction will occur.

Semi-Empirical/AM1 method and the B3LYP/6-31G* method of optimisation will be used on Gaussian to determine particular properites of the reactants to predict the way in which the reaction will proceed.

==='''Reagents'''===

<jmol>
<jmolAppletButton>
<uploadedFileContents>Cis_butadienedh08.mol</uploadedFileContents>
<text>Cis-butadiene</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Ethenedh08.mol</uploadedFileContents>
<text>ethylene</text>
<color>white</color>
</jmolAppletButton>
</jmol> were optimised separately using the AM1 semi-empirical method on Gaussview.The cis-butadiene had an energy of 0.04879725 a. u (1.28 x 102 KJ mol-1) and ethylene had an energy of 0.02619027 a. u (68.76 KJ mol-1).

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10: Frontier molecular orbitals of cis-butadiene and ethylene '''
| '''Reagent''' || '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| cis-butadiene || LUMO || [[Image:LUMO1_cis_butadieneDH08.jpg|100x100px]] || [[Image:LUMO2_cis_butadienedh08.jpg|100x100px]]|| Symmetric || +0.017
|-
| cis-butadiene || HOMO || [[Image:HOMO1_cis_butadienedh08.jpg|100x100px]] || [[Image:HOMO2_cis_butadienedh08.jpg|100x100px]]|| Anti-symmetric || -0.343
|-
| ethylene || LUMO || [[Image:LUMO1_ethenedh08.jpg|100x100px]] || [[Image:LUMO2_ethenedh08.jpg|100x100px]]|| Anti-symmetric || +0.052
|-
| ethylene || HOMO || [[Image:HOMO1_ethenedh08.jpg|100x100px]] || [[Image:HOMO2_ethenedh08.jpg|100x100px]]|| Symmetric || -0.387
|}

 
The symmetries of the LUMO and HOMO orbitals for cis-butadiene are opposite of those for the ethylene. This therefore means that the only interactions that can take place are those between ethylene's LUMO and cis-butadiene's HOMO and in addition to this the interaction between cis-butadiene's LUMO and ethylene's HOMO. This is because orbital symmetry has to be conserved (the conservation of orbital symmetry states that the symmetry of an new orbital formed by the overlap of the 2 original orbitals must be the same as the symmetry of the initial orbitals).

==='''Transition State'''===

A <jmol>
<jmolAppletButton>
<uploadedFileContents>Bicyclic_startingdh08.mol</uploadedFileContents>
<text>bicyclo system</text>
<color>white</color>
</jmolAppletButton>
</jmol> was drawn and one of the -CH2-CH2- fragments were removed. Three C=C bonds were added and the previous one was taken away so a <jmol>
<jmolAppletButton>
<uploadedFileContents>Frozen_coorddh08.mol</uploadedFileContents>
<text>guess transition state</text>
<color>white</color>
</jmolAppletButton>
</jmol> was produced.

After this an optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while bonds between the termini of the fragments were frozen at 2.1Å. Following this the resulting file had two separate optimisation and frequency calculations run on it using the TS (Berny) with one using the semi-empirical/AM1 method and the other using the DFT/B3LYP/6-31G* level of theory. Both of these optimisations had the bonds unfrozen and the "derivative" option was chosen instead.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11: Comparison of two methods of optimsations for the transition state '''
| || '''semi-empirical/AM1 method''' || '''DFT/B3LYP/6-31G* method'''
|-
| Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>SE_AM1_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>DFT_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Energy (a. u) || 0.11165608 || -234.54389573
|-
| Bond lengths || [[Image:DistancesSEdh08.jpg|150x150px]]|| [[Image:DistancesDFTdh08.jpg|150x150px]]
|-
|Imaginary frequency (cm-1) || -957 || -525
|-
|Animation || [[Image:-957dh08.gif]] || [[Image:-525dh08.gif]]
|-
| Lowest real frequency (cm-1) || 148 || 136
|-
|Animation || [[Image: 148dh08.gif]] || [[Image: 136dh08.gif]]

|}

The C-C bond forming length between the termini carbons is 0.15 Å larger for the DFT/B3LYP/6-31G* level of theory and this is because this method gives carries out the calculation to a greater degree of accuracy. This larger bond length leads a lower potential energy gradient to the transition state, which gives a lower total energy for the transition state. Therefore the given values below will be the DFT/B3LYP/6-31G* and not the semi-empirical/AM1 method.

A typical sp3 C-C bond length is 1.54 Å and a typical sp2 C=C bond length is 1.34 Å<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref>. Within the optimised transition molecule above the C-C bond lengths are 1.40 Å and the C=C bond length is 1.38 Å. This suggests that the π-bonds are breaking in order for the C-C bonds to form and additionally to this a π-bond is forming in between the two sp3 carbons.

The Van der Waals radius of a C-atom is 1.72 Å<ref name="ref 3">M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer, D. G. Truhlar, 1995, 346, pp. 63-69.[http://pubs.acs.org/doi/abs/10.1021/jp8111556]</ref>. The partly formed C-C length is 2.27 Å, which is shorter than two times the the van der Waals radius of a carbon atom. Therefore the carbon atoms are within a close enough distance to each other to share electron density and yet the bond has not formed at this point.

From looking at the animations for the imaginary frequency above it can be seen that the formation of the two bonds between the termini of the reactants is synchronous. This is expected of the Diels-Alder reaction since the mechanism is concerted. However the lowest real frequency is asynchronous with respect to the two bonds forming between the termini.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12: Frontier molecular orbitals of the transition state '''
| '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| LUMO || [[Image:LUMO1TSdh08.jpg|100x100px]] || [[Image:LUMO2TSdh08.jpg|100x100px]]|| Symmetric || +0.023
|-
| HOMO || [[Image:HOMO1TSdh08.jpg|100x100px]] || [[Image:HOMO2TSdh08.jpg|100x100px]]|| Anti-symmetric || -0.323
|}

From examining the images from table 12 above it can be deduced that the LUMO of the transition state originates from the cis-butadiene's LUMO and the ethylene's HOMO, which represents the σ*-antibonding orbitals. However the HOMO of the transition state originates from the cis-butadiene's HOMO and the ethylene's LUMO, which represents the σ-bonding orbitals within the two single C-C bonds that are formed within the reaction.

==='''Cyclohexa-1,3-diene reaction with maleic anhydride'''===

The <jmol>
<jmolAppletButton>
<uploadedFileContents>FrozenTS22dh08.mol</uploadedFileContents>
<text>exo</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_frozenTS22dh08.mol</uploadedFileContents>
<text>endo</text>
<color>white</color>
</jmolAppletButton>
</jmol> transition states were drawn on Gaussview and then the optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while the distance of the forming bonds were frozen at 2.1Å between the respective carbons. After this the structures changed slightly (<jmol>
<jmolAppletButton>
<uploadedFileContents>TS2_afteropt1_exo1dh08.mol</uploadedFileContents>
<text>exo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_TS2_afteropt1dh08.mol</uploadedFileContents>
<text>endo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol>) and the resulting files had an optimisation and frequency calculation run on it using the TA (Berny) with using the semi-empirical/AM1 level of theory while the unfrozen option as changed to "derivative". After this the endo and exo optimised transition state structures were re-optimised as they were before but using the DFT/B3LYP/6-31G* level of theory instead with the additional keyword "Opt=NoEigen".

{| class="wikitable" border="1" align="center"
|+ '''Table 13: Comparison of the exo and endo transition states '''
! rowspan="2"|-!! colspan="2"|''exo'' !! colspan="2"|''endo''
|-
| Semi-empirical/AM1 || DFT/B3LYP/6-31G* || Semi-empirical/AM1 || DFT/B3LYP/6-31G*
|-
| align="center"|Structure ||align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>After_opt2_exodh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>Endo_TS2_afteropt2dh081.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| align="center"| Energy (a. u) ||align="center"| -0.05041975 ||align="center"| -612.67931095 ||align="center"| -0.05150470 ||align="center"| -612.68339684
|-
| C-C bond forming length (Å) ||align="center"| 2.17 ||align="center"| 2.29 ||align="center"| 2.16 ||align="center"| 2.27
|-align="center"
| Imaginary frequency (cm-1) ||align="center"| -812 ||align="center"| -448 ||align="center"| -806 ||align="center"| -447
|-align="center"
| Animation ||colspan="2"|[[Image:-812_SEdh08.gif]] || colspan="2"|[[Image:-806_cm-1dh08.gif]]
|}

Again the C-C bond forming length between the termini carbons is larger (by 0.11 and 0.12 Å) for the DFT/B3LYP/6-31G* level of theory and this is due the same reasons talked about above. However the C-C bond forming length between the termini carbons are very similar in both the endo and exo transition states (0.02 Å difference).

From looking at the animations for the imaginary frequency above it can also be seen that the formation of the two bonds between the carbons in this case as well is synchronous. Again this is expected of the Diels-Alder reaction since it is concerted.

{| align=center
|+ '''Strain diagrams for the endo and exo transition states'''
| [[Image:Exostraindh08.jpg|thumb|200x200px|'''Exo TS''']]
| [[Image:Strainendodh08.jpg|thumb|200x200px|'''Endo TS''']]
|}

From studying the energies in table 13 it can be seen that the endo transition state is more stable than the exo by 4.08589 x 10-3 a. u (10.73 KJ mol-1). However by looking at the strain of both transition states one would expect the exo TS to be more stable. This is because the distance between the cyclohexa-1,3-diene ring and the -(C=O)-O-(C=O)- maleic anhydride fragment is 2.99 Å in the endo TS and 3.03 Å in the exo TS. Thus the endo form would have more strain due to steric hindrance.

However within the exo TS the vicinal hydrogens on the CH2-CH2 (carbon numbers 18 and 21) fragment on the cyclohexa-1,3-diene ring are found below the plane and next to the maleic anhydride. This leads to steric hindrance is this is where the strain originates from within the exo TS. Therefore this disfavours the exo form, which makes the endo more stable.

Another factor making the endo TS more favourable is secondary orbital overlap. These orbitals contribute to the bonds forming and they interact with each other through space. Within the endo TS below there are 2 more orbitals interacting (secondary orbitals) than within the exo TS. This therefore makes the endo TS more stable and therefore more favoured.

[[Image:Secondary_orbital_shizzledh08.jpg|500x500px|center]]

{| class="wikitable" border="1" align="center"
|+ '''Table 14: Frontier molecular orbitals of the the transition state for the cyclohexa-1,3-diene reaction '''
! rowspan="2"|Molecular Orbital !! colspan="4"|''exo'' !! colspan="4"|''endo''
|- align="center"
| '''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''||'''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''
|- align="center"
| LUMO || [[Image:LUMO1exoh081.jpg|150x150px]]|| [[Image:LUMO2exodh081.jpg|150x150px]]|| Anti-symmetric || -0.078 || [[Image:LUMO1ENDOdh08.jpg|150x150px]]||[[Image:LUMO2ENDOdh08.jpg|150x150px]] || Anti-symmetric || -0.067
|- align="center"
| HOMO || [[Image:HOMO1exodh081.jpg|150x150px]]||[[Image:HOMO2exodh081.jpg|150x150px]] || Anti-symmetric|| -0.242 || [[Image:HOMO1ENDOdh08.jpg|150x150px]]|| [[Image:HOMO2ENDOdh08.jpg|150x150px]]||Anti-symmetric || -0.242
|}

Digital repositories:

exo: http://hdl.handle.net/10042/to-5596

endo: http://hdl.handle.net/10042/to-5597

From examining table 14 it is clear to see that both the HOMO and LUMO are anti-symmetric for both the transition states. The LUMO of the maleic anhydride fragment is the π* anti-bonding orbital of the C=C bond. Within the reaction then this has electrons transferred to it and the C=C bond breaks, which leaves a C-C bond. The LUMOs for both transition states are both anti-bonding with respect to the two fragments.

The HOMO of each describes the σ-bonding interaction to be formed between the terminal carbons of the reactants and products. It is made up by overlap of the HOMO of cyclohexadiene and the LUMO of the Maleic anhydride. The LUMO of Maleic anhydride is the π* anti-bonding orbital of the C=C bond and so as it is populated during the cycloaddition the C=C bond breaks, leaving a C-C single bond. The LUMOs of both are entirely anti-bonding between the two fragments.

=='''Conclusion'''==

='''References'''=
<references/>

Rep:Mod:hampluscheese

2010-11-10T22:15:07Z

Dh08: /* '''Cyclohexa-1,3-diene reaction with maleic anhydride''' */

='''Module 3:Physical Computational Chemistry Lab'''=

Douglas Hunt

=='''Objectives'''==

The aim of this experiment is to

=='''The Cope Rearrangement'''==

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangement] is a [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement |3,3|-sigmatropic rearrangement] that takes place through a single [http://en.wikipedia.org/wiki/Pericyclic pericyclic], concerted [http://en.wikipedia.org/wiki/Transition_state transition state]. Within this rearrangement a σ bond is formed in concert with another being broken within a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] π system.

The Cope rearrangement we will be studying will be that of 1,5-hexadiene, which is shown below.

[[Image:Cope_rearrangement_schemedh08.jpg|300x300px|center]]

The objective is to locate the low-energy minima and transition structures on the C6H10 potential energy surface and to determine the preferred reaction mechanism by using [http://en.wikipedia.org/wiki/GAUSSIAN Gaussian], with the [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)] methods.

==='''Optimisation of 1,5-hexadiene Conformers'''===

Rough <jmol>
<jmolAppletButton>
<uploadedFileContents>Anti_2_drawndh08.mol</uploadedFileContents>
<text>anti</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche3_drawndh08.mol</uploadedFileContents>
<text>gauche</text>
<color>white</color>
</jmolAppletButton>
</jmol> conformers of 1,5-hexadiene were drawn on Gaussview and their structures were tidied by the clean function. They were then further optimised by the HF/3-21G method and were identified which conformer they were from their point group and appendix 1 on the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3 physical lab page].

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 1: Optimisation and comparison of anti and gauche 1,5-hexadiene conformers using HF/3-21G methods
| ''' Conformer''' || '''Optimised Structure''' || '''Energy (Atomic Units)''' || '''[http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1 Energies (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (Anti2 - Gauche3)''' || '''Point group'''
|-
| '''Anti2''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Anti_2_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692535 || -231.69254 ||-6.08 x 105 || 1.2606 x 10-4 a. u|| Ci
|-
| '''Gauche3''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Gauche3_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692661 || -231.69266 || -6.08 x 105 || 3.31 x 10-1 KJ mol-1 || C1
|}

From studying table 1 it can be seen that the gauche conformer is 0.33 KJ mol-1 more stable than the anti conformation. One would assume that the anti2 conformer would be more stable due to [http://en.wikipedia.org/wiki/Steric steric] factors/repulsion within the gauche3 conformation. However within the gauche3 conformer there is a favourable donation of [http://en.wikipedia.org/wiki/Electron_density electron density] from the C=C π orbital to the C-H σ* orbital from the adjacent proton, which stabilises the conformer<ref name="ref 1">B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446.[http://www.informaworld.com/smpp/content~db=all?content=10.1080/00268970110081412]</ref>.

 

The anti2 conformer was then optimised using the B3LYP 6-31G* (6-31G (d)) method.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 2: Comparison of the anti2 conformer optimisations carried out at different calculation methods
| ''' Method''' || '''Structure and bond distances (Å)''' || '''Dihedral angle (C1 to C4)''' || '''Energy (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (HF/3-21G - DFT/B3LYP/6-31G)'''
|-
| HF/3-21G || [[Image:Anti2_opt1dh08.jpg|300x300px]] || 114.6 o || -231.692535 || -6.08 x 105 || 2.919171 a. u
|-
| DFT/B3LYP/6-31G* || [[Image:Anti2_opt2DH08.jpg|300x300px]] || 118.8 o || -234.611716 || -6.16 x 105 || 7.66 x 103 KJ mol-1
|}

From examining table 2 at first it is possible to see that the bond lengths are very similar to each other and that of the literature bond lengths<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref> (C1=C2: 1.340 Å, C2-C3: 1.508 Å and C3-C4: 1.538 Å). It is also possible to see that the C=C bond length is more accurate for the DFT/B3LYP/6-31G* method by it being 0.01 Å closer to the literature value. Another main difference in terms of geometry is that the DFT/B3LYP/6-31G* method gives a larger [http://en.wikipedia.org/wiki/Dihedral_angle dihedral angle] (and most likely more accurate) of 118.8o compared to that of 114.6o.

In terms of energy the DFT/B3LYP/6-31G* method produces a conformer of a higher stability by it possessing a lower energy with a difference of 7.66 x 103 KJ mol-1.

==='''Frequency analysis of 1,5-hexadiene Conformers'''===

[[Image:Vibrations_anti2dh08.jpg|right|300x300px]]

For the frequency analysis the DFT/B3LYP/6-31G* method was used.

Frequency analysis is essential within this calculation due to it being the second derivative of the potential energy surface. This means that if all the frequencies are all positive the system is at a minimum, if one of them is negative the optimised system is at a transition state, and if more than one are negative then the calculation has failed to find a critical point, which means the optimisation has failed. This therefore gives us a method of seeing whether the calculation has worked.

From looking to the image on the right it can deduced that the optimisation has taken place successfully by there being no negative vibrations and therefore the conformation is at a minimum.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 3: Energies under the thermochemistry section at 298.15 K and 0 K within the frequency logfiles'''
| '''Energies''' || '''298.15 K (atomic units)''' ||| '''0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (KJ mol-1)''' || '''Discussion'''
|-
| Sum of electronic and zero-point energies || -234.469177 || -234.466673 || -2.504 x 10-3 || -6.57 || This is the potential energy at 0 K, which includes the zero-point vibrational energy (E = Eelec + ZPE).

|-
| Sum of electronic and thermal energies || -234.461835 || -234.461085 || -7.5 x 10-4|| -1.97 || This is the energy at 298.15 K (0 K for 0 K calculation) and 1 atm of pressure, which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans).

|-
| Sum of electronic and thermal enthalpies || -234.460891 || -234.460141 || -7.5 x 10-4 || -1.97 || This contains an additional correction for RT (H = E + RT), which is especially important when looking at dissociation reactions.
|-
| Sum of electronic and thermal free energies || -234.500739 || -234.495931 || 4.808 x 10-3 || -12.62 || This includes the entropic contribution to the free energy (G = H - TS).
|}

{|align=right
|+
| [[Image:Anti2_freq_IRdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 298.15 K''']]
| [[Image:Anti2_freq_0Kdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 0 K''']]
|}

From looking at table 3 above it can be deduced that the anti2 conformer is more stable at room temperature to that of 0 K with respect to all the types of energies. The spectra also seem to not change much between the two temperatures but the main change is the change in the sum of electronic and thermal free energies, which is heavily dependent of temperature by it including the entropic contribution to the free energy (G = H - '''T'''S)

 
 
 
 

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

Within the Cope Rearrangement there are two possible transition states, the chair (C2h) and the boat (C2v), which are shown within [https://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_2 appendix two ].

==='''Comparison of optimisation methods for the chair transition state'''===

An <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2hdh08.mol</uploadedFileContents>
<text>allyl fragment</text>
<color>white</color>
</jmolAppletButton>
</jmol>, CH2CHCH2 was drawn on Gaussview and was optimized through the HF/3-21G method. After this the original allyl fragment was copied and pasted into a new window and place exactly above the first fragment with a bond distance of 2.20 Å between the terminal end carbons. This therefore produced the <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2h_fulldh08.mol</uploadedFileContents>
<text>chair transition structure.</text>
<color>white</color>
</jmolAppletButton>
</jmol>

Gaussian was then set up for an optimisation for the transition state with the additional key words "Opt=NoEigen", the optimisation set to TS (Berny), the force constants set to "once" and using the HF/3-21G method.

After this the "chair" transition state was optimised an alternative way by freezing the coordinates of the two sets of terminal carbons (where the bonds form/break during the rearrangement) within the Redundant Coord Editor on Gauusview.
Following this a second calculation was run where the freeze coordinates option was changed to "Derivative" and the force constants were set to "never".

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 4: Comparison of chair transition state optimisation methods
| || '''Optimised through a TS (Berny)''' || '''Optimised through a frozen coordinate method'''
|-
| '''Optimised TS Structure''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>C2h_optfreq_TS_Bdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| '''Terminal C-C bond length (Å)''' || 2.02 || 2.02
|-
|'''Energy (atomic units)''' || -231.61932247 || -231.61932190
|-
| '''ΔE, TS (Berny) - frozen coordinate''' || 5.7 x 10-7 a.u || 1.50 x 10-3 KJ mol-1
|-
| '''Imaginary frequency (cm-1)''' || -818 || -818
|-
| '''Animation''' ||[[Image:818dh08TSB.gif|350x350px]] || [[Image:818cm-1Fdh08.gif]]
|}

The data in table 4 shows that both methods give the same results and they both give the same transition state. They both produced the same energy with a negligible energy difference between them and the bond lengths between the terminal carbons are both 2.02 Å. In addition to this the imaginary frequencies were both -818cm-1 and within the animations above it is possible to see that when one C-C bond is forming the other is breaking. This means it is a asynchronous bond formation, which is expected for the Cope Rearrangement since it is a pericyclic reaction.

The only difference between the two optimisations is the way in which the transition states were calculated. This is due to the TS (Berny) method calculating the force constant matrix within the first optimisation step and is then being recalculated as the optimisation carries on. When the frozen coordinate method freezes the reaction coordinate in order for the rest of the structure to be optimised around it, which does not require the whole hessian (force constant matrix) to be calculated but only that the reaction coordinate is differentiated (second step). The main advantage of this, is that you do not need to calculate the entire force constant matrix and just need to differentiate the reaction coordinate in order to obtain a good enough guess for the initial force constant matrix. The frozen coordinate method will therefore take less time to compute the transition state, which can be very expensive within certain systems. However to carry out this method one needs to have an idea of how far apart the terminal carbons are.

==='''Boat transition state'''===

===='''QST2 optimisation'''====

The optimisation for the boat transition state had a different method and this was the QST2 method. This calculation interpolates between two structures (product and reactant) and tries to find a transition state between them.

The reactants and products were numbered in the same way. Therefore both 1,5-hexadiene molecules had their numbering manually changed for the product molecule so it corresponded to the numbering of the reactant if it rearranged. This is illustrated in the reaction scheme below.

[[Image:Boat_rerrangmentdh08.jpg|300x300px|center]]

The chk file for the optimized anti2 conformer was opened and copied into a new window. A second window was opened within the new window and the same anti2 conformer was copied into the new window. The numbering was changed within the second molecule so the two looked like they do in the image below.

[[Image:TS_(QST2)_numbering_1dh08.jpg|400x400px|center]]

The job type was set as "Opt+Freq" and "TS (QST2)" was chosen to optimise the transition state. However this failed and produced
a structure a bit like a <jmol>
<jmolAppletButton>
<uploadedFileContents>Trialone_BPdh08.mol</uploadedFileContents>
<text>chair transition structure </text>
<color>white</color>
</jmolAppletButton>
</jmol> but more dissociated. This is because it interpolated between the two structures and translated the top allyl fragment and therefore did not consider the possibility of a rotation around the central bonds.

To get around this the C-C-C-C dihedral angle (C2-C3-C4-C5) was set to 0 o and the C-C-C angle (C2-C3-C4 and C3-C4-C5 for the product molecule) was set to 100 o for both molecules. This gave rise to the two structures below.

[[Image:TS_(QST2)_numbering_2dh08.jpg|400x400px|center]]

After this the QST2 calculation was run again and the results in table 5 were produced. The imaginary frequency at -839 cm-1 shows that while one bond is forming the other one is breaking, which is again an asynchronous bond formation.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5: Data for the boat transition state from the QST2 optimisation method
| Optimised TS Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.14
|-
|Energy (atomic units) || -231.60280151
|-
| Energy (KJ mol-1) || -6.08 x 105
|-
| Imaginary frequency (cm-1) || -839
|-
| Animation ||[[Image:-839dh08.gif]]
|}

===='''Intrinsic Reaction Coordinate'''====

It is not possible to predict which product the chair and boat transition state will lead to. To solve this problem within Gaussian one can use the IRC method, which allows one to follow the minimum energy path from a transition structure down to its local minimum on a potential energy surface. This creates a succession of points by taking small geometry steps in the direction to where the gradient of the energy surface is at its steepest until the minimum in energy is reached.

The chk file for the optimized chair transition structure was opened and the job type selected was "IRC", the reaction coordinate was computed in the forward direction (since molecule is symmetric) with 50 points along the IRC. This was done with two type of force constants being calculated, once and always.

Within the "once" method the output showed no bond formation process by the 26th step and therefore the calculation was incomplete, since a complete 1,5-hexadiene structure was supposed to be formed. From looking at the IRC pathway it can be seen that the minimum on the potential energy surface was not reached at that point, since the points on the pathway did not form an asymptote and the gradient did not go close enough to zero.

For the "always" method the computation of force constants were carried out at every step. This therefore lets the IRC calculation distinguish whether it is moving along the minimum energy pathway while going down the potential energy surface. While this is happening it can make changes to the pathway if necessary. Within the "always" case the bond forming arrives at 1,5-hexadiene and this is backed up by the IRC pathway reaching a minimum within its energy by the gradient going close to zero and the pathway reaches an asymptote.

The structure produced through the always method had an energy of -231.69199434 atomic units, which is very close to the energy of the gauche2 conformer in the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 appendix 1]. This suggests that the reaction takes place through the chair transition state, which later produces the gauche2 conformer of 1,5-hexadiene.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 6: Comparison of IRC calculations for the chair structure
| || '''Calculate force constants once''' || '''Calculate force constants always'''
|-
| '''Number of points along IRC''' || 26 || 47
|-
| '''Structure at the last point of IRC''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Once_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Always_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|'''Energy (atomic units)''' || -231.68906923 || -231.69199434
|-
| '''ΔE, once - always''' || 2.92511 x 10-3 a.u || 7.68 KJ mol-1
|-
| '''IRC pathway''' || [[Image:One_pathwaydh08.jpg|400x400px]] || [[Image:Always_pathwaydh08.jpg|400x400px]]
|-
| '''IRC gradient''' || [[Image:One_gradientdh08.jpg|400x400px]]|| [[Image:Always_gradientdh08.jpg|400x400px]]
|}

 
 
 

===='''Comparison of activation energies for the transition states'''====

The HF/3-21G optimised chair and boat transition structures were both reoptimised using the B3LYP/6-31G* (B3LYP/6-31G (d)) method. This gives a more accurate minimisation however this method takes longer to run. The values for the zero-point energies and thermal energies were found within the thermochemistry section of the log file.

The values for the HF/3-21G reactant (anti2) optimisation were calculated and the B3LYP/6-31G* values were taken from the "Frequency analysis of 1,5-hexadiene Conformers" earlier in this module.

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 7: Energy data for chair, boat and the anti2 conformation from the HF/3-21G and B3LYP/6-31G* methods of optimisation '''
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466677
| align="center" | -231.461320
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450936
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402342
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539524
| align="center" | -231.532562
| align="center" | -234.611716
| align="center" | -234.469177
| align="center" | -234.461835
|}
Digital repositories

Chair:
HF/3-21G: http://hdl.handle.net/10042/to-5569
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5568

Boat:
HF/3-21G: http://hdl.handle.net/10042/to-5570
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5571

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 8: Summary of activation energies for the chair and boat conformations (ΔE = chair TS or boat TS minus anti2)'''
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| align="center" | 1 a.u = 627.509 kcal mol-1
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE, chair (a.u)'''
| align="center" | 0.072847
| align="center" | 0.071242
| align="center" | 0.054248
| align="center" | 0.052827
| align="center" | -
|-
| '''ΔE, chair (kcal mol-1)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE, boat (a. u)'''
| align="center" | 0.088588
| align="center" | 0.087259
| align="center" | 0.066835
| align="center" | 0.065827
| align="center" | -
|-
| '''ΔE, boat (kcal mol-1)'''
| align="center" | 55.59
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.31
| align="center" | 44.7 ± 2.0
|}

By examining table 8 it is clear to see that the B3LYP/6-31G* method gives values which are much closer to the experimental ones. This is due to the B3LYP/6-31G* giving a higher level of optimisation than the HF/3-21G method. Table 8 also shows that the chair transition state has a lower activation energy to that of the boat transition state by 8.16 kcal mol-1. This suggests that the Cope rearrangement mechanism would most likely proceed through the chair transition state to that of the boat. Therefore the reaction would take place from the anti2 conformer to the chair transition state and then onto the gauche2 conformer within the Cope rearrangement.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9: Geometric parameters for the optimised chair and boat structures '''
| || ''' HF/3-21G chair TS''' || ''' B3LYP/6-31G* chair TS''' || '''HF/3-21G boat TS''' || '''B3LYP/6-31G* boat TS'''
|-
| Optimised Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Chair_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Boat_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.02 (C5-C10 or C3-C14)|| 1.97 (C5-C10 or C3-C14) || 2.14 (C3-C4 or C1-C6) || 2.21 (C3-C4 or C1-C6)
|-
| Fragment angle || 120 o (C3-C1-C5) || 120 o (C3-C1-C5) || 122 o (C4-C5-C6) || 122 o (C4-C5-C6)
|-
| Dihedral angle || 55 o (C1-C3-C14-C11)|| 54 o (C1-C3-C14-C11)|| 0 o (C2-C1-C6-C5) || 0 o (C2-C1-C6-C5)
|-
| Numbering of carbons || [[Image:Numbering_chairdh08.jpg|200x200px]]|| [[Image:Numbering_chairdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]]
|}

From studying table 9 it can be seen that the C-C bond breaking/forming length has changed between the two different types of optimisation methods. However the angles between the bonds remains unchanged.

=='''The Diels Alder Cycloaddition'''==

The Diels-Alder reaction is a cycloaddition between a conjugated diene and a dienophile (alkene system), which forms a cyclohexene system. It is often referred to as a [4s+2s] cycloaddition and proceeds through a single cyclic, concerted transition state where 2 σ bonds are formed in between the termini carbons of the 2 conjugated п systems. This is shown in the reaction scheme below where ethene is reacting with cis-butadiene.

[[Image:Reaction_scheme2dh08.jpg|300x300px|center]]

The π orbitals of the dienophile form new σ bonds with the π orbitals of the diene. The number of π electrons involved determine whether the reaction proceeds in a concerted stereospecific fashion (allowed) or not (forbidden). The HOMO/LUMO of one fragment interacts with the HOMO/LUMO of the other fragment to form two new bonding and anti-bonding MOs. The nodal properties allow one to predict how the reaction will occur.

Semi-Empirical/AM1 method and the B3LYP/6-31G* method of optimisation will be used on Gaussian to determine particular properites of the reactants to predict the way in which the reaction will proceed.

==='''Reagents'''===

<jmol>
<jmolAppletButton>
<uploadedFileContents>Cis_butadienedh08.mol</uploadedFileContents>
<text>Cis-butadiene</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Ethenedh08.mol</uploadedFileContents>
<text>ethylene</text>
<color>white</color>
</jmolAppletButton>
</jmol> were optimised separately using the AM1 semi-empirical method on Gaussview.The cis-butadiene had an energy of 0.04879725 a. u (1.28 x 102 KJ mol-1) and ethylene had an energy of 0.02619027 a. u (68.76 KJ mol-1).

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10: Frontier molecular orbitals of cis-butadiene and ethylene '''
| '''Reagent''' || '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| cis-butadiene || LUMO || [[Image:LUMO1_cis_butadieneDH08.jpg|100x100px]] || [[Image:LUMO2_cis_butadienedh08.jpg|100x100px]]|| Symmetric || +0.017
|-
| cis-butadiene || HOMO || [[Image:HOMO1_cis_butadienedh08.jpg|100x100px]] || [[Image:HOMO2_cis_butadienedh08.jpg|100x100px]]|| Anti-symmetric || -0.343
|-
| ethylene || LUMO || [[Image:LUMO1_ethenedh08.jpg|100x100px]] || [[Image:LUMO2_ethenedh08.jpg|100x100px]]|| Anti-symmetric || +0.052
|-
| ethylene || HOMO || [[Image:HOMO1_ethenedh08.jpg|100x100px]] || [[Image:HOMO2_ethenedh08.jpg|100x100px]]|| Symmetric || -0.387
|}

 
The symmetries of the LUMO and HOMO orbitals for cis-butadiene are opposite of those for the ethylene. This therefore means that the only interactions that can take place are those between ethylene's LUMO and cis-butadiene's HOMO and in addition to this the interaction between cis-butadiene's LUMO and ethylene's HOMO. This is because orbital symmetry has to be conserved (the conservation of orbital symmetry states that the symmetry of an new orbital formed by the overlap of the 2 original orbitals must be the same as the symmetry of the initial orbitals).

==='''Transition State'''===

A <jmol>
<jmolAppletButton>
<uploadedFileContents>Bicyclic_startingdh08.mol</uploadedFileContents>
<text>bicyclo system</text>
<color>white</color>
</jmolAppletButton>
</jmol> was drawn and one of the -CH2-CH2- fragments were removed. Three C=C bonds were added and the previous one was taken away so a <jmol>
<jmolAppletButton>
<uploadedFileContents>Frozen_coorddh08.mol</uploadedFileContents>
<text>guess transition state</text>
<color>white</color>
</jmolAppletButton>
</jmol> was produced.

After this an optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while bonds between the termini of the fragments were frozen at 2.1Å. Following this the resulting file had two separate optimisation and frequency calculations run on it using the TS (Berny) with one using the semi-empirical/AM1 method and the other using the DFT/B3LYP/6-31G* level of theory. Both of these optimisations had the bonds unfrozen and the "derivative" option was chosen instead.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11: Comparison of two methods of optimsations for the transition state '''
| || '''semi-empirical/AM1 method''' || '''DFT/B3LYP/6-31G* method'''
|-
| Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>SE_AM1_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>DFT_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Energy (a. u) || 0.11165608 || -234.54389573
|-
| Bond lengths || [[Image:DistancesSEdh08.jpg|150x150px]]|| [[Image:DistancesDFTdh08.jpg|150x150px]]
|-
|Imaginary frequency (cm-1) || -957 || -525
|-
|Animation || [[Image:-957dh08.gif]] || [[Image:-525dh08.gif]]
|-
| Lowest real frequency (cm-1) || 148 || 136
|-
|Animation || [[Image: 148dh08.gif]] || [[Image: 136dh08.gif]]

|}

The C-C bond forming length between the termini carbons is 0.15 Å larger for the DFT/B3LYP/6-31G* level of theory and this is because this method gives carries out the calculation to a greater degree of accuracy. This larger bond length leads a lower potential energy gradient to the transition state, which gives a lower total energy for the transition state. Therefore the given values below will be the DFT/B3LYP/6-31G* and not the semi-empirical/AM1 method.

A typical sp3 C-C bond length is 1.54 Å and a typical sp2 C=C bond length is 1.34 Å<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref>. Within the optimised transition molecule above the C-C bond lengths are 1.40 Å and the C=C bond length is 1.38 Å. This suggests that the π-bonds are breaking in order for the C-C bonds to form and additionally to this a π-bond is forming in between the two sp3 carbons.

The Van der Waals radius of a C-atom is 1.72 Å<ref name="ref 3">M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer, D. G. Truhlar, 1995, 346, pp. 63-69.[http://pubs.acs.org/doi/abs/10.1021/jp8111556]</ref>. The partly formed C-C length is 2.27 Å, which is shorter than two times the the van der Waals radius of a carbon atom. Therefore the carbon atoms are within a close enough distance to each other to share electron density and yet the bond has not formed at this point.

From looking at the animations for the imaginary frequency above it can be seen that the formation of the two bonds between the termini of the reactants is synchronous. This is expected of the Diels-Alder reaction since the mechanism is concerted. However the lowest real frequency is asynchronous with respect to the two bonds forming between the termini.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12: Frontier molecular orbitals of the transition state '''
| '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| LUMO || [[Image:LUMO1TSdh08.jpg|100x100px]] || [[Image:LUMO2TSdh08.jpg|100x100px]]|| Symmetric || +0.023
|-
| HOMO || [[Image:HOMO1TSdh08.jpg|100x100px]] || [[Image:HOMO2TSdh08.jpg|100x100px]]|| Anti-symmetric || -0.323
|}

From examining the images from table 12 above it can be deduced that the LUMO of the transition state originates from the cis-butadiene's LUMO and the ethylene's HOMO, which represents the σ*-antibonding orbitals. However the HOMO of the transition state originates from the cis-butadiene's HOMO and the ethylene's LUMO, which represents the σ-bonding orbitals within the two single C-C bonds that are formed within the reaction.

==='''Cyclohexa-1,3-diene reaction with maleic anhydride'''===

The <jmol>
<jmolAppletButton>
<uploadedFileContents>FrozenTS22dh08.mol</uploadedFileContents>
<text>exo</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_frozenTS22dh08.mol</uploadedFileContents>
<text>endo</text>
<color>white</color>
</jmolAppletButton>
</jmol> transition states were drawn on Gaussview and then the optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while the distance of the forming bonds were frozen at 2.1Å between the respective carbons. After this the structures changed slightly (<jmol>
<jmolAppletButton>
<uploadedFileContents>TS2_afteropt1_exo1dh08.mol</uploadedFileContents>
<text>exo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_TS2_afteropt1dh08.mol</uploadedFileContents>
<text>endo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol>) and the resulting files had an optimisation and frequency calculation run on it using the TA (Berny) with using the semi-empirical/AM1 level of theory while the unfrozen option as changed to "derivative". After this the endo and exo optimised transition state structures were re-optimised as they were before but using the DFT/B3LYP/6-31G* level of theory instead with the additional keyword "Opt=NoEigen".

{| class="wikitable" border="1" align="center"
|+ '''Table 13: Comparison of the exo and endo transition states '''
! rowspan="2"|-!! colspan="2"|''exo'' !! colspan="2"|''endo''
|-
| Semi-empirical/AM1 || DFT/B3LYP/6-31G* || Semi-empirical/AM1 || DFT/B3LYP/6-31G*
|-
| align="center"|Structure ||align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>After_opt2_exodh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>Endo_TS2_afteropt2dh081.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| align="center"| Energy (a. u) ||align="center"| -0.05041975 ||align="center"| -612.67931095 ||align="center"| -0.05150470 ||align="center"| -612.68339684
|-
| C-C bond forming length (Å) ||align="center"| 2.17 ||align="center"| 2.29 ||align="center"| 2.16 ||align="center"| 2.27
|-align="center"
| Imaginary frequency (cm-1) ||align="center"| -812 ||align="center"| -448 ||align="center"| -806 ||align="center"| -447
|-align="center"
| Animation ||colspan="2"|[[Image:-812_SEdh08.gif]] || colspan="2"|[[Image:-806_cm-1dh08.gif]]
|}

Again the C-C bond forming length between the termini carbons is larger (by 0.11 and 0.12 Å) for the DFT/B3LYP/6-31G* level of theory and this is due the same reasons talked about above. However the C-C bond forming length between the termini carbons are very similar in both the endo and exo transition states (0.02 Å difference).

From looking at the animations for the imaginary frequency above it can also be seen that the formation of the two bonds between the carbons in this case as well is synchronous. Again this is expected of the Diels-Alder reaction since it is concerted.

{| align=center
|+ '''Strain diagrams for the endo and exo transition states'''
| [[Image:Exostraindh08.jpg|thumb|200x200px|'''Exo TS''']]
| [[Image:Strainendodh08.jpg|thumb|200x200px|'''Endo TS''']]
|}

From studying the energies in table 13 it can be seen that the endo transition state is more stable than the exo by 4.08589 x 10-3 a. u (10.73 KJ mol-1). However by looking at the strain of both transition states one would expect the exo TS to be more stable. This is because the distance between the cyclohexa-1,3-diene ring and the -(C=O)-O-(C=O)- maleic anhydride fragment is 2.99 Å in the endo TS and 3.03 Å in the exo TS. Thus the endo form would have more strain due to steric hindrance.

However within the exo TS the vicinal hydrogens on the CH2-CH2 (carbon numbers 18 and 21) fragment on the cyclohexa-1,3-diene ring are found below the plane and next to the maleic anhydride. This leads to steric hindrance is this is where the strain originates from within the exo TS. Therefore this disfavours the exo form, which makes the endo more stable.

Another factor making the endo TS more favourable is secondary orbital overlap. These orbitals contribute to the bonds forming and they interact with each other through space. Within the endo TS below there are 2 more orbitals interacting (secondary orbitals) than within the exo TS. This therefore makes the endo TS more stable and therefore more favoured.

[[Image:Secondary_orbital_shizzledh08.jpg|500x500px|center]]

{| class="wikitable" border="1" align="center"
|+ '''Table 14: Frontier molecular orbitals of the the transition state for the cyclohexa-1,3-diene reaction '''
! rowspan="2"|Molecular Orbital !! colspan="4"|''exo'' !! colspan="4"|''endo''
|- align="center"
| '''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''||'''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''
|- align="center"
| LUMO || [[Image:LUMO1exoh081.jpg|150x150px]]|| [[Image:LUMO2exodh081.jpg|150x150px]]|| Anti-symmetric || -0.078 || [[Image:LUMO1ENDOdh08.jpg|150x150px]]||[[Image:LUMO2ENDOdh08.jpg|150x150px]] || Anti-symmetric || -0.067
|- align="center"
| HOMO || [[Image:HOMO1exodh081.jpg|150x150px]]||[[Image:HOMO2exodh081.jpg|150x150px]] || Anti-symmetric|| -0.242 || [[Image:HOMO1ENDOdh08.jpg|150x150px]]|| [[Image:HOMO2ENDOdh08.jpg|150x150px]]||Anti-symmetric || -0.242
|}

Digital repositories:

exo: http://hdl.handle.net/10042/to-5596

endo: http://hdl.handle.net/10042/to-5597

Why do you think that the exo form could be more strained?

Examine carefully the nodal properties of the HOMO between the -(C=O)-O-(C=O)- fragment and the remainder of the system.

What can you conclude about the so called “secondary orbital overlap effect”? (There is some discussion of this in Ian Fleming's book 'Frontier Orbitals and Organic Chemical Reactions').

Further discussion:
What effects have been neglected in these calculations of Diels Alder transition states?

=='''Conclusion'''==

='''References'''=
<references/>

Rep:Mod:hampluscheese

2010-11-10T22:09:31Z

Dh08: /* '''Cyclohexa-1,3-diene reaction with maleic anhydride''' */

='''Module 3:Physical Computational Chemistry Lab'''=

Douglas Hunt

=='''Objectives'''==

The aim of this experiment is to

=='''The Cope Rearrangement'''==

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangement] is a [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement |3,3|-sigmatropic rearrangement] that takes place through a single [http://en.wikipedia.org/wiki/Pericyclic pericyclic], concerted [http://en.wikipedia.org/wiki/Transition_state transition state]. Within this rearrangement a σ bond is formed in concert with another being broken within a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] π system.

The Cope rearrangement we will be studying will be that of 1,5-hexadiene, which is shown below.

[[Image:Cope_rearrangement_schemedh08.jpg|300x300px|center]]

The objective is to locate the low-energy minima and transition structures on the C6H10 potential energy surface and to determine the preferred reaction mechanism by using [http://en.wikipedia.org/wiki/GAUSSIAN Gaussian], with the [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)] methods.

==='''Optimisation of 1,5-hexadiene Conformers'''===

Rough <jmol>
<jmolAppletButton>
<uploadedFileContents>Anti_2_drawndh08.mol</uploadedFileContents>
<text>anti</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche3_drawndh08.mol</uploadedFileContents>
<text>gauche</text>
<color>white</color>
</jmolAppletButton>
</jmol> conformers of 1,5-hexadiene were drawn on Gaussview and their structures were tidied by the clean function. They were then further optimised by the HF/3-21G method and were identified which conformer they were from their point group and appendix 1 on the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3 physical lab page].

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 1: Optimisation and comparison of anti and gauche 1,5-hexadiene conformers using HF/3-21G methods
| ''' Conformer''' || '''Optimised Structure''' || '''Energy (Atomic Units)''' || '''[http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1 Energies (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (Anti2 - Gauche3)''' || '''Point group'''
|-
| '''Anti2''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Anti_2_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692535 || -231.69254 ||-6.08 x 105 || 1.2606 x 10-4 a. u|| Ci
|-
| '''Gauche3''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Gauche3_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692661 || -231.69266 || -6.08 x 105 || 3.31 x 10-1 KJ mol-1 || C1
|}

From studying table 1 it can be seen that the gauche conformer is 0.33 KJ mol-1 more stable than the anti conformation. One would assume that the anti2 conformer would be more stable due to [http://en.wikipedia.org/wiki/Steric steric] factors/repulsion within the gauche3 conformation. However within the gauche3 conformer there is a favourable donation of [http://en.wikipedia.org/wiki/Electron_density electron density] from the C=C π orbital to the C-H σ* orbital from the adjacent proton, which stabilises the conformer<ref name="ref 1">B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446.[http://www.informaworld.com/smpp/content~db=all?content=10.1080/00268970110081412]</ref>.

 

The anti2 conformer was then optimised using the B3LYP 6-31G* (6-31G (d)) method.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 2: Comparison of the anti2 conformer optimisations carried out at different calculation methods
| ''' Method''' || '''Structure and bond distances (Å)''' || '''Dihedral angle (C1 to C4)''' || '''Energy (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (HF/3-21G - DFT/B3LYP/6-31G)'''
|-
| HF/3-21G || [[Image:Anti2_opt1dh08.jpg|300x300px]] || 114.6 o || -231.692535 || -6.08 x 105 || 2.919171 a. u
|-
| DFT/B3LYP/6-31G* || [[Image:Anti2_opt2DH08.jpg|300x300px]] || 118.8 o || -234.611716 || -6.16 x 105 || 7.66 x 103 KJ mol-1
|}

From examining table 2 at first it is possible to see that the bond lengths are very similar to each other and that of the literature bond lengths<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref> (C1=C2: 1.340 Å, C2-C3: 1.508 Å and C3-C4: 1.538 Å). It is also possible to see that the C=C bond length is more accurate for the DFT/B3LYP/6-31G* method by it being 0.01 Å closer to the literature value. Another main difference in terms of geometry is that the DFT/B3LYP/6-31G* method gives a larger [http://en.wikipedia.org/wiki/Dihedral_angle dihedral angle] (and most likely more accurate) of 118.8o compared to that of 114.6o.

In terms of energy the DFT/B3LYP/6-31G* method produces a conformer of a higher stability by it possessing a lower energy with a difference of 7.66 x 103 KJ mol-1.

==='''Frequency analysis of 1,5-hexadiene Conformers'''===

[[Image:Vibrations_anti2dh08.jpg|right|300x300px]]

For the frequency analysis the DFT/B3LYP/6-31G* method was used.

Frequency analysis is essential within this calculation due to it being the second derivative of the potential energy surface. This means that if all the frequencies are all positive the system is at a minimum, if one of them is negative the optimised system is at a transition state, and if more than one are negative then the calculation has failed to find a critical point, which means the optimisation has failed. This therefore gives us a method of seeing whether the calculation has worked.

From looking to the image on the right it can deduced that the optimisation has taken place successfully by there being no negative vibrations and therefore the conformation is at a minimum.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 3: Energies under the thermochemistry section at 298.15 K and 0 K within the frequency logfiles'''
| '''Energies''' || '''298.15 K (atomic units)''' ||| '''0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (KJ mol-1)''' || '''Discussion'''
|-
| Sum of electronic and zero-point energies || -234.469177 || -234.466673 || -2.504 x 10-3 || -6.57 || This is the potential energy at 0 K, which includes the zero-point vibrational energy (E = Eelec + ZPE).

|-
| Sum of electronic and thermal energies || -234.461835 || -234.461085 || -7.5 x 10-4|| -1.97 || This is the energy at 298.15 K (0 K for 0 K calculation) and 1 atm of pressure, which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans).

|-
| Sum of electronic and thermal enthalpies || -234.460891 || -234.460141 || -7.5 x 10-4 || -1.97 || This contains an additional correction for RT (H = E + RT), which is especially important when looking at dissociation reactions.
|-
| Sum of electronic and thermal free energies || -234.500739 || -234.495931 || 4.808 x 10-3 || -12.62 || This includes the entropic contribution to the free energy (G = H - TS).
|}

{|align=right
|+
| [[Image:Anti2_freq_IRdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 298.15 K''']]
| [[Image:Anti2_freq_0Kdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 0 K''']]
|}

From looking at table 3 above it can be deduced that the anti2 conformer is more stable at room temperature to that of 0 K with respect to all the types of energies. The spectra also seem to not change much between the two temperatures but the main change is the change in the sum of electronic and thermal free energies, which is heavily dependent of temperature by it including the entropic contribution to the free energy (G = H - '''T'''S)

 
 
 
 

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

Within the Cope Rearrangement there are two possible transition states, the chair (C2h) and the boat (C2v), which are shown within [https://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_2 appendix two ].

==='''Comparison of optimisation methods for the chair transition state'''===

An <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2hdh08.mol</uploadedFileContents>
<text>allyl fragment</text>
<color>white</color>
</jmolAppletButton>
</jmol>, CH2CHCH2 was drawn on Gaussview and was optimized through the HF/3-21G method. After this the original allyl fragment was copied and pasted into a new window and place exactly above the first fragment with a bond distance of 2.20 Å between the terminal end carbons. This therefore produced the <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2h_fulldh08.mol</uploadedFileContents>
<text>chair transition structure.</text>
<color>white</color>
</jmolAppletButton>
</jmol>

Gaussian was then set up for an optimisation for the transition state with the additional key words "Opt=NoEigen", the optimisation set to TS (Berny), the force constants set to "once" and using the HF/3-21G method.

After this the "chair" transition state was optimised an alternative way by freezing the coordinates of the two sets of terminal carbons (where the bonds form/break during the rearrangement) within the Redundant Coord Editor on Gauusview.
Following this a second calculation was run where the freeze coordinates option was changed to "Derivative" and the force constants were set to "never".

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 4: Comparison of chair transition state optimisation methods
| || '''Optimised through a TS (Berny)''' || '''Optimised through a frozen coordinate method'''
|-
| '''Optimised TS Structure''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>C2h_optfreq_TS_Bdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| '''Terminal C-C bond length (Å)''' || 2.02 || 2.02
|-
|'''Energy (atomic units)''' || -231.61932247 || -231.61932190
|-
| '''ΔE, TS (Berny) - frozen coordinate''' || 5.7 x 10-7 a.u || 1.50 x 10-3 KJ mol-1
|-
| '''Imaginary frequency (cm-1)''' || -818 || -818
|-
| '''Animation''' ||[[Image:818dh08TSB.gif|350x350px]] || [[Image:818cm-1Fdh08.gif]]
|}

The data in table 4 shows that both methods give the same results and they both give the same transition state. They both produced the same energy with a negligible energy difference between them and the bond lengths between the terminal carbons are both 2.02 Å. In addition to this the imaginary frequencies were both -818cm-1 and within the animations above it is possible to see that when one C-C bond is forming the other is breaking. This means it is a asynchronous bond formation, which is expected for the Cope Rearrangement since it is a pericyclic reaction.

The only difference between the two optimisations is the way in which the transition states were calculated. This is due to the TS (Berny) method calculating the force constant matrix within the first optimisation step and is then being recalculated as the optimisation carries on. When the frozen coordinate method freezes the reaction coordinate in order for the rest of the structure to be optimised around it, which does not require the whole hessian (force constant matrix) to be calculated but only that the reaction coordinate is differentiated (second step). The main advantage of this, is that you do not need to calculate the entire force constant matrix and just need to differentiate the reaction coordinate in order to obtain a good enough guess for the initial force constant matrix. The frozen coordinate method will therefore take less time to compute the transition state, which can be very expensive within certain systems. However to carry out this method one needs to have an idea of how far apart the terminal carbons are.

==='''Boat transition state'''===

===='''QST2 optimisation'''====

The optimisation for the boat transition state had a different method and this was the QST2 method. This calculation interpolates between two structures (product and reactant) and tries to find a transition state between them.

The reactants and products were numbered in the same way. Therefore both 1,5-hexadiene molecules had their numbering manually changed for the product molecule so it corresponded to the numbering of the reactant if it rearranged. This is illustrated in the reaction scheme below.

[[Image:Boat_rerrangmentdh08.jpg|300x300px|center]]

The chk file for the optimized anti2 conformer was opened and copied into a new window. A second window was opened within the new window and the same anti2 conformer was copied into the new window. The numbering was changed within the second molecule so the two looked like they do in the image below.

[[Image:TS_(QST2)_numbering_1dh08.jpg|400x400px|center]]

The job type was set as "Opt+Freq" and "TS (QST2)" was chosen to optimise the transition state. However this failed and produced
a structure a bit like a <jmol>
<jmolAppletButton>
<uploadedFileContents>Trialone_BPdh08.mol</uploadedFileContents>
<text>chair transition structure </text>
<color>white</color>
</jmolAppletButton>
</jmol> but more dissociated. This is because it interpolated between the two structures and translated the top allyl fragment and therefore did not consider the possibility of a rotation around the central bonds.

To get around this the C-C-C-C dihedral angle (C2-C3-C4-C5) was set to 0 o and the C-C-C angle (C2-C3-C4 and C3-C4-C5 for the product molecule) was set to 100 o for both molecules. This gave rise to the two structures below.

[[Image:TS_(QST2)_numbering_2dh08.jpg|400x400px|center]]

After this the QST2 calculation was run again and the results in table 5 were produced. The imaginary frequency at -839 cm-1 shows that while one bond is forming the other one is breaking, which is again an asynchronous bond formation.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5: Data for the boat transition state from the QST2 optimisation method
| Optimised TS Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.14
|-
|Energy (atomic units) || -231.60280151
|-
| Energy (KJ mol-1) || -6.08 x 105
|-
| Imaginary frequency (cm-1) || -839
|-
| Animation ||[[Image:-839dh08.gif]]
|}

===='''Intrinsic Reaction Coordinate'''====

It is not possible to predict which product the chair and boat transition state will lead to. To solve this problem within Gaussian one can use the IRC method, which allows one to follow the minimum energy path from a transition structure down to its local minimum on a potential energy surface. This creates a succession of points by taking small geometry steps in the direction to where the gradient of the energy surface is at its steepest until the minimum in energy is reached.

The chk file for the optimized chair transition structure was opened and the job type selected was "IRC", the reaction coordinate was computed in the forward direction (since molecule is symmetric) with 50 points along the IRC. This was done with two type of force constants being calculated, once and always.

Within the "once" method the output showed no bond formation process by the 26th step and therefore the calculation was incomplete, since a complete 1,5-hexadiene structure was supposed to be formed. From looking at the IRC pathway it can be seen that the minimum on the potential energy surface was not reached at that point, since the points on the pathway did not form an asymptote and the gradient did not go close enough to zero.

For the "always" method the computation of force constants were carried out at every step. This therefore lets the IRC calculation distinguish whether it is moving along the minimum energy pathway while going down the potential energy surface. While this is happening it can make changes to the pathway if necessary. Within the "always" case the bond forming arrives at 1,5-hexadiene and this is backed up by the IRC pathway reaching a minimum within its energy by the gradient going close to zero and the pathway reaches an asymptote.

The structure produced through the always method had an energy of -231.69199434 atomic units, which is very close to the energy of the gauche2 conformer in the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 appendix 1]. This suggests that the reaction takes place through the chair transition state, which later produces the gauche2 conformer of 1,5-hexadiene.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 6: Comparison of IRC calculations for the chair structure
| || '''Calculate force constants once''' || '''Calculate force constants always'''
|-
| '''Number of points along IRC''' || 26 || 47
|-
| '''Structure at the last point of IRC''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Once_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Always_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|'''Energy (atomic units)''' || -231.68906923 || -231.69199434
|-
| '''ΔE, once - always''' || 2.92511 x 10-3 a.u || 7.68 KJ mol-1
|-
| '''IRC pathway''' || [[Image:One_pathwaydh08.jpg|400x400px]] || [[Image:Always_pathwaydh08.jpg|400x400px]]
|-
| '''IRC gradient''' || [[Image:One_gradientdh08.jpg|400x400px]]|| [[Image:Always_gradientdh08.jpg|400x400px]]
|}

 
 
 

===='''Comparison of activation energies for the transition states'''====

The HF/3-21G optimised chair and boat transition structures were both reoptimised using the B3LYP/6-31G* (B3LYP/6-31G (d)) method. This gives a more accurate minimisation however this method takes longer to run. The values for the zero-point energies and thermal energies were found within the thermochemistry section of the log file.

The values for the HF/3-21G reactant (anti2) optimisation were calculated and the B3LYP/6-31G* values were taken from the "Frequency analysis of 1,5-hexadiene Conformers" earlier in this module.

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 7: Energy data for chair, boat and the anti2 conformation from the HF/3-21G and B3LYP/6-31G* methods of optimisation '''
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466677
| align="center" | -231.461320
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450936
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402342
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539524
| align="center" | -231.532562
| align="center" | -234.611716
| align="center" | -234.469177
| align="center" | -234.461835
|}
Digital repositories

Chair:
HF/3-21G: http://hdl.handle.net/10042/to-5569
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5568

Boat:
HF/3-21G: http://hdl.handle.net/10042/to-5570
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5571

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 8: Summary of activation energies for the chair and boat conformations (ΔE = chair TS or boat TS minus anti2)'''
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| align="center" | 1 a.u = 627.509 kcal mol-1
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE, chair (a.u)'''
| align="center" | 0.072847
| align="center" | 0.071242
| align="center" | 0.054248
| align="center" | 0.052827
| align="center" | -
|-
| '''ΔE, chair (kcal mol-1)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE, boat (a. u)'''
| align="center" | 0.088588
| align="center" | 0.087259
| align="center" | 0.066835
| align="center" | 0.065827
| align="center" | -
|-
| '''ΔE, boat (kcal mol-1)'''
| align="center" | 55.59
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.31
| align="center" | 44.7 ± 2.0
|}

By examining table 8 it is clear to see that the B3LYP/6-31G* method gives values which are much closer to the experimental ones. This is due to the B3LYP/6-31G* giving a higher level of optimisation than the HF/3-21G method. Table 8 also shows that the chair transition state has a lower activation energy to that of the boat transition state by 8.16 kcal mol-1. This suggests that the Cope rearrangement mechanism would most likely proceed through the chair transition state to that of the boat. Therefore the reaction would take place from the anti2 conformer to the chair transition state and then onto the gauche2 conformer within the Cope rearrangement.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9: Geometric parameters for the optimised chair and boat structures '''
| || ''' HF/3-21G chair TS''' || ''' B3LYP/6-31G* chair TS''' || '''HF/3-21G boat TS''' || '''B3LYP/6-31G* boat TS'''
|-
| Optimised Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Chair_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Boat_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.02 (C5-C10 or C3-C14)|| 1.97 (C5-C10 or C3-C14) || 2.14 (C3-C4 or C1-C6) || 2.21 (C3-C4 or C1-C6)
|-
| Fragment angle || 120 o (C3-C1-C5) || 120 o (C3-C1-C5) || 122 o (C4-C5-C6) || 122 o (C4-C5-C6)
|-
| Dihedral angle || 55 o (C1-C3-C14-C11)|| 54 o (C1-C3-C14-C11)|| 0 o (C2-C1-C6-C5) || 0 o (C2-C1-C6-C5)
|-
| Numbering of carbons || [[Image:Numbering_chairdh08.jpg|200x200px]]|| [[Image:Numbering_chairdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]]
|}

From studying table 9 it can be seen that the C-C bond breaking/forming length has changed between the two different types of optimisation methods. However the angles between the bonds remains unchanged.

=='''The Diels Alder Cycloaddition'''==

The Diels-Alder reaction is a cycloaddition between a conjugated diene and a dienophile (alkene system), which forms a cyclohexene system. It is often referred to as a [4s+2s] cycloaddition and proceeds through a single cyclic, concerted transition state where 2 σ bonds are formed in between the termini carbons of the 2 conjugated п systems. This is shown in the reaction scheme below where ethene is reacting with cis-butadiene.

[[Image:Reaction_scheme2dh08.jpg|300x300px|center]]

The π orbitals of the dienophile form new σ bonds with the π orbitals of the diene. The number of π electrons involved determine whether the reaction proceeds in a concerted stereospecific fashion (allowed) or not (forbidden). The HOMO/LUMO of one fragment interacts with the HOMO/LUMO of the other fragment to form two new bonding and anti-bonding MOs. The nodal properties allow one to predict how the reaction will occur.

Semi-Empirical/AM1 method and the B3LYP/6-31G* method of optimisation will be used on Gaussian to determine particular properites of the reactants to predict the way in which the reaction will proceed.

==='''Reagents'''===

<jmol>
<jmolAppletButton>
<uploadedFileContents>Cis_butadienedh08.mol</uploadedFileContents>
<text>Cis-butadiene</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Ethenedh08.mol</uploadedFileContents>
<text>ethylene</text>
<color>white</color>
</jmolAppletButton>
</jmol> were optimised separately using the AM1 semi-empirical method on Gaussview.The cis-butadiene had an energy of 0.04879725 a. u (1.28 x 102 KJ mol-1) and ethylene had an energy of 0.02619027 a. u (68.76 KJ mol-1).

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10: Frontier molecular orbitals of cis-butadiene and ethylene '''
| '''Reagent''' || '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| cis-butadiene || LUMO || [[Image:LUMO1_cis_butadieneDH08.jpg|100x100px]] || [[Image:LUMO2_cis_butadienedh08.jpg|100x100px]]|| Symmetric || +0.017
|-
| cis-butadiene || HOMO || [[Image:HOMO1_cis_butadienedh08.jpg|100x100px]] || [[Image:HOMO2_cis_butadienedh08.jpg|100x100px]]|| Anti-symmetric || -0.343
|-
| ethylene || LUMO || [[Image:LUMO1_ethenedh08.jpg|100x100px]] || [[Image:LUMO2_ethenedh08.jpg|100x100px]]|| Anti-symmetric || +0.052
|-
| ethylene || HOMO || [[Image:HOMO1_ethenedh08.jpg|100x100px]] || [[Image:HOMO2_ethenedh08.jpg|100x100px]]|| Symmetric || -0.387
|}

 
The symmetries of the LUMO and HOMO orbitals for cis-butadiene are opposite of those for the ethylene. This therefore means that the only interactions that can take place are those between ethylene's LUMO and cis-butadiene's HOMO and in addition to this the interaction between cis-butadiene's LUMO and ethylene's HOMO. This is because orbital symmetry has to be conserved (the conservation of orbital symmetry states that the symmetry of an new orbital formed by the overlap of the 2 original orbitals must be the same as the symmetry of the initial orbitals).

==='''Transition State'''===

A <jmol>
<jmolAppletButton>
<uploadedFileContents>Bicyclic_startingdh08.mol</uploadedFileContents>
<text>bicyclo system</text>
<color>white</color>
</jmolAppletButton>
</jmol> was drawn and one of the -CH2-CH2- fragments were removed. Three C=C bonds were added and the previous one was taken away so a <jmol>
<jmolAppletButton>
<uploadedFileContents>Frozen_coorddh08.mol</uploadedFileContents>
<text>guess transition state</text>
<color>white</color>
</jmolAppletButton>
</jmol> was produced.

After this an optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while bonds between the termini of the fragments were frozen at 2.1Å. Following this the resulting file had two separate optimisation and frequency calculations run on it using the TS (Berny) with one using the semi-empirical/AM1 method and the other using the DFT/B3LYP/6-31G* level of theory. Both of these optimisations had the bonds unfrozen and the "derivative" option was chosen instead.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11: Comparison of two methods of optimsations for the transition state '''
| || '''semi-empirical/AM1 method''' || '''DFT/B3LYP/6-31G* method'''
|-
| Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>SE_AM1_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>DFT_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Energy (a. u) || 0.11165608 || -234.54389573
|-
| Bond lengths || [[Image:DistancesSEdh08.jpg|150x150px]]|| [[Image:DistancesDFTdh08.jpg|150x150px]]
|-
|Imaginary frequency (cm-1) || -957 || -525
|-
|Animation || [[Image:-957dh08.gif]] || [[Image:-525dh08.gif]]
|-
| Lowest real frequency (cm-1) || 148 || 136
|-
|Animation || [[Image: 148dh08.gif]] || [[Image: 136dh08.gif]]

|}

The C-C bond forming length between the termini carbons is 0.15 Å larger for the DFT/B3LYP/6-31G* level of theory and this is because this method gives carries out the calculation to a greater degree of accuracy. This larger bond length leads a lower potential energy gradient to the transition state, which gives a lower total energy for the transition state. Therefore the given values below will be the DFT/B3LYP/6-31G* and not the semi-empirical/AM1 method.

A typical sp3 C-C bond length is 1.54 Å and a typical sp2 C=C bond length is 1.34 Å<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref>. Within the optimised transition molecule above the C-C bond lengths are 1.40 Å and the C=C bond length is 1.38 Å. This suggests that the π-bonds are breaking in order for the C-C bonds to form and additionally to this a π-bond is forming in between the two sp3 carbons.

The Van der Waals radius of a C-atom is 1.72 Å<ref name="ref 3">M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer, D. G. Truhlar, 1995, 346, pp. 63-69.[http://pubs.acs.org/doi/abs/10.1021/jp8111556]</ref>. The partly formed C-C length is 2.27 Å, which is shorter than two times the the van der Waals radius of a carbon atom. Therefore the carbon atoms are within a close enough distance to each other to share electron density and yet the bond has not formed at this point.

From looking at the animations for the imaginary frequency above it can be seen that the formation of the two bonds between the termini of the reactants is synchronous. This is expected of the Diels-Alder reaction since the mechanism is concerted. However the lowest real frequency is asynchronous with respect to the two bonds forming between the termini.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12: Frontier molecular orbitals of the transition state '''
| '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| LUMO || [[Image:LUMO1TSdh08.jpg|100x100px]] || [[Image:LUMO2TSdh08.jpg|100x100px]]|| Symmetric || +0.023
|-
| HOMO || [[Image:HOMO1TSdh08.jpg|100x100px]] || [[Image:HOMO2TSdh08.jpg|100x100px]]|| Anti-symmetric || -0.323
|}

From examining the images from table 12 above it can be deduced that the LUMO of the transition state originates from the cis-butadiene's LUMO and the ethylene's HOMO, which represents the σ*-antibonding orbitals. However the HOMO of the transition state originates from the cis-butadiene's HOMO and the ethylene's LUMO, which represents the σ-bonding orbitals within the two single C-C bonds that are formed within the reaction.

==='''Cyclohexa-1,3-diene reaction with maleic anhydride'''===

The <jmol>
<jmolAppletButton>
<uploadedFileContents>FrozenTS22dh08.mol</uploadedFileContents>
<text>exo</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_frozenTS22dh08.mol</uploadedFileContents>
<text>endo</text>
<color>white</color>
</jmolAppletButton>
</jmol> transition states were drawn on Gaussview and then the optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while the distance of the forming bonds were frozen at 2.1Å between the respective carbons. After this the structures changed slightly (<jmol>
<jmolAppletButton>
<uploadedFileContents>TS2_afteropt1_exo1dh08.mol</uploadedFileContents>
<text>exo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_TS2_afteropt1dh08.mol</uploadedFileContents>
<text>endo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol>) and the resulting files had an optimisation and frequency calculation run on it using the TA (Berny) with using the semi-empirical/AM1 level of theory while the unfrozen option as changed to "derivative". After this the endo and exo optimised transition state structures were re-optimised as they were before but using the DFT/B3LYP/6-31G* level of theory instead with the additional keyword "Opt=NoEigen".

{| class="wikitable" border="1" align="center"
|+ '''Table 13: Comparison of the exo and endo transition states '''
! rowspan="2"|-!! colspan="2"|''exo'' !! colspan="2"|''endo''
|-
| Semi-empirical/AM1 || DFT/B3LYP/6-31G* || Semi-empirical/AM1 || DFT/B3LYP/6-31G*
|-
| align="center"|Structure ||align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>After_opt2_exodh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>Endo_TS2_afteropt2dh081.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| align="center"| Energy (a. u) ||align="center"| -0.05041975 ||align="center"| -612.67931095 ||align="center"| -0.05150470 ||align="center"| -612.68339684
|-
| C-C bond forming length (Å) ||align="center"| 2.17 ||align="center"| 2.29 ||align="center"| 2.16 ||align="center"| 2.27
|-align="center"
| Imaginary frequency (cm-1) ||align="center"| -812 ||align="center"| -448 ||align="center"| -806 ||align="center"| -447
|-align="center"
| Animation ||colspan="2"|[[Image:-812_SEdh08.gif]] || colspan="2"|[[Image:-806_cm-1dh08.gif]]
|}

Again the C-C bond forming length between the termini carbons is larger (by 0.11 and 0.12 Å) for the DFT/B3LYP/6-31G* level of theory and this is due the same reasons talked about above. However the C-C bond forming length between the termini carbons are very similar in both the endo and exo transition states (0.02 Å difference).

From looking at the animations for the imaginary frequency above it can also be seen that the formation of the two bonds between the carbons in this case as well is synchronous. Again this is expected of the Diels-Alder reaction since it is concerted.

{| align=center
|+ '''Strain diagrams for the endo and exo transition states'''
| [[Image:Exostraindh08.jpg|thumb|200x200px|'''Exo TS''']]
| [[Image:Strainendodh08.jpg|thumb|200x200px|'''Endo TS''']]
|}

From studying the energies in table 13 it can be seen that the endo transition state is more stable than the exo by 4.08589 x 10-3 a. u (10.73 KJ mol-1). However by looking at the strain of both transition states one would expect the exo TS to be more stable. This is because the distance between the cyclohexa-1,3-diene ring and the -(C=O)-O-(C=O)- maleic anhydride fragment is 2.99 Å in the endo TS and 3.03 Å in the exo TS. Thus the endo form would have more strain due to steric hindrance.

However within the exo TS the vicinal hydrogens on the CH2-CH2 (carbon numbers 18 and 21) fragment on the cyclohexa-1,3-diene ring are found below the plane and next to the maleic anhydride. This leads to steric hindrance is this is where the strain originates from within the exo TS. Therefore this disfavours the exo form, which makes the endo more stable.

[[Image:Secondary_orbital_shizzledh08.jpg|500x500px|center]]

{| class="wikitable" border="1" align="center"
|+ '''Table 14: Frontier molecular orbitals of the the transition state for the cyclohexa-1,3-diene reaction '''
! rowspan="2"|Molecular Orbital !! colspan="4"|''exo'' !! colspan="4"|''endo''
|- align="center"
| '''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''||'''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''
|- align="center"
| LUMO || [[Image:LUMO1exoh081.jpg|150x150px]]|| [[Image:LUMO2exodh081.jpg|150x150px]]|| Anti-symmetric || -0.078 || [[Image:LUMO1ENDOdh08.jpg|150x150px]]||[[Image:LUMO2ENDOdh08.jpg|150x150px]] || Anti-symmetric || -0.067
|- align="center"
| HOMO || [[Image:HOMO1exodh081.jpg|150x150px]]||[[Image:HOMO2exodh081.jpg|150x150px]] || Anti-symmetric|| -0.242 || [[Image:HOMO1ENDOdh08.jpg|150x150px]]|| [[Image:HOMO2ENDOdh08.jpg|150x150px]]||Anti-symmetric || -0.242
|}

Digital repositories:

exo: http://hdl.handle.net/10042/to-5596

endo: http://hdl.handle.net/10042/to-5597

Why do you think that the exo form could be more strained?

Examine carefully the nodal properties of the HOMO between the -(C=O)-O-(C=O)- fragment and the remainder of the system.

What can you conclude about the so called “secondary orbital overlap effect”? (There is some discussion of this in Ian Fleming's book 'Frontier Orbitals and Organic Chemical Reactions').

Further discussion:
What effects have been neglected in these calculations of Diels Alder transition states?

=='''Conclusion'''==

='''References'''=
<references/>

Rep:Mod:hampluscheese

2010-11-10T22:08:50Z

Dh08: /* '''Cyclohexa-1,3-diene reaction with maleic anhydride''' */

='''Module 3:Physical Computational Chemistry Lab'''=

Douglas Hunt

=='''Objectives'''==

The aim of this experiment is to

=='''The Cope Rearrangement'''==

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangement] is a [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement |3,3|-sigmatropic rearrangement] that takes place through a single [http://en.wikipedia.org/wiki/Pericyclic pericyclic], concerted [http://en.wikipedia.org/wiki/Transition_state transition state]. Within this rearrangement a σ bond is formed in concert with another being broken within a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] π system.

The Cope rearrangement we will be studying will be that of 1,5-hexadiene, which is shown below.

[[Image:Cope_rearrangement_schemedh08.jpg|300x300px|center]]

The objective is to locate the low-energy minima and transition structures on the C6H10 potential energy surface and to determine the preferred reaction mechanism by using [http://en.wikipedia.org/wiki/GAUSSIAN Gaussian], with the [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)] methods.

==='''Optimisation of 1,5-hexadiene Conformers'''===

Rough <jmol>
<jmolAppletButton>
<uploadedFileContents>Anti_2_drawndh08.mol</uploadedFileContents>
<text>anti</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche3_drawndh08.mol</uploadedFileContents>
<text>gauche</text>
<color>white</color>
</jmolAppletButton>
</jmol> conformers of 1,5-hexadiene were drawn on Gaussview and their structures were tidied by the clean function. They were then further optimised by the HF/3-21G method and were identified which conformer they were from their point group and appendix 1 on the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3 physical lab page].

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 1: Optimisation and comparison of anti and gauche 1,5-hexadiene conformers using HF/3-21G methods
| ''' Conformer''' || '''Optimised Structure''' || '''Energy (Atomic Units)''' || '''[http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1 Energies (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (Anti2 - Gauche3)''' || '''Point group'''
|-
| '''Anti2''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Anti_2_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692535 || -231.69254 ||-6.08 x 105 || 1.2606 x 10-4 a. u|| Ci
|-
| '''Gauche3''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Gauche3_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692661 || -231.69266 || -6.08 x 105 || 3.31 x 10-1 KJ mol-1 || C1
|}

From studying table 1 it can be seen that the gauche conformer is 0.33 KJ mol-1 more stable than the anti conformation. One would assume that the anti2 conformer would be more stable due to [http://en.wikipedia.org/wiki/Steric steric] factors/repulsion within the gauche3 conformation. However within the gauche3 conformer there is a favourable donation of [http://en.wikipedia.org/wiki/Electron_density electron density] from the C=C π orbital to the C-H σ* orbital from the adjacent proton, which stabilises the conformer<ref name="ref 1">B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446.[http://www.informaworld.com/smpp/content~db=all?content=10.1080/00268970110081412]</ref>.

 

The anti2 conformer was then optimised using the B3LYP 6-31G* (6-31G (d)) method.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 2: Comparison of the anti2 conformer optimisations carried out at different calculation methods
| ''' Method''' || '''Structure and bond distances (Å)''' || '''Dihedral angle (C1 to C4)''' || '''Energy (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (HF/3-21G - DFT/B3LYP/6-31G)'''
|-
| HF/3-21G || [[Image:Anti2_opt1dh08.jpg|300x300px]] || 114.6 o || -231.692535 || -6.08 x 105 || 2.919171 a. u
|-
| DFT/B3LYP/6-31G* || [[Image:Anti2_opt2DH08.jpg|300x300px]] || 118.8 o || -234.611716 || -6.16 x 105 || 7.66 x 103 KJ mol-1
|}

From examining table 2 at first it is possible to see that the bond lengths are very similar to each other and that of the literature bond lengths<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref> (C1=C2: 1.340 Å, C2-C3: 1.508 Å and C3-C4: 1.538 Å). It is also possible to see that the C=C bond length is more accurate for the DFT/B3LYP/6-31G* method by it being 0.01 Å closer to the literature value. Another main difference in terms of geometry is that the DFT/B3LYP/6-31G* method gives a larger [http://en.wikipedia.org/wiki/Dihedral_angle dihedral angle] (and most likely more accurate) of 118.8o compared to that of 114.6o.

In terms of energy the DFT/B3LYP/6-31G* method produces a conformer of a higher stability by it possessing a lower energy with a difference of 7.66 x 103 KJ mol-1.

==='''Frequency analysis of 1,5-hexadiene Conformers'''===

[[Image:Vibrations_anti2dh08.jpg|right|300x300px]]

For the frequency analysis the DFT/B3LYP/6-31G* method was used.

Frequency analysis is essential within this calculation due to it being the second derivative of the potential energy surface. This means that if all the frequencies are all positive the system is at a minimum, if one of them is negative the optimised system is at a transition state, and if more than one are negative then the calculation has failed to find a critical point, which means the optimisation has failed. This therefore gives us a method of seeing whether the calculation has worked.

From looking to the image on the right it can deduced that the optimisation has taken place successfully by there being no negative vibrations and therefore the conformation is at a minimum.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 3: Energies under the thermochemistry section at 298.15 K and 0 K within the frequency logfiles'''
| '''Energies''' || '''298.15 K (atomic units)''' ||| '''0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (KJ mol-1)''' || '''Discussion'''
|-
| Sum of electronic and zero-point energies || -234.469177 || -234.466673 || -2.504 x 10-3 || -6.57 || This is the potential energy at 0 K, which includes the zero-point vibrational energy (E = Eelec + ZPE).

|-
| Sum of electronic and thermal energies || -234.461835 || -234.461085 || -7.5 x 10-4|| -1.97 || This is the energy at 298.15 K (0 K for 0 K calculation) and 1 atm of pressure, which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans).

|-
| Sum of electronic and thermal enthalpies || -234.460891 || -234.460141 || -7.5 x 10-4 || -1.97 || This contains an additional correction for RT (H = E + RT), which is especially important when looking at dissociation reactions.
|-
| Sum of electronic and thermal free energies || -234.500739 || -234.495931 || 4.808 x 10-3 || -12.62 || This includes the entropic contribution to the free energy (G = H - TS).
|}

{|align=right
|+
| [[Image:Anti2_freq_IRdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 298.15 K''']]
| [[Image:Anti2_freq_0Kdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 0 K''']]
|}

From looking at table 3 above it can be deduced that the anti2 conformer is more stable at room temperature to that of 0 K with respect to all the types of energies. The spectra also seem to not change much between the two temperatures but the main change is the change in the sum of electronic and thermal free energies, which is heavily dependent of temperature by it including the entropic contribution to the free energy (G = H - '''T'''S)

 
 
 
 

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

Within the Cope Rearrangement there are two possible transition states, the chair (C2h) and the boat (C2v), which are shown within [https://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_2 appendix two ].

==='''Comparison of optimisation methods for the chair transition state'''===

An <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2hdh08.mol</uploadedFileContents>
<text>allyl fragment</text>
<color>white</color>
</jmolAppletButton>
</jmol>, CH2CHCH2 was drawn on Gaussview and was optimized through the HF/3-21G method. After this the original allyl fragment was copied and pasted into a new window and place exactly above the first fragment with a bond distance of 2.20 Å between the terminal end carbons. This therefore produced the <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2h_fulldh08.mol</uploadedFileContents>
<text>chair transition structure.</text>
<color>white</color>
</jmolAppletButton>
</jmol>

Gaussian was then set up for an optimisation for the transition state with the additional key words "Opt=NoEigen", the optimisation set to TS (Berny), the force constants set to "once" and using the HF/3-21G method.

After this the "chair" transition state was optimised an alternative way by freezing the coordinates of the two sets of terminal carbons (where the bonds form/break during the rearrangement) within the Redundant Coord Editor on Gauusview.
Following this a second calculation was run where the freeze coordinates option was changed to "Derivative" and the force constants were set to "never".

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 4: Comparison of chair transition state optimisation methods
| || '''Optimised through a TS (Berny)''' || '''Optimised through a frozen coordinate method'''
|-
| '''Optimised TS Structure''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>C2h_optfreq_TS_Bdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| '''Terminal C-C bond length (Å)''' || 2.02 || 2.02
|-
|'''Energy (atomic units)''' || -231.61932247 || -231.61932190
|-
| '''ΔE, TS (Berny) - frozen coordinate''' || 5.7 x 10-7 a.u || 1.50 x 10-3 KJ mol-1
|-
| '''Imaginary frequency (cm-1)''' || -818 || -818
|-
| '''Animation''' ||[[Image:818dh08TSB.gif|350x350px]] || [[Image:818cm-1Fdh08.gif]]
|}

The data in table 4 shows that both methods give the same results and they both give the same transition state. They both produced the same energy with a negligible energy difference between them and the bond lengths between the terminal carbons are both 2.02 Å. In addition to this the imaginary frequencies were both -818cm-1 and within the animations above it is possible to see that when one C-C bond is forming the other is breaking. This means it is a asynchronous bond formation, which is expected for the Cope Rearrangement since it is a pericyclic reaction.

The only difference between the two optimisations is the way in which the transition states were calculated. This is due to the TS (Berny) method calculating the force constant matrix within the first optimisation step and is then being recalculated as the optimisation carries on. When the frozen coordinate method freezes the reaction coordinate in order for the rest of the structure to be optimised around it, which does not require the whole hessian (force constant matrix) to be calculated but only that the reaction coordinate is differentiated (second step). The main advantage of this, is that you do not need to calculate the entire force constant matrix and just need to differentiate the reaction coordinate in order to obtain a good enough guess for the initial force constant matrix. The frozen coordinate method will therefore take less time to compute the transition state, which can be very expensive within certain systems. However to carry out this method one needs to have an idea of how far apart the terminal carbons are.

==='''Boat transition state'''===

===='''QST2 optimisation'''====

The optimisation for the boat transition state had a different method and this was the QST2 method. This calculation interpolates between two structures (product and reactant) and tries to find a transition state between them.

The reactants and products were numbered in the same way. Therefore both 1,5-hexadiene molecules had their numbering manually changed for the product molecule so it corresponded to the numbering of the reactant if it rearranged. This is illustrated in the reaction scheme below.

[[Image:Boat_rerrangmentdh08.jpg|300x300px|center]]

The chk file for the optimized anti2 conformer was opened and copied into a new window. A second window was opened within the new window and the same anti2 conformer was copied into the new window. The numbering was changed within the second molecule so the two looked like they do in the image below.

[[Image:TS_(QST2)_numbering_1dh08.jpg|400x400px|center]]

The job type was set as "Opt+Freq" and "TS (QST2)" was chosen to optimise the transition state. However this failed and produced
a structure a bit like a <jmol>
<jmolAppletButton>
<uploadedFileContents>Trialone_BPdh08.mol</uploadedFileContents>
<text>chair transition structure </text>
<color>white</color>
</jmolAppletButton>
</jmol> but more dissociated. This is because it interpolated between the two structures and translated the top allyl fragment and therefore did not consider the possibility of a rotation around the central bonds.

To get around this the C-C-C-C dihedral angle (C2-C3-C4-C5) was set to 0 o and the C-C-C angle (C2-C3-C4 and C3-C4-C5 for the product molecule) was set to 100 o for both molecules. This gave rise to the two structures below.

[[Image:TS_(QST2)_numbering_2dh08.jpg|400x400px|center]]

After this the QST2 calculation was run again and the results in table 5 were produced. The imaginary frequency at -839 cm-1 shows that while one bond is forming the other one is breaking, which is again an asynchronous bond formation.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5: Data for the boat transition state from the QST2 optimisation method
| Optimised TS Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.14
|-
|Energy (atomic units) || -231.60280151
|-
| Energy (KJ mol-1) || -6.08 x 105
|-
| Imaginary frequency (cm-1) || -839
|-
| Animation ||[[Image:-839dh08.gif]]
|}

===='''Intrinsic Reaction Coordinate'''====

It is not possible to predict which product the chair and boat transition state will lead to. To solve this problem within Gaussian one can use the IRC method, which allows one to follow the minimum energy path from a transition structure down to its local minimum on a potential energy surface. This creates a succession of points by taking small geometry steps in the direction to where the gradient of the energy surface is at its steepest until the minimum in energy is reached.

The chk file for the optimized chair transition structure was opened and the job type selected was "IRC", the reaction coordinate was computed in the forward direction (since molecule is symmetric) with 50 points along the IRC. This was done with two type of force constants being calculated, once and always.

Within the "once" method the output showed no bond formation process by the 26th step and therefore the calculation was incomplete, since a complete 1,5-hexadiene structure was supposed to be formed. From looking at the IRC pathway it can be seen that the minimum on the potential energy surface was not reached at that point, since the points on the pathway did not form an asymptote and the gradient did not go close enough to zero.

For the "always" method the computation of force constants were carried out at every step. This therefore lets the IRC calculation distinguish whether it is moving along the minimum energy pathway while going down the potential energy surface. While this is happening it can make changes to the pathway if necessary. Within the "always" case the bond forming arrives at 1,5-hexadiene and this is backed up by the IRC pathway reaching a minimum within its energy by the gradient going close to zero and the pathway reaches an asymptote.

The structure produced through the always method had an energy of -231.69199434 atomic units, which is very close to the energy of the gauche2 conformer in the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 appendix 1]. This suggests that the reaction takes place through the chair transition state, which later produces the gauche2 conformer of 1,5-hexadiene.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 6: Comparison of IRC calculations for the chair structure
| || '''Calculate force constants once''' || '''Calculate force constants always'''
|-
| '''Number of points along IRC''' || 26 || 47
|-
| '''Structure at the last point of IRC''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Once_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Always_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|'''Energy (atomic units)''' || -231.68906923 || -231.69199434
|-
| '''ΔE, once - always''' || 2.92511 x 10-3 a.u || 7.68 KJ mol-1
|-
| '''IRC pathway''' || [[Image:One_pathwaydh08.jpg|400x400px]] || [[Image:Always_pathwaydh08.jpg|400x400px]]
|-
| '''IRC gradient''' || [[Image:One_gradientdh08.jpg|400x400px]]|| [[Image:Always_gradientdh08.jpg|400x400px]]
|}

 
 
 

===='''Comparison of activation energies for the transition states'''====

The HF/3-21G optimised chair and boat transition structures were both reoptimised using the B3LYP/6-31G* (B3LYP/6-31G (d)) method. This gives a more accurate minimisation however this method takes longer to run. The values for the zero-point energies and thermal energies were found within the thermochemistry section of the log file.

The values for the HF/3-21G reactant (anti2) optimisation were calculated and the B3LYP/6-31G* values were taken from the "Frequency analysis of 1,5-hexadiene Conformers" earlier in this module.

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 7: Energy data for chair, boat and the anti2 conformation from the HF/3-21G and B3LYP/6-31G* methods of optimisation '''
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466677
| align="center" | -231.461320
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450936
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402342
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539524
| align="center" | -231.532562
| align="center" | -234.611716
| align="center" | -234.469177
| align="center" | -234.461835
|}
Digital repositories

Chair:
HF/3-21G: http://hdl.handle.net/10042/to-5569
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5568

Boat:
HF/3-21G: http://hdl.handle.net/10042/to-5570
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5571

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 8: Summary of activation energies for the chair and boat conformations (ΔE = chair TS or boat TS minus anti2)'''
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| align="center" | 1 a.u = 627.509 kcal mol-1
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE, chair (a.u)'''
| align="center" | 0.072847
| align="center" | 0.071242
| align="center" | 0.054248
| align="center" | 0.052827
| align="center" | -
|-
| '''ΔE, chair (kcal mol-1)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE, boat (a. u)'''
| align="center" | 0.088588
| align="center" | 0.087259
| align="center" | 0.066835
| align="center" | 0.065827
| align="center" | -
|-
| '''ΔE, boat (kcal mol-1)'''
| align="center" | 55.59
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.31
| align="center" | 44.7 ± 2.0
|}

By examining table 8 it is clear to see that the B3LYP/6-31G* method gives values which are much closer to the experimental ones. This is due to the B3LYP/6-31G* giving a higher level of optimisation than the HF/3-21G method. Table 8 also shows that the chair transition state has a lower activation energy to that of the boat transition state by 8.16 kcal mol-1. This suggests that the Cope rearrangement mechanism would most likely proceed through the chair transition state to that of the boat. Therefore the reaction would take place from the anti2 conformer to the chair transition state and then onto the gauche2 conformer within the Cope rearrangement.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9: Geometric parameters for the optimised chair and boat structures '''
| || ''' HF/3-21G chair TS''' || ''' B3LYP/6-31G* chair TS''' || '''HF/3-21G boat TS''' || '''B3LYP/6-31G* boat TS'''
|-
| Optimised Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Chair_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Boat_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.02 (C5-C10 or C3-C14)|| 1.97 (C5-C10 or C3-C14) || 2.14 (C3-C4 or C1-C6) || 2.21 (C3-C4 or C1-C6)
|-
| Fragment angle || 120 o (C3-C1-C5) || 120 o (C3-C1-C5) || 122 o (C4-C5-C6) || 122 o (C4-C5-C6)
|-
| Dihedral angle || 55 o (C1-C3-C14-C11)|| 54 o (C1-C3-C14-C11)|| 0 o (C2-C1-C6-C5) || 0 o (C2-C1-C6-C5)
|-
| Numbering of carbons || [[Image:Numbering_chairdh08.jpg|200x200px]]|| [[Image:Numbering_chairdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]]
|}

From studying table 9 it can be seen that the C-C bond breaking/forming length has changed between the two different types of optimisation methods. However the angles between the bonds remains unchanged.

=='''The Diels Alder Cycloaddition'''==

The Diels-Alder reaction is a cycloaddition between a conjugated diene and a dienophile (alkene system), which forms a cyclohexene system. It is often referred to as a [4s+2s] cycloaddition and proceeds through a single cyclic, concerted transition state where 2 σ bonds are formed in between the termini carbons of the 2 conjugated п systems. This is shown in the reaction scheme below where ethene is reacting with cis-butadiene.

[[Image:Reaction_scheme2dh08.jpg|300x300px|center]]

The π orbitals of the dienophile form new σ bonds with the π orbitals of the diene. The number of π electrons involved determine whether the reaction proceeds in a concerted stereospecific fashion (allowed) or not (forbidden). The HOMO/LUMO of one fragment interacts with the HOMO/LUMO of the other fragment to form two new bonding and anti-bonding MOs. The nodal properties allow one to predict how the reaction will occur.

Semi-Empirical/AM1 method and the B3LYP/6-31G* method of optimisation will be used on Gaussian to determine particular properites of the reactants to predict the way in which the reaction will proceed.

==='''Reagents'''===

<jmol>
<jmolAppletButton>
<uploadedFileContents>Cis_butadienedh08.mol</uploadedFileContents>
<text>Cis-butadiene</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Ethenedh08.mol</uploadedFileContents>
<text>ethylene</text>
<color>white</color>
</jmolAppletButton>
</jmol> were optimised separately using the AM1 semi-empirical method on Gaussview.The cis-butadiene had an energy of 0.04879725 a. u (1.28 x 102 KJ mol-1) and ethylene had an energy of 0.02619027 a. u (68.76 KJ mol-1).

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10: Frontier molecular orbitals of cis-butadiene and ethylene '''
| '''Reagent''' || '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| cis-butadiene || LUMO || [[Image:LUMO1_cis_butadieneDH08.jpg|100x100px]] || [[Image:LUMO2_cis_butadienedh08.jpg|100x100px]]|| Symmetric || +0.017
|-
| cis-butadiene || HOMO || [[Image:HOMO1_cis_butadienedh08.jpg|100x100px]] || [[Image:HOMO2_cis_butadienedh08.jpg|100x100px]]|| Anti-symmetric || -0.343
|-
| ethylene || LUMO || [[Image:LUMO1_ethenedh08.jpg|100x100px]] || [[Image:LUMO2_ethenedh08.jpg|100x100px]]|| Anti-symmetric || +0.052
|-
| ethylene || HOMO || [[Image:HOMO1_ethenedh08.jpg|100x100px]] || [[Image:HOMO2_ethenedh08.jpg|100x100px]]|| Symmetric || -0.387
|}

 
The symmetries of the LUMO and HOMO orbitals for cis-butadiene are opposite of those for the ethylene. This therefore means that the only interactions that can take place are those between ethylene's LUMO and cis-butadiene's HOMO and in addition to this the interaction between cis-butadiene's LUMO and ethylene's HOMO. This is because orbital symmetry has to be conserved (the conservation of orbital symmetry states that the symmetry of an new orbital formed by the overlap of the 2 original orbitals must be the same as the symmetry of the initial orbitals).

==='''Transition State'''===

A <jmol>
<jmolAppletButton>
<uploadedFileContents>Bicyclic_startingdh08.mol</uploadedFileContents>
<text>bicyclo system</text>
<color>white</color>
</jmolAppletButton>
</jmol> was drawn and one of the -CH2-CH2- fragments were removed. Three C=C bonds were added and the previous one was taken away so a <jmol>
<jmolAppletButton>
<uploadedFileContents>Frozen_coorddh08.mol</uploadedFileContents>
<text>guess transition state</text>
<color>white</color>
</jmolAppletButton>
</jmol> was produced.

After this an optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while bonds between the termini of the fragments were frozen at 2.1Å. Following this the resulting file had two separate optimisation and frequency calculations run on it using the TS (Berny) with one using the semi-empirical/AM1 method and the other using the DFT/B3LYP/6-31G* level of theory. Both of these optimisations had the bonds unfrozen and the "derivative" option was chosen instead.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11: Comparison of two methods of optimsations for the transition state '''
| || '''semi-empirical/AM1 method''' || '''DFT/B3LYP/6-31G* method'''
|-
| Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>SE_AM1_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>DFT_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Energy (a. u) || 0.11165608 || -234.54389573
|-
| Bond lengths || [[Image:DistancesSEdh08.jpg|150x150px]]|| [[Image:DistancesDFTdh08.jpg|150x150px]]
|-
|Imaginary frequency (cm-1) || -957 || -525
|-
|Animation || [[Image:-957dh08.gif]] || [[Image:-525dh08.gif]]
|-
| Lowest real frequency (cm-1) || 148 || 136
|-
|Animation || [[Image: 148dh08.gif]] || [[Image: 136dh08.gif]]

|}

The C-C bond forming length between the termini carbons is 0.15 Å larger for the DFT/B3LYP/6-31G* level of theory and this is because this method gives carries out the calculation to a greater degree of accuracy. This larger bond length leads a lower potential energy gradient to the transition state, which gives a lower total energy for the transition state. Therefore the given values below will be the DFT/B3LYP/6-31G* and not the semi-empirical/AM1 method.

A typical sp3 C-C bond length is 1.54 Å and a typical sp2 C=C bond length is 1.34 Å<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref>. Within the optimised transition molecule above the C-C bond lengths are 1.40 Å and the C=C bond length is 1.38 Å. This suggests that the π-bonds are breaking in order for the C-C bonds to form and additionally to this a π-bond is forming in between the two sp3 carbons.

The Van der Waals radius of a C-atom is 1.72 Å<ref name="ref 3">M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer, D. G. Truhlar, 1995, 346, pp. 63-69.[http://pubs.acs.org/doi/abs/10.1021/jp8111556]</ref>. The partly formed C-C length is 2.27 Å, which is shorter than two times the the van der Waals radius of a carbon atom. Therefore the carbon atoms are within a close enough distance to each other to share electron density and yet the bond has not formed at this point.

From looking at the animations for the imaginary frequency above it can be seen that the formation of the two bonds between the termini of the reactants is synchronous. This is expected of the Diels-Alder reaction since the mechanism is concerted. However the lowest real frequency is asynchronous with respect to the two bonds forming between the termini.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12: Frontier molecular orbitals of the transition state '''
| '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| LUMO || [[Image:LUMO1TSdh08.jpg|100x100px]] || [[Image:LUMO2TSdh08.jpg|100x100px]]|| Symmetric || +0.023
|-
| HOMO || [[Image:HOMO1TSdh08.jpg|100x100px]] || [[Image:HOMO2TSdh08.jpg|100x100px]]|| Anti-symmetric || -0.323
|}

From examining the images from table 12 above it can be deduced that the LUMO of the transition state originates from the cis-butadiene's LUMO and the ethylene's HOMO, which represents the σ*-antibonding orbitals. However the HOMO of the transition state originates from the cis-butadiene's HOMO and the ethylene's LUMO, which represents the σ-bonding orbitals within the two single C-C bonds that are formed within the reaction.

==='''Cyclohexa-1,3-diene reaction with maleic anhydride'''===

The <jmol>
<jmolAppletButton>
<uploadedFileContents>FrozenTS22dh08.mol</uploadedFileContents>
<text>exo</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_frozenTS22dh08.mol</uploadedFileContents>
<text>endo</text>
<color>white</color>
</jmolAppletButton>
</jmol> transition states were drawn on Gaussview and then the optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while the distance of the forming bonds were frozen at 2.1Å between the respective carbons. After this the structures changed slightly (<jmol>
<jmolAppletButton>
<uploadedFileContents>TS2_afteropt1_exo1dh08.mol</uploadedFileContents>
<text>exo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_TS2_afteropt1dh08.mol</uploadedFileContents>
<text>endo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol>) and the resulting files had an optimisation and frequency calculation run on it using the TA (Berny) with using the semi-empirical/AM1 level of theory while the unfrozen option as changed to "derivative". After this the endo and exo optimised transition state structures were re-optimised as they were before but using the DFT/B3LYP/6-31G* level of theory instead with the additional keyword "Opt=NoEigen".

{| class="wikitable" border="1" align="center"
|+ '''Table 13: Comparison of the exo and endo transition states '''
! rowspan="2"|-!! colspan="2"|''exo'' !! colspan="2"|''endo''
|-
| Semi-empirical/AM1 || DFT/B3LYP/6-31G* || Semi-empirical/AM1 || DFT/B3LYP/6-31G*
|-
| align="center"|Structure ||align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>After_opt2_exodh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>Endo_TS2_afteropt2dh081.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| align="center"| Energy (a. u) ||align="center"| -0.05041975 ||align="center"| -612.67931095 ||align="center"| -0.05150470 ||align="center"| -612.68339684
|-
| C-C bond forming length (Å) ||align="center"| 2.17 ||align="center"| 2.29 ||align="center"| 2.16 ||align="center"| 2.27
|-align="center"
| Imaginary frequency (cm-1) ||align="center"| -812 ||align="center"| -448 ||align="center"| -806 ||align="center"| -447
|-align="center"
| Animation ||colspan="2"|[[Image:-812_SEdh08.gif]] || colspan="2"|[[Image:-806_cm-1dh08.gif]]
|}

Again the C-C bond forming length between the termini carbons is larger (by 0.11 and 0.12 Å) for the DFT/B3LYP/6-31G* level of theory and this is due the same reasons talked about above. However the C-C bond forming length between the termini carbons are very similar in both the endo and exo transition states (0.02 Å difference).

From looking at the animations for the imaginary frequency above it can also be seen that the formation of the two bonds between the carbons in this case as well is synchronous. Again this is expected of the Diels-Alder reaction since it is concerted.

{| align=center
|+ '''Strain diagrams for the endo and exo transition states'''
| [[Image:Exostraindh08.jpg|thumb|200x200px|'''Exo TS''']]
| [[Image:Strainendodh08.jpg|thumb|200x200px|'''Endo TS''']]
|}

From studying the energies in table 13 it can be seen that the endo transition state is more stable than the exo by 4.08589 x 10-3 a. u (10.73 KJ mol-1). However by looking at the strain of both transition states one would expect the exo TS to be more stable. This is because the distance between the cyclohexa-1,3-diene ring and the -(C=O)-O-(C=O)- maleic anhydride fragment is 2.99 Å in the endo TS and 3.03 Å in the exo TS. Thus the endo form would have more strain due to steric hindrance.

However within the exo TS the vicinal hydrogens on the CH2-CH2 (carbon numbers 18 and 21) fragment on the cyclohexa-1,3-diene ring are found below the plane and next to the maleic anhydride. This leads to steric hindrance is this is where the strain originates from within the exo TS. Therefore this disfavours the exo form, which makes the endo more stable.

[[Image:Secondary_orbital_shizzledh08.jpg|center]]

{| class="wikitable" border="1" align="center"
|+ '''Table 14: Frontier molecular orbitals of the the transition state for the cyclohexa-1,3-diene reaction '''
! rowspan="2"|Molecular Orbital !! colspan="4"|''exo'' !! colspan="4"|''endo''
|- align="center"
| '''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''||'''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''
|- align="center"
| LUMO || [[Image:LUMO1exoh081.jpg|150x150px]]|| [[Image:LUMO2exodh081.jpg|150x150px]]|| Anti-symmetric || -0.078 || [[Image:LUMO1ENDOdh08.jpg|150x150px]]||[[Image:LUMO2ENDOdh08.jpg|150x150px]] || Anti-symmetric || -0.067
|- align="center"
| HOMO || [[Image:HOMO1exodh081.jpg|150x150px]]||[[Image:HOMO2exodh081.jpg|150x150px]] || Anti-symmetric|| -0.242 || [[Image:HOMO1ENDOdh08.jpg|150x150px]]|| [[Image:HOMO2ENDOdh08.jpg|150x150px]]||Anti-symmetric || -0.242
|}

Digital repositories:

exo: http://hdl.handle.net/10042/to-5596

endo: http://hdl.handle.net/10042/to-5597

Why do you think that the exo form could be more strained?

Examine carefully the nodal properties of the HOMO between the -(C=O)-O-(C=O)- fragment and the remainder of the system.

What can you conclude about the so called “secondary orbital overlap effect”? (There is some discussion of this in Ian Fleming's book 'Frontier Orbitals and Organic Chemical Reactions').

Further discussion:
What effects have been neglected in these calculations of Diels Alder transition states?

=='''Conclusion'''==

='''References'''=
<references/>

File:Secondary orbital shizzledh08.jpg

2010-11-10T22:08:38Z

Dh08:

Rep:Mod:hampluscheese

2010-11-10T21:49:23Z

Dh08: /* '''Cyclohexa-1,3-diene reaction with maleic anhydride''' */

='''Module 3:Physical Computational Chemistry Lab'''=

Douglas Hunt

=='''Objectives'''==

The aim of this experiment is to

=='''The Cope Rearrangement'''==

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangement] is a [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement |3,3|-sigmatropic rearrangement] that takes place through a single [http://en.wikipedia.org/wiki/Pericyclic pericyclic], concerted [http://en.wikipedia.org/wiki/Transition_state transition state]. Within this rearrangement a σ bond is formed in concert with another being broken within a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] π system.

The Cope rearrangement we will be studying will be that of 1,5-hexadiene, which is shown below.

[[Image:Cope_rearrangement_schemedh08.jpg|300x300px|center]]

The objective is to locate the low-energy minima and transition structures on the C6H10 potential energy surface and to determine the preferred reaction mechanism by using [http://en.wikipedia.org/wiki/GAUSSIAN Gaussian], with the [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)] methods.

==='''Optimisation of 1,5-hexadiene Conformers'''===

Rough <jmol>
<jmolAppletButton>
<uploadedFileContents>Anti_2_drawndh08.mol</uploadedFileContents>
<text>anti</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche3_drawndh08.mol</uploadedFileContents>
<text>gauche</text>
<color>white</color>
</jmolAppletButton>
</jmol> conformers of 1,5-hexadiene were drawn on Gaussview and their structures were tidied by the clean function. They were then further optimised by the HF/3-21G method and were identified which conformer they were from their point group and appendix 1 on the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3 physical lab page].

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 1: Optimisation and comparison of anti and gauche 1,5-hexadiene conformers using HF/3-21G methods
| ''' Conformer''' || '''Optimised Structure''' || '''Energy (Atomic Units)''' || '''[http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1 Energies (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (Anti2 - Gauche3)''' || '''Point group'''
|-
| '''Anti2''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Anti_2_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692535 || -231.69254 ||-6.08 x 105 || 1.2606 x 10-4 a. u|| Ci
|-
| '''Gauche3''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Gauche3_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692661 || -231.69266 || -6.08 x 105 || 3.31 x 10-1 KJ mol-1 || C1
|}

From studying table 1 it can be seen that the gauche conformer is 0.33 KJ mol-1 more stable than the anti conformation. One would assume that the anti2 conformer would be more stable due to [http://en.wikipedia.org/wiki/Steric steric] factors/repulsion within the gauche3 conformation. However within the gauche3 conformer there is a favourable donation of [http://en.wikipedia.org/wiki/Electron_density electron density] from the C=C π orbital to the C-H σ* orbital from the adjacent proton, which stabilises the conformer<ref name="ref 1">B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446.[http://www.informaworld.com/smpp/content~db=all?content=10.1080/00268970110081412]</ref>.

 

The anti2 conformer was then optimised using the B3LYP 6-31G* (6-31G (d)) method.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 2: Comparison of the anti2 conformer optimisations carried out at different calculation methods
| ''' Method''' || '''Structure and bond distances (Å)''' || '''Dihedral angle (C1 to C4)''' || '''Energy (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (HF/3-21G - DFT/B3LYP/6-31G)'''
|-
| HF/3-21G || [[Image:Anti2_opt1dh08.jpg|300x300px]] || 114.6 o || -231.692535 || -6.08 x 105 || 2.919171 a. u
|-
| DFT/B3LYP/6-31G* || [[Image:Anti2_opt2DH08.jpg|300x300px]] || 118.8 o || -234.611716 || -6.16 x 105 || 7.66 x 103 KJ mol-1
|}

From examining table 2 at first it is possible to see that the bond lengths are very similar to each other and that of the literature bond lengths<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref> (C1=C2: 1.340 Å, C2-C3: 1.508 Å and C3-C4: 1.538 Å). It is also possible to see that the C=C bond length is more accurate for the DFT/B3LYP/6-31G* method by it being 0.01 Å closer to the literature value. Another main difference in terms of geometry is that the DFT/B3LYP/6-31G* method gives a larger [http://en.wikipedia.org/wiki/Dihedral_angle dihedral angle] (and most likely more accurate) of 118.8o compared to that of 114.6o.

In terms of energy the DFT/B3LYP/6-31G* method produces a conformer of a higher stability by it possessing a lower energy with a difference of 7.66 x 103 KJ mol-1.

==='''Frequency analysis of 1,5-hexadiene Conformers'''===

[[Image:Vibrations_anti2dh08.jpg|right|300x300px]]

For the frequency analysis the DFT/B3LYP/6-31G* method was used.

Frequency analysis is essential within this calculation due to it being the second derivative of the potential energy surface. This means that if all the frequencies are all positive the system is at a minimum, if one of them is negative the optimised system is at a transition state, and if more than one are negative then the calculation has failed to find a critical point, which means the optimisation has failed. This therefore gives us a method of seeing whether the calculation has worked.

From looking to the image on the right it can deduced that the optimisation has taken place successfully by there being no negative vibrations and therefore the conformation is at a minimum.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 3: Energies under the thermochemistry section at 298.15 K and 0 K within the frequency logfiles'''
| '''Energies''' || '''298.15 K (atomic units)''' ||| '''0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (KJ mol-1)''' || '''Discussion'''
|-
| Sum of electronic and zero-point energies || -234.469177 || -234.466673 || -2.504 x 10-3 || -6.57 || This is the potential energy at 0 K, which includes the zero-point vibrational energy (E = Eelec + ZPE).

|-
| Sum of electronic and thermal energies || -234.461835 || -234.461085 || -7.5 x 10-4|| -1.97 || This is the energy at 298.15 K (0 K for 0 K calculation) and 1 atm of pressure, which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans).

|-
| Sum of electronic and thermal enthalpies || -234.460891 || -234.460141 || -7.5 x 10-4 || -1.97 || This contains an additional correction for RT (H = E + RT), which is especially important when looking at dissociation reactions.
|-
| Sum of electronic and thermal free energies || -234.500739 || -234.495931 || 4.808 x 10-3 || -12.62 || This includes the entropic contribution to the free energy (G = H - TS).
|}

{|align=right
|+
| [[Image:Anti2_freq_IRdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 298.15 K''']]
| [[Image:Anti2_freq_0Kdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 0 K''']]
|}

From looking at table 3 above it can be deduced that the anti2 conformer is more stable at room temperature to that of 0 K with respect to all the types of energies. The spectra also seem to not change much between the two temperatures but the main change is the change in the sum of electronic and thermal free energies, which is heavily dependent of temperature by it including the entropic contribution to the free energy (G = H - '''T'''S)

 
 
 
 

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

Within the Cope Rearrangement there are two possible transition states, the chair (C2h) and the boat (C2v), which are shown within [https://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_2 appendix two ].

==='''Comparison of optimisation methods for the chair transition state'''===

An <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2hdh08.mol</uploadedFileContents>
<text>allyl fragment</text>
<color>white</color>
</jmolAppletButton>
</jmol>, CH2CHCH2 was drawn on Gaussview and was optimized through the HF/3-21G method. After this the original allyl fragment was copied and pasted into a new window and place exactly above the first fragment with a bond distance of 2.20 Å between the terminal end carbons. This therefore produced the <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2h_fulldh08.mol</uploadedFileContents>
<text>chair transition structure.</text>
<color>white</color>
</jmolAppletButton>
</jmol>

Gaussian was then set up for an optimisation for the transition state with the additional key words "Opt=NoEigen", the optimisation set to TS (Berny), the force constants set to "once" and using the HF/3-21G method.

After this the "chair" transition state was optimised an alternative way by freezing the coordinates of the two sets of terminal carbons (where the bonds form/break during the rearrangement) within the Redundant Coord Editor on Gauusview.
Following this a second calculation was run where the freeze coordinates option was changed to "Derivative" and the force constants were set to "never".

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 4: Comparison of chair transition state optimisation methods
| || '''Optimised through a TS (Berny)''' || '''Optimised through a frozen coordinate method'''
|-
| '''Optimised TS Structure''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>C2h_optfreq_TS_Bdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| '''Terminal C-C bond length (Å)''' || 2.02 || 2.02
|-
|'''Energy (atomic units)''' || -231.61932247 || -231.61932190
|-
| '''ΔE, TS (Berny) - frozen coordinate''' || 5.7 x 10-7 a.u || 1.50 x 10-3 KJ mol-1
|-
| '''Imaginary frequency (cm-1)''' || -818 || -818
|-
| '''Animation''' ||[[Image:818dh08TSB.gif|350x350px]] || [[Image:818cm-1Fdh08.gif]]
|}

The data in table 4 shows that both methods give the same results and they both give the same transition state. They both produced the same energy with a negligible energy difference between them and the bond lengths between the terminal carbons are both 2.02 Å. In addition to this the imaginary frequencies were both -818cm-1 and within the animations above it is possible to see that when one C-C bond is forming the other is breaking. This means it is a asynchronous bond formation, which is expected for the Cope Rearrangement since it is a pericyclic reaction.

The only difference between the two optimisations is the way in which the transition states were calculated. This is due to the TS (Berny) method calculating the force constant matrix within the first optimisation step and is then being recalculated as the optimisation carries on. When the frozen coordinate method freezes the reaction coordinate in order for the rest of the structure to be optimised around it, which does not require the whole hessian (force constant matrix) to be calculated but only that the reaction coordinate is differentiated (second step). The main advantage of this, is that you do not need to calculate the entire force constant matrix and just need to differentiate the reaction coordinate in order to obtain a good enough guess for the initial force constant matrix. The frozen coordinate method will therefore take less time to compute the transition state, which can be very expensive within certain systems. However to carry out this method one needs to have an idea of how far apart the terminal carbons are.

==='''Boat transition state'''===

===='''QST2 optimisation'''====

The optimisation for the boat transition state had a different method and this was the QST2 method. This calculation interpolates between two structures (product and reactant) and tries to find a transition state between them.

The reactants and products were numbered in the same way. Therefore both 1,5-hexadiene molecules had their numbering manually changed for the product molecule so it corresponded to the numbering of the reactant if it rearranged. This is illustrated in the reaction scheme below.

[[Image:Boat_rerrangmentdh08.jpg|300x300px|center]]

The chk file for the optimized anti2 conformer was opened and copied into a new window. A second window was opened within the new window and the same anti2 conformer was copied into the new window. The numbering was changed within the second molecule so the two looked like they do in the image below.

[[Image:TS_(QST2)_numbering_1dh08.jpg|400x400px|center]]

The job type was set as "Opt+Freq" and "TS (QST2)" was chosen to optimise the transition state. However this failed and produced
a structure a bit like a <jmol>
<jmolAppletButton>
<uploadedFileContents>Trialone_BPdh08.mol</uploadedFileContents>
<text>chair transition structure </text>
<color>white</color>
</jmolAppletButton>
</jmol> but more dissociated. This is because it interpolated between the two structures and translated the top allyl fragment and therefore did not consider the possibility of a rotation around the central bonds.

To get around this the C-C-C-C dihedral angle (C2-C3-C4-C5) was set to 0 o and the C-C-C angle (C2-C3-C4 and C3-C4-C5 for the product molecule) was set to 100 o for both molecules. This gave rise to the two structures below.

[[Image:TS_(QST2)_numbering_2dh08.jpg|400x400px|center]]

After this the QST2 calculation was run again and the results in table 5 were produced. The imaginary frequency at -839 cm-1 shows that while one bond is forming the other one is breaking, which is again an asynchronous bond formation.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5: Data for the boat transition state from the QST2 optimisation method
| Optimised TS Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.14
|-
|Energy (atomic units) || -231.60280151
|-
| Energy (KJ mol-1) || -6.08 x 105
|-
| Imaginary frequency (cm-1) || -839
|-
| Animation ||[[Image:-839dh08.gif]]
|}

===='''Intrinsic Reaction Coordinate'''====

It is not possible to predict which product the chair and boat transition state will lead to. To solve this problem within Gaussian one can use the IRC method, which allows one to follow the minimum energy path from a transition structure down to its local minimum on a potential energy surface. This creates a succession of points by taking small geometry steps in the direction to where the gradient of the energy surface is at its steepest until the minimum in energy is reached.

The chk file for the optimized chair transition structure was opened and the job type selected was "IRC", the reaction coordinate was computed in the forward direction (since molecule is symmetric) with 50 points along the IRC. This was done with two type of force constants being calculated, once and always.

Within the "once" method the output showed no bond formation process by the 26th step and therefore the calculation was incomplete, since a complete 1,5-hexadiene structure was supposed to be formed. From looking at the IRC pathway it can be seen that the minimum on the potential energy surface was not reached at that point, since the points on the pathway did not form an asymptote and the gradient did not go close enough to zero.

For the "always" method the computation of force constants were carried out at every step. This therefore lets the IRC calculation distinguish whether it is moving along the minimum energy pathway while going down the potential energy surface. While this is happening it can make changes to the pathway if necessary. Within the "always" case the bond forming arrives at 1,5-hexadiene and this is backed up by the IRC pathway reaching a minimum within its energy by the gradient going close to zero and the pathway reaches an asymptote.

The structure produced through the always method had an energy of -231.69199434 atomic units, which is very close to the energy of the gauche2 conformer in the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 appendix 1]. This suggests that the reaction takes place through the chair transition state, which later produces the gauche2 conformer of 1,5-hexadiene.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 6: Comparison of IRC calculations for the chair structure
| || '''Calculate force constants once''' || '''Calculate force constants always'''
|-
| '''Number of points along IRC''' || 26 || 47
|-
| '''Structure at the last point of IRC''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Once_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Always_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|'''Energy (atomic units)''' || -231.68906923 || -231.69199434
|-
| '''ΔE, once - always''' || 2.92511 x 10-3 a.u || 7.68 KJ mol-1
|-
| '''IRC pathway''' || [[Image:One_pathwaydh08.jpg|400x400px]] || [[Image:Always_pathwaydh08.jpg|400x400px]]
|-
| '''IRC gradient''' || [[Image:One_gradientdh08.jpg|400x400px]]|| [[Image:Always_gradientdh08.jpg|400x400px]]
|}

 
 
 

===='''Comparison of activation energies for the transition states'''====

The HF/3-21G optimised chair and boat transition structures were both reoptimised using the B3LYP/6-31G* (B3LYP/6-31G (d)) method. This gives a more accurate minimisation however this method takes longer to run. The values for the zero-point energies and thermal energies were found within the thermochemistry section of the log file.

The values for the HF/3-21G reactant (anti2) optimisation were calculated and the B3LYP/6-31G* values were taken from the "Frequency analysis of 1,5-hexadiene Conformers" earlier in this module.

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 7: Energy data for chair, boat and the anti2 conformation from the HF/3-21G and B3LYP/6-31G* methods of optimisation '''
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466677
| align="center" | -231.461320
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450936
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402342
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539524
| align="center" | -231.532562
| align="center" | -234.611716
| align="center" | -234.469177
| align="center" | -234.461835
|}
Digital repositories

Chair:
HF/3-21G: http://hdl.handle.net/10042/to-5569
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5568

Boat:
HF/3-21G: http://hdl.handle.net/10042/to-5570
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5571

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 8: Summary of activation energies for the chair and boat conformations (ΔE = chair TS or boat TS minus anti2)'''
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| align="center" | 1 a.u = 627.509 kcal mol-1
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE, chair (a.u)'''
| align="center" | 0.072847
| align="center" | 0.071242
| align="center" | 0.054248
| align="center" | 0.052827
| align="center" | -
|-
| '''ΔE, chair (kcal mol-1)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE, boat (a. u)'''
| align="center" | 0.088588
| align="center" | 0.087259
| align="center" | 0.066835
| align="center" | 0.065827
| align="center" | -
|-
| '''ΔE, boat (kcal mol-1)'''
| align="center" | 55.59
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.31
| align="center" | 44.7 ± 2.0
|}

By examining table 8 it is clear to see that the B3LYP/6-31G* method gives values which are much closer to the experimental ones. This is due to the B3LYP/6-31G* giving a higher level of optimisation than the HF/3-21G method. Table 8 also shows that the chair transition state has a lower activation energy to that of the boat transition state by 8.16 kcal mol-1. This suggests that the Cope rearrangement mechanism would most likely proceed through the chair transition state to that of the boat. Therefore the reaction would take place from the anti2 conformer to the chair transition state and then onto the gauche2 conformer within the Cope rearrangement.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9: Geometric parameters for the optimised chair and boat structures '''
| || ''' HF/3-21G chair TS''' || ''' B3LYP/6-31G* chair TS''' || '''HF/3-21G boat TS''' || '''B3LYP/6-31G* boat TS'''
|-
| Optimised Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Chair_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Boat_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.02 (C5-C10 or C3-C14)|| 1.97 (C5-C10 or C3-C14) || 2.14 (C3-C4 or C1-C6) || 2.21 (C3-C4 or C1-C6)
|-
| Fragment angle || 120 o (C3-C1-C5) || 120 o (C3-C1-C5) || 122 o (C4-C5-C6) || 122 o (C4-C5-C6)
|-
| Dihedral angle || 55 o (C1-C3-C14-C11)|| 54 o (C1-C3-C14-C11)|| 0 o (C2-C1-C6-C5) || 0 o (C2-C1-C6-C5)
|-
| Numbering of carbons || [[Image:Numbering_chairdh08.jpg|200x200px]]|| [[Image:Numbering_chairdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]]
|}

From studying table 9 it can be seen that the C-C bond breaking/forming length has changed between the two different types of optimisation methods. However the angles between the bonds remains unchanged.

=='''The Diels Alder Cycloaddition'''==

The Diels-Alder reaction is a cycloaddition between a conjugated diene and a dienophile (alkene system), which forms a cyclohexene system. It is often referred to as a [4s+2s] cycloaddition and proceeds through a single cyclic, concerted transition state where 2 σ bonds are formed in between the termini carbons of the 2 conjugated п systems. This is shown in the reaction scheme below where ethene is reacting with cis-butadiene.

[[Image:Reaction_scheme2dh08.jpg|300x300px|center]]

The π orbitals of the dienophile form new σ bonds with the π orbitals of the diene. The number of π electrons involved determine whether the reaction proceeds in a concerted stereospecific fashion (allowed) or not (forbidden). The HOMO/LUMO of one fragment interacts with the HOMO/LUMO of the other fragment to form two new bonding and anti-bonding MOs. The nodal properties allow one to predict how the reaction will occur.

Semi-Empirical/AM1 method and the B3LYP/6-31G* method of optimisation will be used on Gaussian to determine particular properites of the reactants to predict the way in which the reaction will proceed.

==='''Reagents'''===

<jmol>
<jmolAppletButton>
<uploadedFileContents>Cis_butadienedh08.mol</uploadedFileContents>
<text>Cis-butadiene</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Ethenedh08.mol</uploadedFileContents>
<text>ethylene</text>
<color>white</color>
</jmolAppletButton>
</jmol> were optimised separately using the AM1 semi-empirical method on Gaussview.The cis-butadiene had an energy of 0.04879725 a. u (1.28 x 102 KJ mol-1) and ethylene had an energy of 0.02619027 a. u (68.76 KJ mol-1).

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10: Frontier molecular orbitals of cis-butadiene and ethylene '''
| '''Reagent''' || '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| cis-butadiene || LUMO || [[Image:LUMO1_cis_butadieneDH08.jpg|100x100px]] || [[Image:LUMO2_cis_butadienedh08.jpg|100x100px]]|| Symmetric || +0.017
|-
| cis-butadiene || HOMO || [[Image:HOMO1_cis_butadienedh08.jpg|100x100px]] || [[Image:HOMO2_cis_butadienedh08.jpg|100x100px]]|| Anti-symmetric || -0.343
|-
| ethylene || LUMO || [[Image:LUMO1_ethenedh08.jpg|100x100px]] || [[Image:LUMO2_ethenedh08.jpg|100x100px]]|| Anti-symmetric || +0.052
|-
| ethylene || HOMO || [[Image:HOMO1_ethenedh08.jpg|100x100px]] || [[Image:HOMO2_ethenedh08.jpg|100x100px]]|| Symmetric || -0.387
|}

 
The symmetries of the LUMO and HOMO orbitals for cis-butadiene are opposite of those for the ethylene. This therefore means that the only interactions that can take place are those between ethylene's LUMO and cis-butadiene's HOMO and in addition to this the interaction between cis-butadiene's LUMO and ethylene's HOMO. This is because orbital symmetry has to be conserved (the conservation of orbital symmetry states that the symmetry of an new orbital formed by the overlap of the 2 original orbitals must be the same as the symmetry of the initial orbitals).

==='''Transition State'''===

A <jmol>
<jmolAppletButton>
<uploadedFileContents>Bicyclic_startingdh08.mol</uploadedFileContents>
<text>bicyclo system</text>
<color>white</color>
</jmolAppletButton>
</jmol> was drawn and one of the -CH2-CH2- fragments were removed. Three C=C bonds were added and the previous one was taken away so a <jmol>
<jmolAppletButton>
<uploadedFileContents>Frozen_coorddh08.mol</uploadedFileContents>
<text>guess transition state</text>
<color>white</color>
</jmolAppletButton>
</jmol> was produced.

After this an optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while bonds between the termini of the fragments were frozen at 2.1Å. Following this the resulting file had two separate optimisation and frequency calculations run on it using the TS (Berny) with one using the semi-empirical/AM1 method and the other using the DFT/B3LYP/6-31G* level of theory. Both of these optimisations had the bonds unfrozen and the "derivative" option was chosen instead.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11: Comparison of two methods of optimsations for the transition state '''
| || '''semi-empirical/AM1 method''' || '''DFT/B3LYP/6-31G* method'''
|-
| Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>SE_AM1_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>DFT_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Energy (a. u) || 0.11165608 || -234.54389573
|-
| Bond lengths || [[Image:DistancesSEdh08.jpg|150x150px]]|| [[Image:DistancesDFTdh08.jpg|150x150px]]
|-
|Imaginary frequency (cm-1) || -957 || -525
|-
|Animation || [[Image:-957dh08.gif]] || [[Image:-525dh08.gif]]
|-
| Lowest real frequency (cm-1) || 148 || 136
|-
|Animation || [[Image: 148dh08.gif]] || [[Image: 136dh08.gif]]

|}

The C-C bond forming length between the termini carbons is 0.15 Å larger for the DFT/B3LYP/6-31G* level of theory and this is because this method gives carries out the calculation to a greater degree of accuracy. This larger bond length leads a lower potential energy gradient to the transition state, which gives a lower total energy for the transition state. Therefore the given values below will be the DFT/B3LYP/6-31G* and not the semi-empirical/AM1 method.

A typical sp3 C-C bond length is 1.54 Å and a typical sp2 C=C bond length is 1.34 Å<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref>. Within the optimised transition molecule above the C-C bond lengths are 1.40 Å and the C=C bond length is 1.38 Å. This suggests that the π-bonds are breaking in order for the C-C bonds to form and additionally to this a π-bond is forming in between the two sp3 carbons.

The Van der Waals radius of a C-atom is 1.72 Å<ref name="ref 3">M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer, D. G. Truhlar, 1995, 346, pp. 63-69.[http://pubs.acs.org/doi/abs/10.1021/jp8111556]</ref>. The partly formed C-C length is 2.27 Å, which is shorter than two times the the van der Waals radius of a carbon atom. Therefore the carbon atoms are within a close enough distance to each other to share electron density and yet the bond has not formed at this point.

From looking at the animations for the imaginary frequency above it can be seen that the formation of the two bonds between the termini of the reactants is synchronous. This is expected of the Diels-Alder reaction since the mechanism is concerted. However the lowest real frequency is asynchronous with respect to the two bonds forming between the termini.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12: Frontier molecular orbitals of the transition state '''
| '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| LUMO || [[Image:LUMO1TSdh08.jpg|100x100px]] || [[Image:LUMO2TSdh08.jpg|100x100px]]|| Symmetric || +0.023
|-
| HOMO || [[Image:HOMO1TSdh08.jpg|100x100px]] || [[Image:HOMO2TSdh08.jpg|100x100px]]|| Anti-symmetric || -0.323
|}

From examining the images from table 12 above it can be deduced that the LUMO of the transition state originates from the cis-butadiene's LUMO and the ethylene's HOMO, which represents the σ*-antibonding orbitals. However the HOMO of the transition state originates from the cis-butadiene's HOMO and the ethylene's LUMO, which represents the σ-bonding orbitals within the two single C-C bonds that are formed within the reaction.

==='''Cyclohexa-1,3-diene reaction with maleic anhydride'''===

The <jmol>
<jmolAppletButton>
<uploadedFileContents>FrozenTS22dh08.mol</uploadedFileContents>
<text>exo</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_frozenTS22dh08.mol</uploadedFileContents>
<text>endo</text>
<color>white</color>
</jmolAppletButton>
</jmol> transition states were drawn on Gaussview and then the optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while the distance of the forming bonds were frozen at 2.1Å between the respective carbons. After this the structures changed slightly (<jmol>
<jmolAppletButton>
<uploadedFileContents>TS2_afteropt1_exo1dh08.mol</uploadedFileContents>
<text>exo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_TS2_afteropt1dh08.mol</uploadedFileContents>
<text>endo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol>) and the resulting files had an optimisation and frequency calculation run on it using the TA (Berny) with using the semi-empirical/AM1 level of theory while the unfrozen option as changed to "derivative". After this the endo and exo optimised transition state structures were re-optimised as they were before but using the DFT/B3LYP/6-31G* level of theory instead with the additional keyword "Opt=NoEigen".

{| class="wikitable" border="1" align="center"
|+ '''Table 13: Comparison of the exo and endo transition states '''
! rowspan="2"|-!! colspan="2"|''exo'' !! colspan="2"|''endo''
|-
| Semi-empirical/AM1 || DFT/B3LYP/6-31G* || Semi-empirical/AM1 || DFT/B3LYP/6-31G*
|-
| align="center"|Structure ||align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>After_opt2_exodh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>Endo_TS2_afteropt2dh081.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| align="center"| Energy (a. u) ||align="center"| -0.05041975 ||align="center"| -612.67931095 ||align="center"| -0.05150470 ||align="center"| -612.68339684
|-
| C-C bond forming length (Å) ||align="center"| 2.17 ||align="center"| 2.29 ||align="center"| 2.16 ||align="center"| 2.27
|-align="center"
| Imaginary frequency (cm-1) ||align="center"| -812 ||align="center"| -448 ||align="center"| -806 ||align="center"| -447
|-align="center"
| Animation ||colspan="2"|[[Image:-812_SEdh08.gif]] || colspan="2"|[[Image:-806_cm-1dh08.gif]]
|}

Again the C-C bond forming length between the termini carbons is larger (by 0.11 and 0.12 Å) for the DFT/B3LYP/6-31G* level of theory and this is due the same reasons talked about above. However the C-C bond forming length between the termini carbons are very similar in both the endo and exo transition states (0.02 Å difference).

From looking at the animations for the imaginary frequency above it can also be seen that the formation of the two bonds between the carbons in this case as well is synchronous. Again this is expected of the Diels-Alder reaction since it is concerted.

{| align=center
|+ '''Strain diagrams for the endo and exo transition states'''
| [[Image:Exostraindh08.jpg|thumb|200x200px|'''Exo TS''']]
| [[Image:Strainendodh08.jpg|thumb|200x200px|'''Endo TS''']]
|}

From studying the energies in table 13 it can be seen that the endo transition state is more stable than the exo by 4.08589 x 10-3 a. u (10.73 KJ mol-1). However by looking at the strain of both transition states one would expect the exo TS to be more stable. This is because the distance between the cyclohexa-1,3-diene ring and the -(C=O)-O-(C=O)- maleic anhydride fragment is 2.99 Å in the endo TS and 3.03 Å in the exo TS. Thus the endo form would have more strain due to steric hindrance.

However within the exo TS the vicinal hydrogens on the CH2-CH2 (carbon numbers 18 and 21) fragment on the cyclohexa-1,3-diene ring are found below the plane and next to the maleic anhydride. This leads to steric hindrance is this is where the strain originates from within the exo TS. Therefore this disfavours the exo form, which makes the endo more stable.

{| class="wikitable" border="1" align="center"
|+ '''Table 14: Frontier molecular orbitals of the the transition state for the cyclohexa-1,3-diene reaction '''
! rowspan="2"|Molecular Orbital !! colspan="4"|''exo'' !! colspan="4"|''endo''
|- align="center"
| '''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''||'''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''
|- align="center"
| LUMO || [[Image:LUMO1exoh081.jpg|150x150px]]|| [[Image:LUMO2exodh081.jpg|150x150px]]|| Anti-symmetric || -0.078 || [[Image:LUMO1ENDOdh08.jpg|150x150px]]||[[Image:LUMO2ENDOdh08.jpg|150x150px]] || Anti-symmetric || -0.067
|- align="center"
| HOMO || [[Image:HOMO1exodh081.jpg|150x150px]]||[[Image:HOMO2exodh081.jpg|150x150px]] || Anti-symmetric|| -0.242 || [[Image:HOMO1ENDOdh08.jpg|150x150px]]|| [[Image:HOMO2ENDOdh08.jpg|150x150px]]||Anti-symmetric || -0.242
|}

Digital repositories:

exo: http://hdl.handle.net/10042/to-5596

endo: http://hdl.handle.net/10042/to-5597

Why do you think that the exo form could be more strained?

Examine carefully the nodal properties of the HOMO between the -(C=O)-O-(C=O)- fragment and the remainder of the system.

What can you conclude about the so called “secondary orbital overlap effect”? (There is some discussion of this in Ian Fleming's book 'Frontier Orbitals and Organic Chemical Reactions').

Further discussion:
What effects have been neglected in these calculations of Diels Alder transition states?

=='''Conclusion'''==

='''References'''=
<references/>

Rep:Mod:hampluscheese

2010-11-10T21:46:52Z

Dh08: /* '''Cyclohexa-1,3-diene reaction with maleic anhydride''' */

='''Module 3:Physical Computational Chemistry Lab'''=

Douglas Hunt

=='''Objectives'''==

The aim of this experiment is to

=='''The Cope Rearrangement'''==

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangement] is a [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement |3,3|-sigmatropic rearrangement] that takes place through a single [http://en.wikipedia.org/wiki/Pericyclic pericyclic], concerted [http://en.wikipedia.org/wiki/Transition_state transition state]. Within this rearrangement a σ bond is formed in concert with another being broken within a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] π system.

The Cope rearrangement we will be studying will be that of 1,5-hexadiene, which is shown below.

[[Image:Cope_rearrangement_schemedh08.jpg|300x300px|center]]

The objective is to locate the low-energy minima and transition structures on the C6H10 potential energy surface and to determine the preferred reaction mechanism by using [http://en.wikipedia.org/wiki/GAUSSIAN Gaussian], with the [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)] methods.

==='''Optimisation of 1,5-hexadiene Conformers'''===

Rough <jmol>
<jmolAppletButton>
<uploadedFileContents>Anti_2_drawndh08.mol</uploadedFileContents>
<text>anti</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche3_drawndh08.mol</uploadedFileContents>
<text>gauche</text>
<color>white</color>
</jmolAppletButton>
</jmol> conformers of 1,5-hexadiene were drawn on Gaussview and their structures were tidied by the clean function. They were then further optimised by the HF/3-21G method and were identified which conformer they were from their point group and appendix 1 on the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3 physical lab page].

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 1: Optimisation and comparison of anti and gauche 1,5-hexadiene conformers using HF/3-21G methods
| ''' Conformer''' || '''Optimised Structure''' || '''Energy (Atomic Units)''' || '''[http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1 Energies (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (Anti2 - Gauche3)''' || '''Point group'''
|-
| '''Anti2''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Anti_2_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692535 || -231.69254 ||-6.08 x 105 || 1.2606 x 10-4 a. u|| Ci
|-
| '''Gauche3''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Gauche3_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692661 || -231.69266 || -6.08 x 105 || 3.31 x 10-1 KJ mol-1 || C1
|}

From studying table 1 it can be seen that the gauche conformer is 0.33 KJ mol-1 more stable than the anti conformation. One would assume that the anti2 conformer would be more stable due to [http://en.wikipedia.org/wiki/Steric steric] factors/repulsion within the gauche3 conformation. However within the gauche3 conformer there is a favourable donation of [http://en.wikipedia.org/wiki/Electron_density electron density] from the C=C π orbital to the C-H σ* orbital from the adjacent proton, which stabilises the conformer<ref name="ref 1">B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446.[http://www.informaworld.com/smpp/content~db=all?content=10.1080/00268970110081412]</ref>.

 

The anti2 conformer was then optimised using the B3LYP 6-31G* (6-31G (d)) method.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 2: Comparison of the anti2 conformer optimisations carried out at different calculation methods
| ''' Method''' || '''Structure and bond distances (Å)''' || '''Dihedral angle (C1 to C4)''' || '''Energy (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (HF/3-21G - DFT/B3LYP/6-31G)'''
|-
| HF/3-21G || [[Image:Anti2_opt1dh08.jpg|300x300px]] || 114.6 o || -231.692535 || -6.08 x 105 || 2.919171 a. u
|-
| DFT/B3LYP/6-31G* || [[Image:Anti2_opt2DH08.jpg|300x300px]] || 118.8 o || -234.611716 || -6.16 x 105 || 7.66 x 103 KJ mol-1
|}

From examining table 2 at first it is possible to see that the bond lengths are very similar to each other and that of the literature bond lengths<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref> (C1=C2: 1.340 Å, C2-C3: 1.508 Å and C3-C4: 1.538 Å). It is also possible to see that the C=C bond length is more accurate for the DFT/B3LYP/6-31G* method by it being 0.01 Å closer to the literature value. Another main difference in terms of geometry is that the DFT/B3LYP/6-31G* method gives a larger [http://en.wikipedia.org/wiki/Dihedral_angle dihedral angle] (and most likely more accurate) of 118.8o compared to that of 114.6o.

In terms of energy the DFT/B3LYP/6-31G* method produces a conformer of a higher stability by it possessing a lower energy with a difference of 7.66 x 103 KJ mol-1.

==='''Frequency analysis of 1,5-hexadiene Conformers'''===

[[Image:Vibrations_anti2dh08.jpg|right|300x300px]]

For the frequency analysis the DFT/B3LYP/6-31G* method was used.

Frequency analysis is essential within this calculation due to it being the second derivative of the potential energy surface. This means that if all the frequencies are all positive the system is at a minimum, if one of them is negative the optimised system is at a transition state, and if more than one are negative then the calculation has failed to find a critical point, which means the optimisation has failed. This therefore gives us a method of seeing whether the calculation has worked.

From looking to the image on the right it can deduced that the optimisation has taken place successfully by there being no negative vibrations and therefore the conformation is at a minimum.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 3: Energies under the thermochemistry section at 298.15 K and 0 K within the frequency logfiles'''
| '''Energies''' || '''298.15 K (atomic units)''' ||| '''0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (KJ mol-1)''' || '''Discussion'''
|-
| Sum of electronic and zero-point energies || -234.469177 || -234.466673 || -2.504 x 10-3 || -6.57 || This is the potential energy at 0 K, which includes the zero-point vibrational energy (E = Eelec + ZPE).

|-
| Sum of electronic and thermal energies || -234.461835 || -234.461085 || -7.5 x 10-4|| -1.97 || This is the energy at 298.15 K (0 K for 0 K calculation) and 1 atm of pressure, which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans).

|-
| Sum of electronic and thermal enthalpies || -234.460891 || -234.460141 || -7.5 x 10-4 || -1.97 || This contains an additional correction for RT (H = E + RT), which is especially important when looking at dissociation reactions.
|-
| Sum of electronic and thermal free energies || -234.500739 || -234.495931 || 4.808 x 10-3 || -12.62 || This includes the entropic contribution to the free energy (G = H - TS).
|}

{|align=right
|+
| [[Image:Anti2_freq_IRdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 298.15 K''']]
| [[Image:Anti2_freq_0Kdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 0 K''']]
|}

From looking at table 3 above it can be deduced that the anti2 conformer is more stable at room temperature to that of 0 K with respect to all the types of energies. The spectra also seem to not change much between the two temperatures but the main change is the change in the sum of electronic and thermal free energies, which is heavily dependent of temperature by it including the entropic contribution to the free energy (G = H - '''T'''S)

 
 
 
 

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

Within the Cope Rearrangement there are two possible transition states, the chair (C2h) and the boat (C2v), which are shown within [https://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_2 appendix two ].

==='''Comparison of optimisation methods for the chair transition state'''===

An <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2hdh08.mol</uploadedFileContents>
<text>allyl fragment</text>
<color>white</color>
</jmolAppletButton>
</jmol>, CH2CHCH2 was drawn on Gaussview and was optimized through the HF/3-21G method. After this the original allyl fragment was copied and pasted into a new window and place exactly above the first fragment with a bond distance of 2.20 Å between the terminal end carbons. This therefore produced the <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2h_fulldh08.mol</uploadedFileContents>
<text>chair transition structure.</text>
<color>white</color>
</jmolAppletButton>
</jmol>

Gaussian was then set up for an optimisation for the transition state with the additional key words "Opt=NoEigen", the optimisation set to TS (Berny), the force constants set to "once" and using the HF/3-21G method.

After this the "chair" transition state was optimised an alternative way by freezing the coordinates of the two sets of terminal carbons (where the bonds form/break during the rearrangement) within the Redundant Coord Editor on Gauusview.
Following this a second calculation was run where the freeze coordinates option was changed to "Derivative" and the force constants were set to "never".

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 4: Comparison of chair transition state optimisation methods
| || '''Optimised through a TS (Berny)''' || '''Optimised through a frozen coordinate method'''
|-
| '''Optimised TS Structure''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>C2h_optfreq_TS_Bdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| '''Terminal C-C bond length (Å)''' || 2.02 || 2.02
|-
|'''Energy (atomic units)''' || -231.61932247 || -231.61932190
|-
| '''ΔE, TS (Berny) - frozen coordinate''' || 5.7 x 10-7 a.u || 1.50 x 10-3 KJ mol-1
|-
| '''Imaginary frequency (cm-1)''' || -818 || -818
|-
| '''Animation''' ||[[Image:818dh08TSB.gif|350x350px]] || [[Image:818cm-1Fdh08.gif]]
|}

The data in table 4 shows that both methods give the same results and they both give the same transition state. They both produced the same energy with a negligible energy difference between them and the bond lengths between the terminal carbons are both 2.02 Å. In addition to this the imaginary frequencies were both -818cm-1 and within the animations above it is possible to see that when one C-C bond is forming the other is breaking. This means it is a asynchronous bond formation, which is expected for the Cope Rearrangement since it is a pericyclic reaction.

The only difference between the two optimisations is the way in which the transition states were calculated. This is due to the TS (Berny) method calculating the force constant matrix within the first optimisation step and is then being recalculated as the optimisation carries on. When the frozen coordinate method freezes the reaction coordinate in order for the rest of the structure to be optimised around it, which does not require the whole hessian (force constant matrix) to be calculated but only that the reaction coordinate is differentiated (second step). The main advantage of this, is that you do not need to calculate the entire force constant matrix and just need to differentiate the reaction coordinate in order to obtain a good enough guess for the initial force constant matrix. The frozen coordinate method will therefore take less time to compute the transition state, which can be very expensive within certain systems. However to carry out this method one needs to have an idea of how far apart the terminal carbons are.

==='''Boat transition state'''===

===='''QST2 optimisation'''====

The optimisation for the boat transition state had a different method and this was the QST2 method. This calculation interpolates between two structures (product and reactant) and tries to find a transition state between them.

The reactants and products were numbered in the same way. Therefore both 1,5-hexadiene molecules had their numbering manually changed for the product molecule so it corresponded to the numbering of the reactant if it rearranged. This is illustrated in the reaction scheme below.

[[Image:Boat_rerrangmentdh08.jpg|300x300px|center]]

The chk file for the optimized anti2 conformer was opened and copied into a new window. A second window was opened within the new window and the same anti2 conformer was copied into the new window. The numbering was changed within the second molecule so the two looked like they do in the image below.

[[Image:TS_(QST2)_numbering_1dh08.jpg|400x400px|center]]

The job type was set as "Opt+Freq" and "TS (QST2)" was chosen to optimise the transition state. However this failed and produced
a structure a bit like a <jmol>
<jmolAppletButton>
<uploadedFileContents>Trialone_BPdh08.mol</uploadedFileContents>
<text>chair transition structure </text>
<color>white</color>
</jmolAppletButton>
</jmol> but more dissociated. This is because it interpolated between the two structures and translated the top allyl fragment and therefore did not consider the possibility of a rotation around the central bonds.

To get around this the C-C-C-C dihedral angle (C2-C3-C4-C5) was set to 0 o and the C-C-C angle (C2-C3-C4 and C3-C4-C5 for the product molecule) was set to 100 o for both molecules. This gave rise to the two structures below.

[[Image:TS_(QST2)_numbering_2dh08.jpg|400x400px|center]]

After this the QST2 calculation was run again and the results in table 5 were produced. The imaginary frequency at -839 cm-1 shows that while one bond is forming the other one is breaking, which is again an asynchronous bond formation.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5: Data for the boat transition state from the QST2 optimisation method
| Optimised TS Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.14
|-
|Energy (atomic units) || -231.60280151
|-
| Energy (KJ mol-1) || -6.08 x 105
|-
| Imaginary frequency (cm-1) || -839
|-
| Animation ||[[Image:-839dh08.gif]]
|}

===='''Intrinsic Reaction Coordinate'''====

It is not possible to predict which product the chair and boat transition state will lead to. To solve this problem within Gaussian one can use the IRC method, which allows one to follow the minimum energy path from a transition structure down to its local minimum on a potential energy surface. This creates a succession of points by taking small geometry steps in the direction to where the gradient of the energy surface is at its steepest until the minimum in energy is reached.

The chk file for the optimized chair transition structure was opened and the job type selected was "IRC", the reaction coordinate was computed in the forward direction (since molecule is symmetric) with 50 points along the IRC. This was done with two type of force constants being calculated, once and always.

Within the "once" method the output showed no bond formation process by the 26th step and therefore the calculation was incomplete, since a complete 1,5-hexadiene structure was supposed to be formed. From looking at the IRC pathway it can be seen that the minimum on the potential energy surface was not reached at that point, since the points on the pathway did not form an asymptote and the gradient did not go close enough to zero.

For the "always" method the computation of force constants were carried out at every step. This therefore lets the IRC calculation distinguish whether it is moving along the minimum energy pathway while going down the potential energy surface. While this is happening it can make changes to the pathway if necessary. Within the "always" case the bond forming arrives at 1,5-hexadiene and this is backed up by the IRC pathway reaching a minimum within its energy by the gradient going close to zero and the pathway reaches an asymptote.

The structure produced through the always method had an energy of -231.69199434 atomic units, which is very close to the energy of the gauche2 conformer in the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 appendix 1]. This suggests that the reaction takes place through the chair transition state, which later produces the gauche2 conformer of 1,5-hexadiene.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 6: Comparison of IRC calculations for the chair structure
| || '''Calculate force constants once''' || '''Calculate force constants always'''
|-
| '''Number of points along IRC''' || 26 || 47
|-
| '''Structure at the last point of IRC''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Once_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Always_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|'''Energy (atomic units)''' || -231.68906923 || -231.69199434
|-
| '''ΔE, once - always''' || 2.92511 x 10-3 a.u || 7.68 KJ mol-1
|-
| '''IRC pathway''' || [[Image:One_pathwaydh08.jpg|400x400px]] || [[Image:Always_pathwaydh08.jpg|400x400px]]
|-
| '''IRC gradient''' || [[Image:One_gradientdh08.jpg|400x400px]]|| [[Image:Always_gradientdh08.jpg|400x400px]]
|}

 
 
 

===='''Comparison of activation energies for the transition states'''====

The HF/3-21G optimised chair and boat transition structures were both reoptimised using the B3LYP/6-31G* (B3LYP/6-31G (d)) method. This gives a more accurate minimisation however this method takes longer to run. The values for the zero-point energies and thermal energies were found within the thermochemistry section of the log file.

The values for the HF/3-21G reactant (anti2) optimisation were calculated and the B3LYP/6-31G* values were taken from the "Frequency analysis of 1,5-hexadiene Conformers" earlier in this module.

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 7: Energy data for chair, boat and the anti2 conformation from the HF/3-21G and B3LYP/6-31G* methods of optimisation '''
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466677
| align="center" | -231.461320
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450936
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402342
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539524
| align="center" | -231.532562
| align="center" | -234.611716
| align="center" | -234.469177
| align="center" | -234.461835
|}
Digital repositories

Chair:
HF/3-21G: http://hdl.handle.net/10042/to-5569
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5568

Boat:
HF/3-21G: http://hdl.handle.net/10042/to-5570
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5571

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 8: Summary of activation energies for the chair and boat conformations (ΔE = chair TS or boat TS minus anti2)'''
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| align="center" | 1 a.u = 627.509 kcal mol-1
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE, chair (a.u)'''
| align="center" | 0.072847
| align="center" | 0.071242
| align="center" | 0.054248
| align="center" | 0.052827
| align="center" | -
|-
| '''ΔE, chair (kcal mol-1)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE, boat (a. u)'''
| align="center" | 0.088588
| align="center" | 0.087259
| align="center" | 0.066835
| align="center" | 0.065827
| align="center" | -
|-
| '''ΔE, boat (kcal mol-1)'''
| align="center" | 55.59
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.31
| align="center" | 44.7 ± 2.0
|}

By examining table 8 it is clear to see that the B3LYP/6-31G* method gives values which are much closer to the experimental ones. This is due to the B3LYP/6-31G* giving a higher level of optimisation than the HF/3-21G method. Table 8 also shows that the chair transition state has a lower activation energy to that of the boat transition state by 8.16 kcal mol-1. This suggests that the Cope rearrangement mechanism would most likely proceed through the chair transition state to that of the boat. Therefore the reaction would take place from the anti2 conformer to the chair transition state and then onto the gauche2 conformer within the Cope rearrangement.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9: Geometric parameters for the optimised chair and boat structures '''
| || ''' HF/3-21G chair TS''' || ''' B3LYP/6-31G* chair TS''' || '''HF/3-21G boat TS''' || '''B3LYP/6-31G* boat TS'''
|-
| Optimised Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Chair_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Boat_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.02 (C5-C10 or C3-C14)|| 1.97 (C5-C10 or C3-C14) || 2.14 (C3-C4 or C1-C6) || 2.21 (C3-C4 or C1-C6)
|-
| Fragment angle || 120 o (C3-C1-C5) || 120 o (C3-C1-C5) || 122 o (C4-C5-C6) || 122 o (C4-C5-C6)
|-
| Dihedral angle || 55 o (C1-C3-C14-C11)|| 54 o (C1-C3-C14-C11)|| 0 o (C2-C1-C6-C5) || 0 o (C2-C1-C6-C5)
|-
| Numbering of carbons || [[Image:Numbering_chairdh08.jpg|200x200px]]|| [[Image:Numbering_chairdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]]
|}

From studying table 9 it can be seen that the C-C bond breaking/forming length has changed between the two different types of optimisation methods. However the angles between the bonds remains unchanged.

=='''The Diels Alder Cycloaddition'''==

The Diels-Alder reaction is a cycloaddition between a conjugated diene and a dienophile (alkene system), which forms a cyclohexene system. It is often referred to as a [4s+2s] cycloaddition and proceeds through a single cyclic, concerted transition state where 2 σ bonds are formed in between the termini carbons of the 2 conjugated п systems. This is shown in the reaction scheme below where ethene is reacting with cis-butadiene.

[[Image:Reaction_scheme2dh08.jpg|300x300px|center]]

The π orbitals of the dienophile form new σ bonds with the π orbitals of the diene. The number of π electrons involved determine whether the reaction proceeds in a concerted stereospecific fashion (allowed) or not (forbidden). The HOMO/LUMO of one fragment interacts with the HOMO/LUMO of the other fragment to form two new bonding and anti-bonding MOs. The nodal properties allow one to predict how the reaction will occur.

Semi-Empirical/AM1 method and the B3LYP/6-31G* method of optimisation will be used on Gaussian to determine particular properites of the reactants to predict the way in which the reaction will proceed.

==='''Reagents'''===

<jmol>
<jmolAppletButton>
<uploadedFileContents>Cis_butadienedh08.mol</uploadedFileContents>
<text>Cis-butadiene</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Ethenedh08.mol</uploadedFileContents>
<text>ethylene</text>
<color>white</color>
</jmolAppletButton>
</jmol> were optimised separately using the AM1 semi-empirical method on Gaussview.The cis-butadiene had an energy of 0.04879725 a. u (1.28 x 102 KJ mol-1) and ethylene had an energy of 0.02619027 a. u (68.76 KJ mol-1).

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10: Frontier molecular orbitals of cis-butadiene and ethylene '''
| '''Reagent''' || '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| cis-butadiene || LUMO || [[Image:LUMO1_cis_butadieneDH08.jpg|100x100px]] || [[Image:LUMO2_cis_butadienedh08.jpg|100x100px]]|| Symmetric || +0.017
|-
| cis-butadiene || HOMO || [[Image:HOMO1_cis_butadienedh08.jpg|100x100px]] || [[Image:HOMO2_cis_butadienedh08.jpg|100x100px]]|| Anti-symmetric || -0.343
|-
| ethylene || LUMO || [[Image:LUMO1_ethenedh08.jpg|100x100px]] || [[Image:LUMO2_ethenedh08.jpg|100x100px]]|| Anti-symmetric || +0.052
|-
| ethylene || HOMO || [[Image:HOMO1_ethenedh08.jpg|100x100px]] || [[Image:HOMO2_ethenedh08.jpg|100x100px]]|| Symmetric || -0.387
|}

 
The symmetries of the LUMO and HOMO orbitals for cis-butadiene are opposite of those for the ethylene. This therefore means that the only interactions that can take place are those between ethylene's LUMO and cis-butadiene's HOMO and in addition to this the interaction between cis-butadiene's LUMO and ethylene's HOMO. This is because orbital symmetry has to be conserved (the conservation of orbital symmetry states that the symmetry of an new orbital formed by the overlap of the 2 original orbitals must be the same as the symmetry of the initial orbitals).

==='''Transition State'''===

A <jmol>
<jmolAppletButton>
<uploadedFileContents>Bicyclic_startingdh08.mol</uploadedFileContents>
<text>bicyclo system</text>
<color>white</color>
</jmolAppletButton>
</jmol> was drawn and one of the -CH2-CH2- fragments were removed. Three C=C bonds were added and the previous one was taken away so a <jmol>
<jmolAppletButton>
<uploadedFileContents>Frozen_coorddh08.mol</uploadedFileContents>
<text>guess transition state</text>
<color>white</color>
</jmolAppletButton>
</jmol> was produced.

After this an optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while bonds between the termini of the fragments were frozen at 2.1Å. Following this the resulting file had two separate optimisation and frequency calculations run on it using the TS (Berny) with one using the semi-empirical/AM1 method and the other using the DFT/B3LYP/6-31G* level of theory. Both of these optimisations had the bonds unfrozen and the "derivative" option was chosen instead.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11: Comparison of two methods of optimsations for the transition state '''
| || '''semi-empirical/AM1 method''' || '''DFT/B3LYP/6-31G* method'''
|-
| Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>SE_AM1_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>DFT_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Energy (a. u) || 0.11165608 || -234.54389573
|-
| Bond lengths || [[Image:DistancesSEdh08.jpg|150x150px]]|| [[Image:DistancesDFTdh08.jpg|150x150px]]
|-
|Imaginary frequency (cm-1) || -957 || -525
|-
|Animation || [[Image:-957dh08.gif]] || [[Image:-525dh08.gif]]
|-
| Lowest real frequency (cm-1) || 148 || 136
|-
|Animation || [[Image: 148dh08.gif]] || [[Image: 136dh08.gif]]

|}

The C-C bond forming length between the termini carbons is 0.15 Å larger for the DFT/B3LYP/6-31G* level of theory and this is because this method gives carries out the calculation to a greater degree of accuracy. This larger bond length leads a lower potential energy gradient to the transition state, which gives a lower total energy for the transition state. Therefore the given values below will be the DFT/B3LYP/6-31G* and not the semi-empirical/AM1 method.

A typical sp3 C-C bond length is 1.54 Å and a typical sp2 C=C bond length is 1.34 Å<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref>. Within the optimised transition molecule above the C-C bond lengths are 1.40 Å and the C=C bond length is 1.38 Å. This suggests that the π-bonds are breaking in order for the C-C bonds to form and additionally to this a π-bond is forming in between the two sp3 carbons.

The Van der Waals radius of a C-atom is 1.72 Å<ref name="ref 3">M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer, D. G. Truhlar, 1995, 346, pp. 63-69.[http://pubs.acs.org/doi/abs/10.1021/jp8111556]</ref>. The partly formed C-C length is 2.27 Å, which is shorter than two times the the van der Waals radius of a carbon atom. Therefore the carbon atoms are within a close enough distance to each other to share electron density and yet the bond has not formed at this point.

From looking at the animations for the imaginary frequency above it can be seen that the formation of the two bonds between the termini of the reactants is synchronous. This is expected of the Diels-Alder reaction since the mechanism is concerted. However the lowest real frequency is asynchronous with respect to the two bonds forming between the termini.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12: Frontier molecular orbitals of the transition state '''
| '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| LUMO || [[Image:LUMO1TSdh08.jpg|100x100px]] || [[Image:LUMO2TSdh08.jpg|100x100px]]|| Symmetric || +0.023
|-
| HOMO || [[Image:HOMO1TSdh08.jpg|100x100px]] || [[Image:HOMO2TSdh08.jpg|100x100px]]|| Anti-symmetric || -0.323
|}

From examining the images from table 12 above it can be deduced that the LUMO of the transition state originates from the cis-butadiene's LUMO and the ethylene's HOMO, which represents the σ*-antibonding orbitals. However the HOMO of the transition state originates from the cis-butadiene's HOMO and the ethylene's LUMO, which represents the σ-bonding orbitals within the two single C-C bonds that are formed within the reaction.

==='''Cyclohexa-1,3-diene reaction with maleic anhydride'''===

The <jmol>
<jmolAppletButton>
<uploadedFileContents>FrozenTS22dh08.mol</uploadedFileContents>
<text>exo</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_frozenTS22dh08.mol</uploadedFileContents>
<text>endo</text>
<color>white</color>
</jmolAppletButton>
</jmol> transition states were drawn on Gaussview and then the optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while the distance of the forming bonds were frozen at 2.1Å between the respective carbons. After this the structures changed slightly (<jmol>
<jmolAppletButton>
<uploadedFileContents>TS2_afteropt1_exo1dh08.mol</uploadedFileContents>
<text>exo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_TS2_afteropt1dh08.mol</uploadedFileContents>
<text>endo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol>) and the resulting files had an optimisation and frequency calculation run on it using the TA (Berny) with using the semi-empirical/AM1 level of theory while the unfrozen option as changed to "derivative". After this the endo and exo optimised transition state structures were re-optimised as they were before but using the DFT/B3LYP/6-31G* level of theory instead with the additional keyword "Opt=NoEigen".

{| class="wikitable" border="1" align="center"
|+ '''Table 13: Comparison of the exo and endo transition states '''
! rowspan="2"|-!! colspan="2"|''exo'' !! colspan="2"|''endo''
|-
| Semi-empirical/AM1 || DFT/B3LYP/6-31G* || Semi-empirical/AM1 || DFT/B3LYP/6-31G*
|-
| align="center"|Structure ||align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>After_opt2_exodh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>Endo_TS2_afteropt2dh081.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| align="center"| Energy (a. u) ||align="center"| -0.05041975 ||align="center"| -612.67931095 ||align="center"| -0.05150470 ||align="center"| -612.68339684
|-
| C-C bond forming length (Å) ||align="center"| 2.17 ||align="center"| 2.29 ||align="center"| 2.16 ||align="center"| 2.27
|-align="center"
| Imaginary frequency (cm-1) ||align="center"| -812 ||align="center"| -448 ||align="center"| -806 ||align="center"| -447
|-align="center"
| Animation ||colspan="2"|[[Image:-812_SEdh08.gif]] || colspan="2"|[[Image:-806_cm-1dh08.gif]]
|}

Again the C-C bond forming length between the termini carbons is larger (by 0.11 and 0.12 Å) for the DFT/B3LYP/6-31G* level of theory and this is due the same reasons talked about above.

From looking at the animations for the imaginary frequency above it can also be seen that the formation of the two bonds between the carbons in this case as well is synchronous. Again this is expected of the Diels-Alder reaction since it is concerted.

{| align=center
|+ '''Strain diagrams for the endo and exo transition states'''
| [[Image:Exostraindh08.jpg|thumb|200x200px|'''Exo TS''']]
| [[Image:Strainendodh08.jpg|thumb|200x200px|'''Endo TS''']]
|}

From studying the energies in table 13 it can be seen that the endo transition state is more stable than the exo by 4.08589 x 10-3 a. u (10.73 KJ mol-1). However by looking at the strain of both transition states one would expect the exo TS to be more stable. This is because the distance between the cyclohexa-1,3-diene ring and the -(C=O)-O-(C=O)- maleic anhydride fragment is 2.99 Å in the endo TS and 3.03 Å in the exo TS. Thus the endo form would have more strain due to steric hindrance.

However within the exo TS the vicinal hydrogens on the CH2-CH2 (carbon numbers 18 and 21) fragment on the cyclohexa-1,3-diene ring are found below the plane and next to the maleic anhydride. This leads to steric hindrance is this is where the strain originates from within the exo TS. Therefore this disfavours the exo form, which makes the endo more stable.

{| class="wikitable" border="1" align="center"
|+ '''Table 14: Frontier molecular orbitals of the the transition state for the cyclohexa-1,3-diene reaction '''
! rowspan="2"|Molecular Orbital !! colspan="4"|''exo'' !! colspan="4"|''endo''
|- align="center"
| '''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''||'''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''
|- align="center"
| LUMO || [[Image:LUMO1exoh081.jpg|150x150px]]|| [[Image:LUMO2exodh081.jpg|150x150px]]|| Anti-symmetric || -0.078 || [[Image:LUMO1ENDOdh08.jpg|150x150px]]||[[Image:LUMO2ENDOdh08.jpg|150x150px]] || Anti-symmetric || -0.067
|- align="center"
| HOMO || [[Image:HOMO1exodh081.jpg|150x150px]]||[[Image:HOMO2exodh081.jpg|150x150px]] || Anti-symmetric|| -0.242 || [[Image:HOMO1ENDOdh08.jpg|150x150px]]|| [[Image:HOMO2ENDOdh08.jpg|150x150px]]||Anti-symmetric || -0.242
|}

Digital repositories:

exo: http://hdl.handle.net/10042/to-5596

endo: http://hdl.handle.net/10042/to-5597

Why do you think that the exo form could be more strained?

Examine carefully the nodal properties of the HOMO between the -(C=O)-O-(C=O)- fragment and the remainder of the system.

What can you conclude about the so called “secondary orbital overlap effect”? (There is some discussion of this in Ian Fleming's book 'Frontier Orbitals and Organic Chemical Reactions').

Further discussion:
What effects have been neglected in these calculations of Diels Alder transition states?

=='''Conclusion'''==

='''References'''=
<references/>

Rep:Mod:hampluscheese

2010-11-10T21:45:23Z

Dh08: /* '''Cyclohexa-1,3-diene reaction with maleic anhydride''' */

='''Module 3:Physical Computational Chemistry Lab'''=

Douglas Hunt

=='''Objectives'''==

The aim of this experiment is to

=='''The Cope Rearrangement'''==

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangement] is a [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement |3,3|-sigmatropic rearrangement] that takes place through a single [http://en.wikipedia.org/wiki/Pericyclic pericyclic], concerted [http://en.wikipedia.org/wiki/Transition_state transition state]. Within this rearrangement a σ bond is formed in concert with another being broken within a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] π system.

The Cope rearrangement we will be studying will be that of 1,5-hexadiene, which is shown below.

[[Image:Cope_rearrangement_schemedh08.jpg|300x300px|center]]

The objective is to locate the low-energy minima and transition structures on the C6H10 potential energy surface and to determine the preferred reaction mechanism by using [http://en.wikipedia.org/wiki/GAUSSIAN Gaussian], with the [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)] methods.

==='''Optimisation of 1,5-hexadiene Conformers'''===

Rough <jmol>
<jmolAppletButton>
<uploadedFileContents>Anti_2_drawndh08.mol</uploadedFileContents>
<text>anti</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche3_drawndh08.mol</uploadedFileContents>
<text>gauche</text>
<color>white</color>
</jmolAppletButton>
</jmol> conformers of 1,5-hexadiene were drawn on Gaussview and their structures were tidied by the clean function. They were then further optimised by the HF/3-21G method and were identified which conformer they were from their point group and appendix 1 on the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3 physical lab page].

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 1: Optimisation and comparison of anti and gauche 1,5-hexadiene conformers using HF/3-21G methods
| ''' Conformer''' || '''Optimised Structure''' || '''Energy (Atomic Units)''' || '''[http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1 Energies (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (Anti2 - Gauche3)''' || '''Point group'''
|-
| '''Anti2''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Anti_2_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692535 || -231.69254 ||-6.08 x 105 || 1.2606 x 10-4 a. u|| Ci
|-
| '''Gauche3''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Gauche3_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692661 || -231.69266 || -6.08 x 105 || 3.31 x 10-1 KJ mol-1 || C1
|}

From studying table 1 it can be seen that the gauche conformer is 0.33 KJ mol-1 more stable than the anti conformation. One would assume that the anti2 conformer would be more stable due to [http://en.wikipedia.org/wiki/Steric steric] factors/repulsion within the gauche3 conformation. However within the gauche3 conformer there is a favourable donation of [http://en.wikipedia.org/wiki/Electron_density electron density] from the C=C π orbital to the C-H σ* orbital from the adjacent proton, which stabilises the conformer<ref name="ref 1">B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446.[http://www.informaworld.com/smpp/content~db=all?content=10.1080/00268970110081412]</ref>.

 

The anti2 conformer was then optimised using the B3LYP 6-31G* (6-31G (d)) method.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 2: Comparison of the anti2 conformer optimisations carried out at different calculation methods
| ''' Method''' || '''Structure and bond distances (Å)''' || '''Dihedral angle (C1 to C4)''' || '''Energy (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (HF/3-21G - DFT/B3LYP/6-31G)'''
|-
| HF/3-21G || [[Image:Anti2_opt1dh08.jpg|300x300px]] || 114.6 o || -231.692535 || -6.08 x 105 || 2.919171 a. u
|-
| DFT/B3LYP/6-31G* || [[Image:Anti2_opt2DH08.jpg|300x300px]] || 118.8 o || -234.611716 || -6.16 x 105 || 7.66 x 103 KJ mol-1
|}

From examining table 2 at first it is possible to see that the bond lengths are very similar to each other and that of the literature bond lengths<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref> (C1=C2: 1.340 Å, C2-C3: 1.508 Å and C3-C4: 1.538 Å). It is also possible to see that the C=C bond length is more accurate for the DFT/B3LYP/6-31G* method by it being 0.01 Å closer to the literature value. Another main difference in terms of geometry is that the DFT/B3LYP/6-31G* method gives a larger [http://en.wikipedia.org/wiki/Dihedral_angle dihedral angle] (and most likely more accurate) of 118.8o compared to that of 114.6o.

In terms of energy the DFT/B3LYP/6-31G* method produces a conformer of a higher stability by it possessing a lower energy with a difference of 7.66 x 103 KJ mol-1.

==='''Frequency analysis of 1,5-hexadiene Conformers'''===

[[Image:Vibrations_anti2dh08.jpg|right|300x300px]]

For the frequency analysis the DFT/B3LYP/6-31G* method was used.

Frequency analysis is essential within this calculation due to it being the second derivative of the potential energy surface. This means that if all the frequencies are all positive the system is at a minimum, if one of them is negative the optimised system is at a transition state, and if more than one are negative then the calculation has failed to find a critical point, which means the optimisation has failed. This therefore gives us a method of seeing whether the calculation has worked.

From looking to the image on the right it can deduced that the optimisation has taken place successfully by there being no negative vibrations and therefore the conformation is at a minimum.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 3: Energies under the thermochemistry section at 298.15 K and 0 K within the frequency logfiles'''
| '''Energies''' || '''298.15 K (atomic units)''' ||| '''0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (KJ mol-1)''' || '''Discussion'''
|-
| Sum of electronic and zero-point energies || -234.469177 || -234.466673 || -2.504 x 10-3 || -6.57 || This is the potential energy at 0 K, which includes the zero-point vibrational energy (E = Eelec + ZPE).

|-
| Sum of electronic and thermal energies || -234.461835 || -234.461085 || -7.5 x 10-4|| -1.97 || This is the energy at 298.15 K (0 K for 0 K calculation) and 1 atm of pressure, which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans).

|-
| Sum of electronic and thermal enthalpies || -234.460891 || -234.460141 || -7.5 x 10-4 || -1.97 || This contains an additional correction for RT (H = E + RT), which is especially important when looking at dissociation reactions.
|-
| Sum of electronic and thermal free energies || -234.500739 || -234.495931 || 4.808 x 10-3 || -12.62 || This includes the entropic contribution to the free energy (G = H - TS).
|}

{|align=right
|+
| [[Image:Anti2_freq_IRdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 298.15 K''']]
| [[Image:Anti2_freq_0Kdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 0 K''']]
|}

From looking at table 3 above it can be deduced that the anti2 conformer is more stable at room temperature to that of 0 K with respect to all the types of energies. The spectra also seem to not change much between the two temperatures but the main change is the change in the sum of electronic and thermal free energies, which is heavily dependent of temperature by it including the entropic contribution to the free energy (G = H - '''T'''S)

 
 
 
 

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

Within the Cope Rearrangement there are two possible transition states, the chair (C2h) and the boat (C2v), which are shown within [https://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_2 appendix two ].

==='''Comparison of optimisation methods for the chair transition state'''===

An <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2hdh08.mol</uploadedFileContents>
<text>allyl fragment</text>
<color>white</color>
</jmolAppletButton>
</jmol>, CH2CHCH2 was drawn on Gaussview and was optimized through the HF/3-21G method. After this the original allyl fragment was copied and pasted into a new window and place exactly above the first fragment with a bond distance of 2.20 Å between the terminal end carbons. This therefore produced the <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2h_fulldh08.mol</uploadedFileContents>
<text>chair transition structure.</text>
<color>white</color>
</jmolAppletButton>
</jmol>

Gaussian was then set up for an optimisation for the transition state with the additional key words "Opt=NoEigen", the optimisation set to TS (Berny), the force constants set to "once" and using the HF/3-21G method.

After this the "chair" transition state was optimised an alternative way by freezing the coordinates of the two sets of terminal carbons (where the bonds form/break during the rearrangement) within the Redundant Coord Editor on Gauusview.
Following this a second calculation was run where the freeze coordinates option was changed to "Derivative" and the force constants were set to "never".

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 4: Comparison of chair transition state optimisation methods
| || '''Optimised through a TS (Berny)''' || '''Optimised through a frozen coordinate method'''
|-
| '''Optimised TS Structure''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>C2h_optfreq_TS_Bdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| '''Terminal C-C bond length (Å)''' || 2.02 || 2.02
|-
|'''Energy (atomic units)''' || -231.61932247 || -231.61932190
|-
| '''ΔE, TS (Berny) - frozen coordinate''' || 5.7 x 10-7 a.u || 1.50 x 10-3 KJ mol-1
|-
| '''Imaginary frequency (cm-1)''' || -818 || -818
|-
| '''Animation''' ||[[Image:818dh08TSB.gif|350x350px]] || [[Image:818cm-1Fdh08.gif]]
|}

The data in table 4 shows that both methods give the same results and they both give the same transition state. They both produced the same energy with a negligible energy difference between them and the bond lengths between the terminal carbons are both 2.02 Å. In addition to this the imaginary frequencies were both -818cm-1 and within the animations above it is possible to see that when one C-C bond is forming the other is breaking. This means it is a asynchronous bond formation, which is expected for the Cope Rearrangement since it is a pericyclic reaction.

The only difference between the two optimisations is the way in which the transition states were calculated. This is due to the TS (Berny) method calculating the force constant matrix within the first optimisation step and is then being recalculated as the optimisation carries on. When the frozen coordinate method freezes the reaction coordinate in order for the rest of the structure to be optimised around it, which does not require the whole hessian (force constant matrix) to be calculated but only that the reaction coordinate is differentiated (second step). The main advantage of this, is that you do not need to calculate the entire force constant matrix and just need to differentiate the reaction coordinate in order to obtain a good enough guess for the initial force constant matrix. The frozen coordinate method will therefore take less time to compute the transition state, which can be very expensive within certain systems. However to carry out this method one needs to have an idea of how far apart the terminal carbons are.

==='''Boat transition state'''===

===='''QST2 optimisation'''====

The optimisation for the boat transition state had a different method and this was the QST2 method. This calculation interpolates between two structures (product and reactant) and tries to find a transition state between them.

The reactants and products were numbered in the same way. Therefore both 1,5-hexadiene molecules had their numbering manually changed for the product molecule so it corresponded to the numbering of the reactant if it rearranged. This is illustrated in the reaction scheme below.

[[Image:Boat_rerrangmentdh08.jpg|300x300px|center]]

The chk file for the optimized anti2 conformer was opened and copied into a new window. A second window was opened within the new window and the same anti2 conformer was copied into the new window. The numbering was changed within the second molecule so the two looked like they do in the image below.

[[Image:TS_(QST2)_numbering_1dh08.jpg|400x400px|center]]

The job type was set as "Opt+Freq" and "TS (QST2)" was chosen to optimise the transition state. However this failed and produced
a structure a bit like a <jmol>
<jmolAppletButton>
<uploadedFileContents>Trialone_BPdh08.mol</uploadedFileContents>
<text>chair transition structure </text>
<color>white</color>
</jmolAppletButton>
</jmol> but more dissociated. This is because it interpolated between the two structures and translated the top allyl fragment and therefore did not consider the possibility of a rotation around the central bonds.

To get around this the C-C-C-C dihedral angle (C2-C3-C4-C5) was set to 0 o and the C-C-C angle (C2-C3-C4 and C3-C4-C5 for the product molecule) was set to 100 o for both molecules. This gave rise to the two structures below.

[[Image:TS_(QST2)_numbering_2dh08.jpg|400x400px|center]]

After this the QST2 calculation was run again and the results in table 5 were produced. The imaginary frequency at -839 cm-1 shows that while one bond is forming the other one is breaking, which is again an asynchronous bond formation.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5: Data for the boat transition state from the QST2 optimisation method
| Optimised TS Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.14
|-
|Energy (atomic units) || -231.60280151
|-
| Energy (KJ mol-1) || -6.08 x 105
|-
| Imaginary frequency (cm-1) || -839
|-
| Animation ||[[Image:-839dh08.gif]]
|}

===='''Intrinsic Reaction Coordinate'''====

It is not possible to predict which product the chair and boat transition state will lead to. To solve this problem within Gaussian one can use the IRC method, which allows one to follow the minimum energy path from a transition structure down to its local minimum on a potential energy surface. This creates a succession of points by taking small geometry steps in the direction to where the gradient of the energy surface is at its steepest until the minimum in energy is reached.

The chk file for the optimized chair transition structure was opened and the job type selected was "IRC", the reaction coordinate was computed in the forward direction (since molecule is symmetric) with 50 points along the IRC. This was done with two type of force constants being calculated, once and always.

Within the "once" method the output showed no bond formation process by the 26th step and therefore the calculation was incomplete, since a complete 1,5-hexadiene structure was supposed to be formed. From looking at the IRC pathway it can be seen that the minimum on the potential energy surface was not reached at that point, since the points on the pathway did not form an asymptote and the gradient did not go close enough to zero.

For the "always" method the computation of force constants were carried out at every step. This therefore lets the IRC calculation distinguish whether it is moving along the minimum energy pathway while going down the potential energy surface. While this is happening it can make changes to the pathway if necessary. Within the "always" case the bond forming arrives at 1,5-hexadiene and this is backed up by the IRC pathway reaching a minimum within its energy by the gradient going close to zero and the pathway reaches an asymptote.

The structure produced through the always method had an energy of -231.69199434 atomic units, which is very close to the energy of the gauche2 conformer in the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 appendix 1]. This suggests that the reaction takes place through the chair transition state, which later produces the gauche2 conformer of 1,5-hexadiene.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 6: Comparison of IRC calculations for the chair structure
| || '''Calculate force constants once''' || '''Calculate force constants always'''
|-
| '''Number of points along IRC''' || 26 || 47
|-
| '''Structure at the last point of IRC''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Once_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Always_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|'''Energy (atomic units)''' || -231.68906923 || -231.69199434
|-
| '''ΔE, once - always''' || 2.92511 x 10-3 a.u || 7.68 KJ mol-1
|-
| '''IRC pathway''' || [[Image:One_pathwaydh08.jpg|400x400px]] || [[Image:Always_pathwaydh08.jpg|400x400px]]
|-
| '''IRC gradient''' || [[Image:One_gradientdh08.jpg|400x400px]]|| [[Image:Always_gradientdh08.jpg|400x400px]]
|}

 
 
 

===='''Comparison of activation energies for the transition states'''====

The HF/3-21G optimised chair and boat transition structures were both reoptimised using the B3LYP/6-31G* (B3LYP/6-31G (d)) method. This gives a more accurate minimisation however this method takes longer to run. The values for the zero-point energies and thermal energies were found within the thermochemistry section of the log file.

The values for the HF/3-21G reactant (anti2) optimisation were calculated and the B3LYP/6-31G* values were taken from the "Frequency analysis of 1,5-hexadiene Conformers" earlier in this module.

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 7: Energy data for chair, boat and the anti2 conformation from the HF/3-21G and B3LYP/6-31G* methods of optimisation '''
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466677
| align="center" | -231.461320
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450936
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402342
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539524
| align="center" | -231.532562
| align="center" | -234.611716
| align="center" | -234.469177
| align="center" | -234.461835
|}
Digital repositories

Chair:
HF/3-21G: http://hdl.handle.net/10042/to-5569
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5568

Boat:
HF/3-21G: http://hdl.handle.net/10042/to-5570
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5571

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 8: Summary of activation energies for the chair and boat conformations (ΔE = chair TS or boat TS minus anti2)'''
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| align="center" | 1 a.u = 627.509 kcal mol-1
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE, chair (a.u)'''
| align="center" | 0.072847
| align="center" | 0.071242
| align="center" | 0.054248
| align="center" | 0.052827
| align="center" | -
|-
| '''ΔE, chair (kcal mol-1)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE, boat (a. u)'''
| align="center" | 0.088588
| align="center" | 0.087259
| align="center" | 0.066835
| align="center" | 0.065827
| align="center" | -
|-
| '''ΔE, boat (kcal mol-1)'''
| align="center" | 55.59
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.31
| align="center" | 44.7 ± 2.0
|}

By examining table 8 it is clear to see that the B3LYP/6-31G* method gives values which are much closer to the experimental ones. This is due to the B3LYP/6-31G* giving a higher level of optimisation than the HF/3-21G method. Table 8 also shows that the chair transition state has a lower activation energy to that of the boat transition state by 8.16 kcal mol-1. This suggests that the Cope rearrangement mechanism would most likely proceed through the chair transition state to that of the boat. Therefore the reaction would take place from the anti2 conformer to the chair transition state and then onto the gauche2 conformer within the Cope rearrangement.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9: Geometric parameters for the optimised chair and boat structures '''
| || ''' HF/3-21G chair TS''' || ''' B3LYP/6-31G* chair TS''' || '''HF/3-21G boat TS''' || '''B3LYP/6-31G* boat TS'''
|-
| Optimised Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Chair_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Boat_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.02 (C5-C10 or C3-C14)|| 1.97 (C5-C10 or C3-C14) || 2.14 (C3-C4 or C1-C6) || 2.21 (C3-C4 or C1-C6)
|-
| Fragment angle || 120 o (C3-C1-C5) || 120 o (C3-C1-C5) || 122 o (C4-C5-C6) || 122 o (C4-C5-C6)
|-
| Dihedral angle || 55 o (C1-C3-C14-C11)|| 54 o (C1-C3-C14-C11)|| 0 o (C2-C1-C6-C5) || 0 o (C2-C1-C6-C5)
|-
| Numbering of carbons || [[Image:Numbering_chairdh08.jpg|200x200px]]|| [[Image:Numbering_chairdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]]
|}

From studying table 9 it can be seen that the C-C bond breaking/forming length has changed between the two different types of optimisation methods. However the angles between the bonds remains unchanged.

=='''The Diels Alder Cycloaddition'''==

The Diels-Alder reaction is a cycloaddition between a conjugated diene and a dienophile (alkene system), which forms a cyclohexene system. It is often referred to as a [4s+2s] cycloaddition and proceeds through a single cyclic, concerted transition state where 2 σ bonds are formed in between the termini carbons of the 2 conjugated п systems. This is shown in the reaction scheme below where ethene is reacting with cis-butadiene.

[[Image:Reaction_scheme2dh08.jpg|300x300px|center]]

The π orbitals of the dienophile form new σ bonds with the π orbitals of the diene. The number of π electrons involved determine whether the reaction proceeds in a concerted stereospecific fashion (allowed) or not (forbidden). The HOMO/LUMO of one fragment interacts with the HOMO/LUMO of the other fragment to form two new bonding and anti-bonding MOs. The nodal properties allow one to predict how the reaction will occur.

Semi-Empirical/AM1 method and the B3LYP/6-31G* method of optimisation will be used on Gaussian to determine particular properites of the reactants to predict the way in which the reaction will proceed.

==='''Reagents'''===

<jmol>
<jmolAppletButton>
<uploadedFileContents>Cis_butadienedh08.mol</uploadedFileContents>
<text>Cis-butadiene</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Ethenedh08.mol</uploadedFileContents>
<text>ethylene</text>
<color>white</color>
</jmolAppletButton>
</jmol> were optimised separately using the AM1 semi-empirical method on Gaussview.The cis-butadiene had an energy of 0.04879725 a. u (1.28 x 102 KJ mol-1) and ethylene had an energy of 0.02619027 a. u (68.76 KJ mol-1).

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10: Frontier molecular orbitals of cis-butadiene and ethylene '''
| '''Reagent''' || '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| cis-butadiene || LUMO || [[Image:LUMO1_cis_butadieneDH08.jpg|100x100px]] || [[Image:LUMO2_cis_butadienedh08.jpg|100x100px]]|| Symmetric || +0.017
|-
| cis-butadiene || HOMO || [[Image:HOMO1_cis_butadienedh08.jpg|100x100px]] || [[Image:HOMO2_cis_butadienedh08.jpg|100x100px]]|| Anti-symmetric || -0.343
|-
| ethylene || LUMO || [[Image:LUMO1_ethenedh08.jpg|100x100px]] || [[Image:LUMO2_ethenedh08.jpg|100x100px]]|| Anti-symmetric || +0.052
|-
| ethylene || HOMO || [[Image:HOMO1_ethenedh08.jpg|100x100px]] || [[Image:HOMO2_ethenedh08.jpg|100x100px]]|| Symmetric || -0.387
|}

 
The symmetries of the LUMO and HOMO orbitals for cis-butadiene are opposite of those for the ethylene. This therefore means that the only interactions that can take place are those between ethylene's LUMO and cis-butadiene's HOMO and in addition to this the interaction between cis-butadiene's LUMO and ethylene's HOMO. This is because orbital symmetry has to be conserved (the conservation of orbital symmetry states that the symmetry of an new orbital formed by the overlap of the 2 original orbitals must be the same as the symmetry of the initial orbitals).

==='''Transition State'''===

A <jmol>
<jmolAppletButton>
<uploadedFileContents>Bicyclic_startingdh08.mol</uploadedFileContents>
<text>bicyclo system</text>
<color>white</color>
</jmolAppletButton>
</jmol> was drawn and one of the -CH2-CH2- fragments were removed. Three C=C bonds were added and the previous one was taken away so a <jmol>
<jmolAppletButton>
<uploadedFileContents>Frozen_coorddh08.mol</uploadedFileContents>
<text>guess transition state</text>
<color>white</color>
</jmolAppletButton>
</jmol> was produced.

After this an optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while bonds between the termini of the fragments were frozen at 2.1Å. Following this the resulting file had two separate optimisation and frequency calculations run on it using the TS (Berny) with one using the semi-empirical/AM1 method and the other using the DFT/B3LYP/6-31G* level of theory. Both of these optimisations had the bonds unfrozen and the "derivative" option was chosen instead.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11: Comparison of two methods of optimsations for the transition state '''
| || '''semi-empirical/AM1 method''' || '''DFT/B3LYP/6-31G* method'''
|-
| Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>SE_AM1_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>DFT_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Energy (a. u) || 0.11165608 || -234.54389573
|-
| Bond lengths || [[Image:DistancesSEdh08.jpg|150x150px]]|| [[Image:DistancesDFTdh08.jpg|150x150px]]
|-
|Imaginary frequency (cm-1) || -957 || -525
|-
|Animation || [[Image:-957dh08.gif]] || [[Image:-525dh08.gif]]
|-
| Lowest real frequency (cm-1) || 148 || 136
|-
|Animation || [[Image: 148dh08.gif]] || [[Image: 136dh08.gif]]

|}

The C-C bond forming length between the termini carbons is 0.15 Å larger for the DFT/B3LYP/6-31G* level of theory and this is because this method gives carries out the calculation to a greater degree of accuracy. This larger bond length leads a lower potential energy gradient to the transition state, which gives a lower total energy for the transition state. Therefore the given values below will be the DFT/B3LYP/6-31G* and not the semi-empirical/AM1 method.

A typical sp3 C-C bond length is 1.54 Å and a typical sp2 C=C bond length is 1.34 Å<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref>. Within the optimised transition molecule above the C-C bond lengths are 1.40 Å and the C=C bond length is 1.38 Å. This suggests that the π-bonds are breaking in order for the C-C bonds to form and additionally to this a π-bond is forming in between the two sp3 carbons.

The Van der Waals radius of a C-atom is 1.72 Å<ref name="ref 3">M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer, D. G. Truhlar, 1995, 346, pp. 63-69.[http://pubs.acs.org/doi/abs/10.1021/jp8111556]</ref>. The partly formed C-C length is 2.27 Å, which is shorter than two times the the van der Waals radius of a carbon atom. Therefore the carbon atoms are within a close enough distance to each other to share electron density and yet the bond has not formed at this point.

From looking at the animations for the imaginary frequency above it can be seen that the formation of the two bonds between the termini of the reactants is synchronous. This is expected of the Diels-Alder reaction since the mechanism is concerted. However the lowest real frequency is asynchronous with respect to the two bonds forming between the termini.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12: Frontier molecular orbitals of the transition state '''
| '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| LUMO || [[Image:LUMO1TSdh08.jpg|100x100px]] || [[Image:LUMO2TSdh08.jpg|100x100px]]|| Symmetric || +0.023
|-
| HOMO || [[Image:HOMO1TSdh08.jpg|100x100px]] || [[Image:HOMO2TSdh08.jpg|100x100px]]|| Anti-symmetric || -0.323
|}

From examining the images from table 12 above it can be deduced that the LUMO of the transition state originates from the cis-butadiene's LUMO and the ethylene's HOMO, which represents the σ*-antibonding orbitals. However the HOMO of the transition state originates from the cis-butadiene's HOMO and the ethylene's LUMO, which represents the σ-bonding orbitals within the two single C-C bonds that are formed within the reaction.

==='''Cyclohexa-1,3-diene reaction with maleic anhydride'''===

The <jmol>
<jmolAppletButton>
<uploadedFileContents>FrozenTS22dh08.mol</uploadedFileContents>
<text>exo</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_frozenTS22dh08.mol</uploadedFileContents>
<text>endo</text>
<color>white</color>
</jmolAppletButton>
</jmol> transition states were drawn on Gaussview and then the optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while the distance of the forming bonds were frozen at 2.1Å between the respective carbons. After this the structures changed slightly (<jmol>
<jmolAppletButton>
<uploadedFileContents>TS2_afteropt1_exo1dh08.mol</uploadedFileContents>
<text>exo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_TS2_afteropt1dh08.mol</uploadedFileContents>
<text>endo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol>) and the resulting files had an optimisation and frequency calculation run on it using the TA (Berny) with using the semi-empirical/AM1 level of theory while the unfrozen option as changed to "derivative". After this the endo and exo optimised transition state structures were re-optimised as they were before but using the DFT/B3LYP/6-31G* level of theory instead with the additional keyword "Opt=NoEigen".

{| class="wikitable" border="1" align="center"
|+ '''Table 13: Comparison of the exo and endo transition states '''
! rowspan="2"|-!! colspan="2"|''exo'' !! colspan="2"|''endo''
|-
| Semi-empirical/AM1 || DFT/B3LYP/6-31G* || Semi-empirical/AM1 || DFT/B3LYP/6-31G*
|-
| align="center"|Structure ||align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>After_opt2_exodh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>Endo_TS2_afteropt2dh081.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| align="center"| Energy (a. u) ||align="center"| -0.05041975 ||align="center"| -612.67931095 ||align="center"| -0.05150470 ||align="center"| -612.68339684
|-
| C-C bond forming length (Å) ||align="center"| 2.17 ||align="center"| 2.29 ||align="center"| 2.16 ||align="center"| 2.27
|-align="center"
| Imaginary frequency (cm-1) ||align="center"| -812 ||align="center"| -448 ||align="center"| -806 ||align="center"| -447
|-align="center"
| Animation ||colspan="2"|[[Image:-812_SEdh08.gif]] || colspan="2"|[[Image:-806_cm-1dh08.gif]]
|}

Again the C-C bond forming length between the termini carbons is larger (by 0.11 and 0.12 Å) for the DFT/B3LYP/6-31G* level of theory and this is due the same reasons talked about above.

From looking at the animations for the imaginary frequency above it can also be seen that the formation of the two bonds between the carbons in this case as well is synchronous. Again this is expected of the Diels-Alder reaction since it is concerted.

{| align=center
|+ '''Strain diagrams for the endo and exo transition states'''
| [[Image:Exostraindh08.jpg|thumb|200x200px|'''Exo TS''']]
| [[Image:Strainendodh08.jpg|thumb|200x200px|'''Endo TS''']]
|}

From studying the energies in table 13 it can be seen that the endo transition state is more stable than the exo by 4.08589 x 10-3 a. u (10.73 KJ mol-1). However by looking at the strain of both transition states one would expect the exo TS to be more stable. This is because the distance between the cyclohexa-1,3-diene ring and the -(C=O)-O-(C=O)- maleic anhydride fragment is 2.99 Å in the endo TS and 3.03 Å in the exo TS. Thus the endo form would have more strain due to steric hindrance.

However within the exo TS the vicinal hydrogens on the CH2-CH2 fragment on the cyclohexa-1,3-diene ring are found below the plane and next to the maleic anhydride. This leads to steric hindrance is this is where the strain originates from within the exo TS. Therefore this disfavours the exo form, which makes the endo more stable.

{| class="wikitable" border="1" align="center"
|+ '''Table 14: Frontier molecular orbitals of the the transition state for the cyclohexa-1,3-diene reaction '''
! rowspan="2"|Molecular Orbital !! colspan="4"|''exo'' !! colspan="4"|''endo''
|- align="center"
| '''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''||'''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''
|- align="center"
| LUMO || [[Image:LUMO1exoh081.jpg|150x150px]]|| [[Image:LUMO2exodh081.jpg|150x150px]]|| Anti-symmetric || -0.078 || [[Image:LUMO1ENDOdh08.jpg|150x150px]]||[[Image:LUMO2ENDOdh08.jpg|150x150px]] || Anti-symmetric || -0.067
|- align="center"
| HOMO || [[Image:HOMO1exodh081.jpg|150x150px]]||[[Image:HOMO2exodh081.jpg|150x150px]] || Anti-symmetric|| -0.242 || [[Image:HOMO1ENDOdh08.jpg|150x150px]]|| [[Image:HOMO2ENDOdh08.jpg|150x150px]]||Anti-symmetric || -0.242
|}

Digital repositories:

exo: http://hdl.handle.net/10042/to-5596

endo: http://hdl.handle.net/10042/to-5597

Why do you think that the exo form could be more strained?

Examine carefully the nodal properties of the HOMO between the -(C=O)-O-(C=O)- fragment and the remainder of the system.

What can you conclude about the so called “secondary orbital overlap effect”? (There is some discussion of this in Ian Fleming's book 'Frontier Orbitals and Organic Chemical Reactions').

Further discussion:
What effects have been neglected in these calculations of Diels Alder transition states?

=='''Conclusion'''==

='''References'''=
<references/>

Rep:Mod:hampluscheese

2010-11-10T21:44:20Z

Dh08: /* '''Cyclohexa-1,3-diene reaction with maleic anhydride''' */

='''Module 3:Physical Computational Chemistry Lab'''=

Douglas Hunt

=='''Objectives'''==

The aim of this experiment is to

=='''The Cope Rearrangement'''==

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangement] is a [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement |3,3|-sigmatropic rearrangement] that takes place through a single [http://en.wikipedia.org/wiki/Pericyclic pericyclic], concerted [http://en.wikipedia.org/wiki/Transition_state transition state]. Within this rearrangement a σ bond is formed in concert with another being broken within a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] π system.

The Cope rearrangement we will be studying will be that of 1,5-hexadiene, which is shown below.

[[Image:Cope_rearrangement_schemedh08.jpg|300x300px|center]]

The objective is to locate the low-energy minima and transition structures on the C6H10 potential energy surface and to determine the preferred reaction mechanism by using [http://en.wikipedia.org/wiki/GAUSSIAN Gaussian], with the [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)] methods.

==='''Optimisation of 1,5-hexadiene Conformers'''===

Rough <jmol>
<jmolAppletButton>
<uploadedFileContents>Anti_2_drawndh08.mol</uploadedFileContents>
<text>anti</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche3_drawndh08.mol</uploadedFileContents>
<text>gauche</text>
<color>white</color>
</jmolAppletButton>
</jmol> conformers of 1,5-hexadiene were drawn on Gaussview and their structures were tidied by the clean function. They were then further optimised by the HF/3-21G method and were identified which conformer they were from their point group and appendix 1 on the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3 physical lab page].

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 1: Optimisation and comparison of anti and gauche 1,5-hexadiene conformers using HF/3-21G methods
| ''' Conformer''' || '''Optimised Structure''' || '''Energy (Atomic Units)''' || '''[http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1 Energies (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (Anti2 - Gauche3)''' || '''Point group'''
|-
| '''Anti2''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Anti_2_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692535 || -231.69254 ||-6.08 x 105 || 1.2606 x 10-4 a. u|| Ci
|-
| '''Gauche3''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Gauche3_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692661 || -231.69266 || -6.08 x 105 || 3.31 x 10-1 KJ mol-1 || C1
|}

From studying table 1 it can be seen that the gauche conformer is 0.33 KJ mol-1 more stable than the anti conformation. One would assume that the anti2 conformer would be more stable due to [http://en.wikipedia.org/wiki/Steric steric] factors/repulsion within the gauche3 conformation. However within the gauche3 conformer there is a favourable donation of [http://en.wikipedia.org/wiki/Electron_density electron density] from the C=C π orbital to the C-H σ* orbital from the adjacent proton, which stabilises the conformer<ref name="ref 1">B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446.[http://www.informaworld.com/smpp/content~db=all?content=10.1080/00268970110081412]</ref>.

 

The anti2 conformer was then optimised using the B3LYP 6-31G* (6-31G (d)) method.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 2: Comparison of the anti2 conformer optimisations carried out at different calculation methods
| ''' Method''' || '''Structure and bond distances (Å)''' || '''Dihedral angle (C1 to C4)''' || '''Energy (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (HF/3-21G - DFT/B3LYP/6-31G)'''
|-
| HF/3-21G || [[Image:Anti2_opt1dh08.jpg|300x300px]] || 114.6 o || -231.692535 || -6.08 x 105 || 2.919171 a. u
|-
| DFT/B3LYP/6-31G* || [[Image:Anti2_opt2DH08.jpg|300x300px]] || 118.8 o || -234.611716 || -6.16 x 105 || 7.66 x 103 KJ mol-1
|}

From examining table 2 at first it is possible to see that the bond lengths are very similar to each other and that of the literature bond lengths<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref> (C1=C2: 1.340 Å, C2-C3: 1.508 Å and C3-C4: 1.538 Å). It is also possible to see that the C=C bond length is more accurate for the DFT/B3LYP/6-31G* method by it being 0.01 Å closer to the literature value. Another main difference in terms of geometry is that the DFT/B3LYP/6-31G* method gives a larger [http://en.wikipedia.org/wiki/Dihedral_angle dihedral angle] (and most likely more accurate) of 118.8o compared to that of 114.6o.

In terms of energy the DFT/B3LYP/6-31G* method produces a conformer of a higher stability by it possessing a lower energy with a difference of 7.66 x 103 KJ mol-1.

==='''Frequency analysis of 1,5-hexadiene Conformers'''===

[[Image:Vibrations_anti2dh08.jpg|right|300x300px]]

For the frequency analysis the DFT/B3LYP/6-31G* method was used.

Frequency analysis is essential within this calculation due to it being the second derivative of the potential energy surface. This means that if all the frequencies are all positive the system is at a minimum, if one of them is negative the optimised system is at a transition state, and if more than one are negative then the calculation has failed to find a critical point, which means the optimisation has failed. This therefore gives us a method of seeing whether the calculation has worked.

From looking to the image on the right it can deduced that the optimisation has taken place successfully by there being no negative vibrations and therefore the conformation is at a minimum.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 3: Energies under the thermochemistry section at 298.15 K and 0 K within the frequency logfiles'''
| '''Energies''' || '''298.15 K (atomic units)''' ||| '''0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (KJ mol-1)''' || '''Discussion'''
|-
| Sum of electronic and zero-point energies || -234.469177 || -234.466673 || -2.504 x 10-3 || -6.57 || This is the potential energy at 0 K, which includes the zero-point vibrational energy (E = Eelec + ZPE).

|-
| Sum of electronic and thermal energies || -234.461835 || -234.461085 || -7.5 x 10-4|| -1.97 || This is the energy at 298.15 K (0 K for 0 K calculation) and 1 atm of pressure, which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans).

|-
| Sum of electronic and thermal enthalpies || -234.460891 || -234.460141 || -7.5 x 10-4 || -1.97 || This contains an additional correction for RT (H = E + RT), which is especially important when looking at dissociation reactions.
|-
| Sum of electronic and thermal free energies || -234.500739 || -234.495931 || 4.808 x 10-3 || -12.62 || This includes the entropic contribution to the free energy (G = H - TS).
|}

{|align=right
|+
| [[Image:Anti2_freq_IRdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 298.15 K''']]
| [[Image:Anti2_freq_0Kdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 0 K''']]
|}

From looking at table 3 above it can be deduced that the anti2 conformer is more stable at room temperature to that of 0 K with respect to all the types of energies. The spectra also seem to not change much between the two temperatures but the main change is the change in the sum of electronic and thermal free energies, which is heavily dependent of temperature by it including the entropic contribution to the free energy (G = H - '''T'''S)

 
 
 
 

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

Within the Cope Rearrangement there are two possible transition states, the chair (C2h) and the boat (C2v), which are shown within [https://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_2 appendix two ].

==='''Comparison of optimisation methods for the chair transition state'''===

An <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2hdh08.mol</uploadedFileContents>
<text>allyl fragment</text>
<color>white</color>
</jmolAppletButton>
</jmol>, CH2CHCH2 was drawn on Gaussview and was optimized through the HF/3-21G method. After this the original allyl fragment was copied and pasted into a new window and place exactly above the first fragment with a bond distance of 2.20 Å between the terminal end carbons. This therefore produced the <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2h_fulldh08.mol</uploadedFileContents>
<text>chair transition structure.</text>
<color>white</color>
</jmolAppletButton>
</jmol>

Gaussian was then set up for an optimisation for the transition state with the additional key words "Opt=NoEigen", the optimisation set to TS (Berny), the force constants set to "once" and using the HF/3-21G method.

After this the "chair" transition state was optimised an alternative way by freezing the coordinates of the two sets of terminal carbons (where the bonds form/break during the rearrangement) within the Redundant Coord Editor on Gauusview.
Following this a second calculation was run where the freeze coordinates option was changed to "Derivative" and the force constants were set to "never".

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 4: Comparison of chair transition state optimisation methods
| || '''Optimised through a TS (Berny)''' || '''Optimised through a frozen coordinate method'''
|-
| '''Optimised TS Structure''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>C2h_optfreq_TS_Bdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| '''Terminal C-C bond length (Å)''' || 2.02 || 2.02
|-
|'''Energy (atomic units)''' || -231.61932247 || -231.61932190
|-
| '''ΔE, TS (Berny) - frozen coordinate''' || 5.7 x 10-7 a.u || 1.50 x 10-3 KJ mol-1
|-
| '''Imaginary frequency (cm-1)''' || -818 || -818
|-
| '''Animation''' ||[[Image:818dh08TSB.gif|350x350px]] || [[Image:818cm-1Fdh08.gif]]
|}

The data in table 4 shows that both methods give the same results and they both give the same transition state. They both produced the same energy with a negligible energy difference between them and the bond lengths between the terminal carbons are both 2.02 Å. In addition to this the imaginary frequencies were both -818cm-1 and within the animations above it is possible to see that when one C-C bond is forming the other is breaking. This means it is a asynchronous bond formation, which is expected for the Cope Rearrangement since it is a pericyclic reaction.

The only difference between the two optimisations is the way in which the transition states were calculated. This is due to the TS (Berny) method calculating the force constant matrix within the first optimisation step and is then being recalculated as the optimisation carries on. When the frozen coordinate method freezes the reaction coordinate in order for the rest of the structure to be optimised around it, which does not require the whole hessian (force constant matrix) to be calculated but only that the reaction coordinate is differentiated (second step). The main advantage of this, is that you do not need to calculate the entire force constant matrix and just need to differentiate the reaction coordinate in order to obtain a good enough guess for the initial force constant matrix. The frozen coordinate method will therefore take less time to compute the transition state, which can be very expensive within certain systems. However to carry out this method one needs to have an idea of how far apart the terminal carbons are.

==='''Boat transition state'''===

===='''QST2 optimisation'''====

The optimisation for the boat transition state had a different method and this was the QST2 method. This calculation interpolates between two structures (product and reactant) and tries to find a transition state between them.

The reactants and products were numbered in the same way. Therefore both 1,5-hexadiene molecules had their numbering manually changed for the product molecule so it corresponded to the numbering of the reactant if it rearranged. This is illustrated in the reaction scheme below.

[[Image:Boat_rerrangmentdh08.jpg|300x300px|center]]

The chk file for the optimized anti2 conformer was opened and copied into a new window. A second window was opened within the new window and the same anti2 conformer was copied into the new window. The numbering was changed within the second molecule so the two looked like they do in the image below.

[[Image:TS_(QST2)_numbering_1dh08.jpg|400x400px|center]]

The job type was set as "Opt+Freq" and "TS (QST2)" was chosen to optimise the transition state. However this failed and produced
a structure a bit like a <jmol>
<jmolAppletButton>
<uploadedFileContents>Trialone_BPdh08.mol</uploadedFileContents>
<text>chair transition structure </text>
<color>white</color>
</jmolAppletButton>
</jmol> but more dissociated. This is because it interpolated between the two structures and translated the top allyl fragment and therefore did not consider the possibility of a rotation around the central bonds.

To get around this the C-C-C-C dihedral angle (C2-C3-C4-C5) was set to 0 o and the C-C-C angle (C2-C3-C4 and C3-C4-C5 for the product molecule) was set to 100 o for both molecules. This gave rise to the two structures below.

[[Image:TS_(QST2)_numbering_2dh08.jpg|400x400px|center]]

After this the QST2 calculation was run again and the results in table 5 were produced. The imaginary frequency at -839 cm-1 shows that while one bond is forming the other one is breaking, which is again an asynchronous bond formation.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5: Data for the boat transition state from the QST2 optimisation method
| Optimised TS Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.14
|-
|Energy (atomic units) || -231.60280151
|-
| Energy (KJ mol-1) || -6.08 x 105
|-
| Imaginary frequency (cm-1) || -839
|-
| Animation ||[[Image:-839dh08.gif]]
|}

===='''Intrinsic Reaction Coordinate'''====

It is not possible to predict which product the chair and boat transition state will lead to. To solve this problem within Gaussian one can use the IRC method, which allows one to follow the minimum energy path from a transition structure down to its local minimum on a potential energy surface. This creates a succession of points by taking small geometry steps in the direction to where the gradient of the energy surface is at its steepest until the minimum in energy is reached.

The chk file for the optimized chair transition structure was opened and the job type selected was "IRC", the reaction coordinate was computed in the forward direction (since molecule is symmetric) with 50 points along the IRC. This was done with two type of force constants being calculated, once and always.

Within the "once" method the output showed no bond formation process by the 26th step and therefore the calculation was incomplete, since a complete 1,5-hexadiene structure was supposed to be formed. From looking at the IRC pathway it can be seen that the minimum on the potential energy surface was not reached at that point, since the points on the pathway did not form an asymptote and the gradient did not go close enough to zero.

For the "always" method the computation of force constants were carried out at every step. This therefore lets the IRC calculation distinguish whether it is moving along the minimum energy pathway while going down the potential energy surface. While this is happening it can make changes to the pathway if necessary. Within the "always" case the bond forming arrives at 1,5-hexadiene and this is backed up by the IRC pathway reaching a minimum within its energy by the gradient going close to zero and the pathway reaches an asymptote.

The structure produced through the always method had an energy of -231.69199434 atomic units, which is very close to the energy of the gauche2 conformer in the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 appendix 1]. This suggests that the reaction takes place through the chair transition state, which later produces the gauche2 conformer of 1,5-hexadiene.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 6: Comparison of IRC calculations for the chair structure
| || '''Calculate force constants once''' || '''Calculate force constants always'''
|-
| '''Number of points along IRC''' || 26 || 47
|-
| '''Structure at the last point of IRC''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Once_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Always_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|'''Energy (atomic units)''' || -231.68906923 || -231.69199434
|-
| '''ΔE, once - always''' || 2.92511 x 10-3 a.u || 7.68 KJ mol-1
|-
| '''IRC pathway''' || [[Image:One_pathwaydh08.jpg|400x400px]] || [[Image:Always_pathwaydh08.jpg|400x400px]]
|-
| '''IRC gradient''' || [[Image:One_gradientdh08.jpg|400x400px]]|| [[Image:Always_gradientdh08.jpg|400x400px]]
|}

 
 
 

===='''Comparison of activation energies for the transition states'''====

The HF/3-21G optimised chair and boat transition structures were both reoptimised using the B3LYP/6-31G* (B3LYP/6-31G (d)) method. This gives a more accurate minimisation however this method takes longer to run. The values for the zero-point energies and thermal energies were found within the thermochemistry section of the log file.

The values for the HF/3-21G reactant (anti2) optimisation were calculated and the B3LYP/6-31G* values were taken from the "Frequency analysis of 1,5-hexadiene Conformers" earlier in this module.

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 7: Energy data for chair, boat and the anti2 conformation from the HF/3-21G and B3LYP/6-31G* methods of optimisation '''
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466677
| align="center" | -231.461320
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450936
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402342
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539524
| align="center" | -231.532562
| align="center" | -234.611716
| align="center" | -234.469177
| align="center" | -234.461835
|}
Digital repositories

Chair:
HF/3-21G: http://hdl.handle.net/10042/to-5569
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5568

Boat:
HF/3-21G: http://hdl.handle.net/10042/to-5570
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5571

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 8: Summary of activation energies for the chair and boat conformations (ΔE = chair TS or boat TS minus anti2)'''
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| align="center" | 1 a.u = 627.509 kcal mol-1
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE, chair (a.u)'''
| align="center" | 0.072847
| align="center" | 0.071242
| align="center" | 0.054248
| align="center" | 0.052827
| align="center" | -
|-
| '''ΔE, chair (kcal mol-1)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE, boat (a. u)'''
| align="center" | 0.088588
| align="center" | 0.087259
| align="center" | 0.066835
| align="center" | 0.065827
| align="center" | -
|-
| '''ΔE, boat (kcal mol-1)'''
| align="center" | 55.59
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.31
| align="center" | 44.7 ± 2.0
|}

By examining table 8 it is clear to see that the B3LYP/6-31G* method gives values which are much closer to the experimental ones. This is due to the B3LYP/6-31G* giving a higher level of optimisation than the HF/3-21G method. Table 8 also shows that the chair transition state has a lower activation energy to that of the boat transition state by 8.16 kcal mol-1. This suggests that the Cope rearrangement mechanism would most likely proceed through the chair transition state to that of the boat. Therefore the reaction would take place from the anti2 conformer to the chair transition state and then onto the gauche2 conformer within the Cope rearrangement.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9: Geometric parameters for the optimised chair and boat structures '''
| || ''' HF/3-21G chair TS''' || ''' B3LYP/6-31G* chair TS''' || '''HF/3-21G boat TS''' || '''B3LYP/6-31G* boat TS'''
|-
| Optimised Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Chair_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Boat_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.02 (C5-C10 or C3-C14)|| 1.97 (C5-C10 or C3-C14) || 2.14 (C3-C4 or C1-C6) || 2.21 (C3-C4 or C1-C6)
|-
| Fragment angle || 120 o (C3-C1-C5) || 120 o (C3-C1-C5) || 122 o (C4-C5-C6) || 122 o (C4-C5-C6)
|-
| Dihedral angle || 55 o (C1-C3-C14-C11)|| 54 o (C1-C3-C14-C11)|| 0 o (C2-C1-C6-C5) || 0 o (C2-C1-C6-C5)
|-
| Numbering of carbons || [[Image:Numbering_chairdh08.jpg|200x200px]]|| [[Image:Numbering_chairdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]]
|}

From studying table 9 it can be seen that the C-C bond breaking/forming length has changed between the two different types of optimisation methods. However the angles between the bonds remains unchanged.

=='''The Diels Alder Cycloaddition'''==

The Diels-Alder reaction is a cycloaddition between a conjugated diene and a dienophile (alkene system), which forms a cyclohexene system. It is often referred to as a [4s+2s] cycloaddition and proceeds through a single cyclic, concerted transition state where 2 σ bonds are formed in between the termini carbons of the 2 conjugated п systems. This is shown in the reaction scheme below where ethene is reacting with cis-butadiene.

[[Image:Reaction_scheme2dh08.jpg|300x300px|center]]

The π orbitals of the dienophile form new σ bonds with the π orbitals of the diene. The number of π electrons involved determine whether the reaction proceeds in a concerted stereospecific fashion (allowed) or not (forbidden). The HOMO/LUMO of one fragment interacts with the HOMO/LUMO of the other fragment to form two new bonding and anti-bonding MOs. The nodal properties allow one to predict how the reaction will occur.

Semi-Empirical/AM1 method and the B3LYP/6-31G* method of optimisation will be used on Gaussian to determine particular properites of the reactants to predict the way in which the reaction will proceed.

==='''Reagents'''===

<jmol>
<jmolAppletButton>
<uploadedFileContents>Cis_butadienedh08.mol</uploadedFileContents>
<text>Cis-butadiene</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Ethenedh08.mol</uploadedFileContents>
<text>ethylene</text>
<color>white</color>
</jmolAppletButton>
</jmol> were optimised separately using the AM1 semi-empirical method on Gaussview.The cis-butadiene had an energy of 0.04879725 a. u (1.28 x 102 KJ mol-1) and ethylene had an energy of 0.02619027 a. u (68.76 KJ mol-1).

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10: Frontier molecular orbitals of cis-butadiene and ethylene '''
| '''Reagent''' || '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| cis-butadiene || LUMO || [[Image:LUMO1_cis_butadieneDH08.jpg|100x100px]] || [[Image:LUMO2_cis_butadienedh08.jpg|100x100px]]|| Symmetric || +0.017
|-
| cis-butadiene || HOMO || [[Image:HOMO1_cis_butadienedh08.jpg|100x100px]] || [[Image:HOMO2_cis_butadienedh08.jpg|100x100px]]|| Anti-symmetric || -0.343
|-
| ethylene || LUMO || [[Image:LUMO1_ethenedh08.jpg|100x100px]] || [[Image:LUMO2_ethenedh08.jpg|100x100px]]|| Anti-symmetric || +0.052
|-
| ethylene || HOMO || [[Image:HOMO1_ethenedh08.jpg|100x100px]] || [[Image:HOMO2_ethenedh08.jpg|100x100px]]|| Symmetric || -0.387
|}

 
The symmetries of the LUMO and HOMO orbitals for cis-butadiene are opposite of those for the ethylene. This therefore means that the only interactions that can take place are those between ethylene's LUMO and cis-butadiene's HOMO and in addition to this the interaction between cis-butadiene's LUMO and ethylene's HOMO. This is because orbital symmetry has to be conserved (the conservation of orbital symmetry states that the symmetry of an new orbital formed by the overlap of the 2 original orbitals must be the same as the symmetry of the initial orbitals).

==='''Transition State'''===

A <jmol>
<jmolAppletButton>
<uploadedFileContents>Bicyclic_startingdh08.mol</uploadedFileContents>
<text>bicyclo system</text>
<color>white</color>
</jmolAppletButton>
</jmol> was drawn and one of the -CH2-CH2- fragments were removed. Three C=C bonds were added and the previous one was taken away so a <jmol>
<jmolAppletButton>
<uploadedFileContents>Frozen_coorddh08.mol</uploadedFileContents>
<text>guess transition state</text>
<color>white</color>
</jmolAppletButton>
</jmol> was produced.

After this an optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while bonds between the termini of the fragments were frozen at 2.1Å. Following this the resulting file had two separate optimisation and frequency calculations run on it using the TS (Berny) with one using the semi-empirical/AM1 method and the other using the DFT/B3LYP/6-31G* level of theory. Both of these optimisations had the bonds unfrozen and the "derivative" option was chosen instead.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11: Comparison of two methods of optimsations for the transition state '''
| || '''semi-empirical/AM1 method''' || '''DFT/B3LYP/6-31G* method'''
|-
| Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>SE_AM1_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>DFT_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Energy (a. u) || 0.11165608 || -234.54389573
|-
| Bond lengths || [[Image:DistancesSEdh08.jpg|150x150px]]|| [[Image:DistancesDFTdh08.jpg|150x150px]]
|-
|Imaginary frequency (cm-1) || -957 || -525
|-
|Animation || [[Image:-957dh08.gif]] || [[Image:-525dh08.gif]]
|-
| Lowest real frequency (cm-1) || 148 || 136
|-
|Animation || [[Image: 148dh08.gif]] || [[Image: 136dh08.gif]]

|}

The C-C bond forming length between the termini carbons is 0.15 Å larger for the DFT/B3LYP/6-31G* level of theory and this is because this method gives carries out the calculation to a greater degree of accuracy. This larger bond length leads a lower potential energy gradient to the transition state, which gives a lower total energy for the transition state. Therefore the given values below will be the DFT/B3LYP/6-31G* and not the semi-empirical/AM1 method.

A typical sp3 C-C bond length is 1.54 Å and a typical sp2 C=C bond length is 1.34 Å<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref>. Within the optimised transition molecule above the C-C bond lengths are 1.40 Å and the C=C bond length is 1.38 Å. This suggests that the π-bonds are breaking in order for the C-C bonds to form and additionally to this a π-bond is forming in between the two sp3 carbons.

The Van der Waals radius of a C-atom is 1.72 Å<ref name="ref 3">M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer, D. G. Truhlar, 1995, 346, pp. 63-69.[http://pubs.acs.org/doi/abs/10.1021/jp8111556]</ref>. The partly formed C-C length is 2.27 Å, which is shorter than two times the the van der Waals radius of a carbon atom. Therefore the carbon atoms are within a close enough distance to each other to share electron density and yet the bond has not formed at this point.

From looking at the animations for the imaginary frequency above it can be seen that the formation of the two bonds between the termini of the reactants is synchronous. This is expected of the Diels-Alder reaction since the mechanism is concerted. However the lowest real frequency is asynchronous with respect to the two bonds forming between the termini.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12: Frontier molecular orbitals of the transition state '''
| '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| LUMO || [[Image:LUMO1TSdh08.jpg|100x100px]] || [[Image:LUMO2TSdh08.jpg|100x100px]]|| Symmetric || +0.023
|-
| HOMO || [[Image:HOMO1TSdh08.jpg|100x100px]] || [[Image:HOMO2TSdh08.jpg|100x100px]]|| Anti-symmetric || -0.323
|}

From examining the images from table 12 above it can be deduced that the LUMO of the transition state originates from the cis-butadiene's LUMO and the ethylene's HOMO, which represents the σ*-antibonding orbitals. However the HOMO of the transition state originates from the cis-butadiene's HOMO and the ethylene's LUMO, which represents the σ-bonding orbitals within the two single C-C bonds that are formed within the reaction.

==='''Cyclohexa-1,3-diene reaction with maleic anhydride'''===

The <jmol>
<jmolAppletButton>
<uploadedFileContents>FrozenTS22dh08.mol</uploadedFileContents>
<text>exo</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_frozenTS22dh08.mol</uploadedFileContents>
<text>endo</text>
<color>white</color>
</jmolAppletButton>
</jmol> transition states were drawn on Gaussview and then the optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while the distance of the forming bonds were frozen at 2.1Å between the respective carbons. After this the structures changed slightly (<jmol>
<jmolAppletButton>
<uploadedFileContents>TS2_afteropt1_exo1dh08.mol</uploadedFileContents>
<text>exo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_TS2_afteropt1dh08.mol</uploadedFileContents>
<text>endo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol>) and the resulting files had an optimisation and frequency calculation run on it using the TA (Berny) with using the semi-empirical/AM1 level of theory while the unfrozen option as changed to "derivative". After this the endo and exo optimised transition state structures were re-optimised as they were before but using the DFT/B3LYP/6-31G* level of theory instead with the additional keyword "Opt=NoEigen".

{| class="wikitable" border="1" align="center"
|+ '''Table 13: Comparison of the exo and endo transition states '''
! rowspan="2"|-!! colspan="2"|''exo'' !! colspan="2"|''endo''
|-
| Semi-empirical/AM1 || DFT/B3LYP/6-31G* || Semi-empirical/AM1 || DFT/B3LYP/6-31G*
|-
| align="center"|Structure ||align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>After_opt2_exodh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>Endo_TS2_afteropt2dh081.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| align="center"| Energy (a. u) ||align="center"| -0.05041975 ||align="center"| -612.67931095 ||align="center"| -0.05150470 ||align="center"| -612.68339684
|-
| C-C bond forming length (Å) ||align="center"| 2.17 ||align="center"| 2.29 ||align="center"| 2.16 ||align="center"| 2.27
|-align="center"
| Imaginary frequency (cm-1) ||align="center"| -812 ||align="center"| -448 ||align="center"| -806 ||align="center"| -447
|-align="center"
| Animation ||colspan="2"|[[Image:-812_SEdh08.gif]] || colspan="2"|[[Image:-806_cm-1dh08.gif]]
|}

Again the C-C bond forming length between the termini carbons is larger (by 0.11 and 0.12 Å) for the DFT/B3LYP/6-31G* level of theory and this is due the same reasons talked about above.

From looking at the animations for the imaginary frequency above it can also be seen that the formation of the two bonds between the carbons in this case as well is synchronous. Again this is expected of the Diels-Alder reaction since it is concerted.

{| align=center
|+ '''Strain diagrams for the endo and exo transition states'''
| [[Image:Exostraindh08.jpg|thumb|200x200px|'''Exo TS''']]
| [[Image:Strainendodh08.jpg|thumb|200x200px|'''Endo TS''']]
|}

From studying the energies in table 13 it can be seen that the endo transition state is more stable than the exo by 4.08589 x 10-3 a. u (10.73 KJ mol-1). However by looking at the strain of both transition states one would expect the exo TS to be more stable. This is because the distance between the cyclohexa-1,3-diene ring and the -(C=O)-O-(C=O)- maleic anhydride fragment is 2.99 Å in the endo TS and 3.03 Å in the exo TS. Thus the endo form would have more strain due to steric hindrance.

However within the exo TS the vicinal hydrogens on the CH2-CH2 fragment on the cyclohexa-1,3-diene ring are found below the plane and next to the maleic anhydride. This is where the strain originates from within the exo TS and therefore disfavours the exo form, which makes the endo more stable.

{| class="wikitable" border="1" align="center"
|+ '''Table 14: Frontier molecular orbitals of the the transition state for the cyclohexa-1,3-diene reaction '''
! rowspan="2"|Molecular Orbital !! colspan="4"|''exo'' !! colspan="4"|''endo''
|- align="center"
| '''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''||'''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''
|- align="center"
| LUMO || [[Image:LUMO1exoh081.jpg|150x150px]]|| [[Image:LUMO2exodh081.jpg|150x150px]]|| Anti-symmetric || -0.078 || [[Image:LUMO1ENDOdh08.jpg|150x150px]]||[[Image:LUMO2ENDOdh08.jpg|150x150px]] || Anti-symmetric || -0.067
|- align="center"
| HOMO || [[Image:HOMO1exodh081.jpg|150x150px]]||[[Image:HOMO2exodh081.jpg|150x150px]] || Anti-symmetric|| -0.242 || [[Image:HOMO1ENDOdh08.jpg|150x150px]]|| [[Image:HOMO2ENDOdh08.jpg|150x150px]]||Anti-symmetric || -0.242
|}

Digital repositories:

exo: http://hdl.handle.net/10042/to-5596

endo: http://hdl.handle.net/10042/to-5597

Why do you think that the exo form could be more strained?

Examine carefully the nodal properties of the HOMO between the -(C=O)-O-(C=O)- fragment and the remainder of the system.

What can you conclude about the so called “secondary orbital overlap effect”? (There is some discussion of this in Ian Fleming's book 'Frontier Orbitals and Organic Chemical Reactions').

Further discussion:
What effects have been neglected in these calculations of Diels Alder transition states?

=='''Conclusion'''==

='''References'''=
<references/>

Rep:Mod:hampluscheese

2010-11-10T21:32:26Z

Dh08: /* '''Cyclohexa-1,3-diene reaction with maleic anhydride''' */

='''Module 3:Physical Computational Chemistry Lab'''=

Douglas Hunt

=='''Objectives'''==

The aim of this experiment is to

=='''The Cope Rearrangement'''==

The [http://en.wikipedia.org/wiki/Cope_rearrangement Cope rearrangement] is a [http://en.wikipedia.org/wiki/Sigmatropic_rearrangement |3,3|-sigmatropic rearrangement] that takes place through a single [http://en.wikipedia.org/wiki/Pericyclic pericyclic], concerted [http://en.wikipedia.org/wiki/Transition_state transition state]. Within this rearrangement a σ bond is formed in concert with another being broken within a [http://en.wikipedia.org/wiki/Conjugated_system conjugated] π system.

The Cope rearrangement we will be studying will be that of 1,5-hexadiene, which is shown below.

[[Image:Cope_rearrangement_schemedh08.jpg|300x300px|center]]

The objective is to locate the low-energy minima and transition structures on the C6H10 potential energy surface and to determine the preferred reaction mechanism by using [http://en.wikipedia.org/wiki/GAUSSIAN Gaussian], with the [http://en.wikipedia.org/wiki/Hartree-Fock Hartree-Fock (HF)] and [http://en.wikipedia.org/wiki/Density_Functional_Theory density functional theory (DFT)] methods.

==='''Optimisation of 1,5-hexadiene Conformers'''===

Rough <jmol>
<jmolAppletButton>
<uploadedFileContents>Anti_2_drawndh08.mol</uploadedFileContents>
<text>anti</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Gauche3_drawndh08.mol</uploadedFileContents>
<text>gauche</text>
<color>white</color>
</jmolAppletButton>
</jmol> conformers of 1,5-hexadiene were drawn on Gaussview and their structures were tidied by the clean function. They were then further optimised by the HF/3-21G method and were identified which conformer they were from their point group and appendix 1 on the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3 physical lab page].

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 1: Optimisation and comparison of anti and gauche 1,5-hexadiene conformers using HF/3-21G methods
| ''' Conformer''' || '''Optimised Structure''' || '''Energy (Atomic Units)''' || '''[http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 Appendix 1 Energies (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (Anti2 - Gauche3)''' || '''Point group'''
|-
| '''Anti2''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Anti_2_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692535 || -231.69254 ||-6.08 x 105 || 1.2606 x 10-4 a. u|| Ci
|-
| '''Gauche3''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Gauche3_opt1dh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || -231.692661 || -231.69266 || -6.08 x 105 || 3.31 x 10-1 KJ mol-1 || C1
|}

From studying table 1 it can be seen that the gauche conformer is 0.33 KJ mol-1 more stable than the anti conformation. One would assume that the anti2 conformer would be more stable due to [http://en.wikipedia.org/wiki/Steric steric] factors/repulsion within the gauche3 conformation. However within the gauche3 conformer there is a favourable donation of [http://en.wikipedia.org/wiki/Electron_density electron density] from the C=C π orbital to the C-H σ* orbital from the adjacent proton, which stabilises the conformer<ref name="ref 1">B. G. Rocque, J. M. Gonzales, H. F. Schaefer, Mol. Phys., 2002, 100, pp. 441-446.[http://www.informaworld.com/smpp/content~db=all?content=10.1080/00268970110081412]</ref>.

 

The anti2 conformer was then optimised using the B3LYP 6-31G* (6-31G (d)) method.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 2: Comparison of the anti2 conformer optimisations carried out at different calculation methods
| ''' Method''' || '''Structure and bond distances (Å)''' || '''Dihedral angle (C1 to C4)''' || '''Energy (Atomic Units)]''' || '''Energy (KJ mol-1)''' ||''' ΔE (HF/3-21G - DFT/B3LYP/6-31G)'''
|-
| HF/3-21G || [[Image:Anti2_opt1dh08.jpg|300x300px]] || 114.6 o || -231.692535 || -6.08 x 105 || 2.919171 a. u
|-
| DFT/B3LYP/6-31G* || [[Image:Anti2_opt2DH08.jpg|300x300px]] || 118.8 o || -234.611716 || -6.16 x 105 || 7.66 x 103 KJ mol-1
|}

From examining table 2 at first it is possible to see that the bond lengths are very similar to each other and that of the literature bond lengths<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref> (C1=C2: 1.340 Å, C2-C3: 1.508 Å and C3-C4: 1.538 Å). It is also possible to see that the C=C bond length is more accurate for the DFT/B3LYP/6-31G* method by it being 0.01 Å closer to the literature value. Another main difference in terms of geometry is that the DFT/B3LYP/6-31G* method gives a larger [http://en.wikipedia.org/wiki/Dihedral_angle dihedral angle] (and most likely more accurate) of 118.8o compared to that of 114.6o.

In terms of energy the DFT/B3LYP/6-31G* method produces a conformer of a higher stability by it possessing a lower energy with a difference of 7.66 x 103 KJ mol-1.

==='''Frequency analysis of 1,5-hexadiene Conformers'''===

[[Image:Vibrations_anti2dh08.jpg|right|300x300px]]

For the frequency analysis the DFT/B3LYP/6-31G* method was used.

Frequency analysis is essential within this calculation due to it being the second derivative of the potential energy surface. This means that if all the frequencies are all positive the system is at a minimum, if one of them is negative the optimised system is at a transition state, and if more than one are negative then the calculation has failed to find a critical point, which means the optimisation has failed. This therefore gives us a method of seeing whether the calculation has worked.

From looking to the image on the right it can deduced that the optimisation has taken place successfully by there being no negative vibrations and therefore the conformation is at a minimum.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 3: Energies under the thermochemistry section at 298.15 K and 0 K within the frequency logfiles'''
| '''Energies''' || '''298.15 K (atomic units)''' ||| '''0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (atomic units)''' || '''ΔE, 298.15 K - 0 K (KJ mol-1)''' || '''Discussion'''
|-
| Sum of electronic and zero-point energies || -234.469177 || -234.466673 || -2.504 x 10-3 || -6.57 || This is the potential energy at 0 K, which includes the zero-point vibrational energy (E = Eelec + ZPE).

|-
| Sum of electronic and thermal energies || -234.461835 || -234.461085 || -7.5 x 10-4|| -1.97 || This is the energy at 298.15 K (0 K for 0 K calculation) and 1 atm of pressure, which includes contributions from the translational, rotational, and vibrational energy modes at this temperature (E = E + Evib + Erot + Etrans).

|-
| Sum of electronic and thermal enthalpies || -234.460891 || -234.460141 || -7.5 x 10-4 || -1.97 || This contains an additional correction for RT (H = E + RT), which is especially important when looking at dissociation reactions.
|-
| Sum of electronic and thermal free energies || -234.500739 || -234.495931 || 4.808 x 10-3 || -12.62 || This includes the entropic contribution to the free energy (G = H - TS).
|}

{|align=right
|+
| [[Image:Anti2_freq_IRdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 298.15 K''']]
| [[Image:Anti2_freq_0Kdh08.jpg|thumb|300x250px|'''IR spectrum of the anti2 conformer at 0 K''']]
|}

From looking at table 3 above it can be deduced that the anti2 conformer is more stable at room temperature to that of 0 K with respect to all the types of energies. The spectra also seem to not change much between the two temperatures but the main change is the change in the sum of electronic and thermal free energies, which is heavily dependent of temperature by it including the entropic contribution to the free energy (G = H - '''T'''S)

 
 
 
 

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

Within the Cope Rearrangement there are two possible transition states, the chair (C2h) and the boat (C2v), which are shown within [https://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_2 appendix two ].

==='''Comparison of optimisation methods for the chair transition state'''===

An <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2hdh08.mol</uploadedFileContents>
<text>allyl fragment</text>
<color>white</color>
</jmolAppletButton>
</jmol>, CH2CHCH2 was drawn on Gaussview and was optimized through the HF/3-21G method. After this the original allyl fragment was copied and pasted into a new window and place exactly above the first fragment with a bond distance of 2.20 Å between the terminal end carbons. This therefore produced the <jmol>
<jmolAppletButton>
<uploadedFileContents>Drawn_c2h_fulldh08.mol</uploadedFileContents>
<text>chair transition structure.</text>
<color>white</color>
</jmolAppletButton>
</jmol>

Gaussian was then set up for an optimisation for the transition state with the additional key words "Opt=NoEigen", the optimisation set to TS (Berny), the force constants set to "once" and using the HF/3-21G method.

After this the "chair" transition state was optimised an alternative way by freezing the coordinates of the two sets of terminal carbons (where the bonds form/break during the rearrangement) within the Redundant Coord Editor on Gauusview.
Following this a second calculation was run where the freeze coordinates option was changed to "Derivative" and the force constants were set to "never".

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 4: Comparison of chair transition state optimisation methods
| || '''Optimised through a TS (Berny)''' || '''Optimised through a frozen coordinate method'''
|-
| '''Optimised TS Structure''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>C2h_optfreq_TS_Bdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| '''Terminal C-C bond length (Å)''' || 2.02 || 2.02
|-
|'''Energy (atomic units)''' || -231.61932247 || -231.61932190
|-
| '''ΔE, TS (Berny) - frozen coordinate''' || 5.7 x 10-7 a.u || 1.50 x 10-3 KJ mol-1
|-
| '''Imaginary frequency (cm-1)''' || -818 || -818
|-
| '''Animation''' ||[[Image:818dh08TSB.gif|350x350px]] || [[Image:818cm-1Fdh08.gif]]
|}

The data in table 4 shows that both methods give the same results and they both give the same transition state. They both produced the same energy with a negligible energy difference between them and the bond lengths between the terminal carbons are both 2.02 Å. In addition to this the imaginary frequencies were both -818cm-1 and within the animations above it is possible to see that when one C-C bond is forming the other is breaking. This means it is a asynchronous bond formation, which is expected for the Cope Rearrangement since it is a pericyclic reaction.

The only difference between the two optimisations is the way in which the transition states were calculated. This is due to the TS (Berny) method calculating the force constant matrix within the first optimisation step and is then being recalculated as the optimisation carries on. When the frozen coordinate method freezes the reaction coordinate in order for the rest of the structure to be optimised around it, which does not require the whole hessian (force constant matrix) to be calculated but only that the reaction coordinate is differentiated (second step). The main advantage of this, is that you do not need to calculate the entire force constant matrix and just need to differentiate the reaction coordinate in order to obtain a good enough guess for the initial force constant matrix. The frozen coordinate method will therefore take less time to compute the transition state, which can be very expensive within certain systems. However to carry out this method one needs to have an idea of how far apart the terminal carbons are.

==='''Boat transition state'''===

===='''QST2 optimisation'''====

The optimisation for the boat transition state had a different method and this was the QST2 method. This calculation interpolates between two structures (product and reactant) and tries to find a transition state between them.

The reactants and products were numbered in the same way. Therefore both 1,5-hexadiene molecules had their numbering manually changed for the product molecule so it corresponded to the numbering of the reactant if it rearranged. This is illustrated in the reaction scheme below.

[[Image:Boat_rerrangmentdh08.jpg|300x300px|center]]

The chk file for the optimized anti2 conformer was opened and copied into a new window. A second window was opened within the new window and the same anti2 conformer was copied into the new window. The numbering was changed within the second molecule so the two looked like they do in the image below.

[[Image:TS_(QST2)_numbering_1dh08.jpg|400x400px|center]]

The job type was set as "Opt+Freq" and "TS (QST2)" was chosen to optimise the transition state. However this failed and produced
a structure a bit like a <jmol>
<jmolAppletButton>
<uploadedFileContents>Trialone_BPdh08.mol</uploadedFileContents>
<text>chair transition structure </text>
<color>white</color>
</jmolAppletButton>
</jmol> but more dissociated. This is because it interpolated between the two structures and translated the top allyl fragment and therefore did not consider the possibility of a rotation around the central bonds.

To get around this the C-C-C-C dihedral angle (C2-C3-C4-C5) was set to 0 o and the C-C-C angle (C2-C3-C4 and C3-C4-C5 for the product molecule) was set to 100 o for both molecules. This gave rise to the two structures below.

[[Image:TS_(QST2)_numbering_2dh08.jpg|400x400px|center]]

After this the QST2 calculation was run again and the results in table 5 were produced. The imaginary frequency at -839 cm-1 shows that while one bond is forming the other one is breaking, which is again an asynchronous bond formation.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 5: Data for the boat transition state from the QST2 optimisation method
| Optimised TS Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.14
|-
|Energy (atomic units) || -231.60280151
|-
| Energy (KJ mol-1) || -6.08 x 105
|-
| Imaginary frequency (cm-1) || -839
|-
| Animation ||[[Image:-839dh08.gif]]
|}

===='''Intrinsic Reaction Coordinate'''====

It is not possible to predict which product the chair and boat transition state will lead to. To solve this problem within Gaussian one can use the IRC method, which allows one to follow the minimum energy path from a transition structure down to its local minimum on a potential energy surface. This creates a succession of points by taking small geometry steps in the direction to where the gradient of the energy surface is at its steepest until the minimum in energy is reached.

The chk file for the optimized chair transition structure was opened and the job type selected was "IRC", the reaction coordinate was computed in the forward direction (since molecule is symmetric) with 50 points along the IRC. This was done with two type of force constants being calculated, once and always.

Within the "once" method the output showed no bond formation process by the 26th step and therefore the calculation was incomplete, since a complete 1,5-hexadiene structure was supposed to be formed. From looking at the IRC pathway it can be seen that the minimum on the potential energy surface was not reached at that point, since the points on the pathway did not form an asymptote and the gradient did not go close enough to zero.

For the "always" method the computation of force constants were carried out at every step. This therefore lets the IRC calculation distinguish whether it is moving along the minimum energy pathway while going down the potential energy surface. While this is happening it can make changes to the pathway if necessary. Within the "always" case the bond forming arrives at 1,5-hexadiene and this is backed up by the IRC pathway reaching a minimum within its energy by the gradient going close to zero and the pathway reaches an asymptote.

The structure produced through the always method had an energy of -231.69199434 atomic units, which is very close to the energy of the gauche2 conformer in the [http://www.ch.imperial.ac.uk/wiki/index.php/Mod:phys3#Appendix_1 appendix 1]. This suggests that the reaction takes place through the chair transition state, which later produces the gauche2 conformer of 1,5-hexadiene.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 6: Comparison of IRC calculations for the chair structure
| || '''Calculate force constants once''' || '''Calculate force constants always'''
|-
| '''Number of points along IRC''' || 26 || 47
|-
| '''Structure at the last point of IRC''' || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Once_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>200</size>
<script>zoom 100</script>
<uploadedFileContents>Always_chairdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
|'''Energy (atomic units)''' || -231.68906923 || -231.69199434
|-
| '''ΔE, once - always''' || 2.92511 x 10-3 a.u || 7.68 KJ mol-1
|-
| '''IRC pathway''' || [[Image:One_pathwaydh08.jpg|400x400px]] || [[Image:Always_pathwaydh08.jpg|400x400px]]
|-
| '''IRC gradient''' || [[Image:One_gradientdh08.jpg|400x400px]]|| [[Image:Always_gradientdh08.jpg|400x400px]]
|}

 
 
 

===='''Comparison of activation energies for the transition states'''====

The HF/3-21G optimised chair and boat transition structures were both reoptimised using the B3LYP/6-31G* (B3LYP/6-31G (d)) method. This gives a more accurate minimisation however this method takes longer to run. The values for the zero-point energies and thermal energies were found within the thermochemistry section of the log file.

The values for the HF/3-21G reactant (anti2) optimisation were calculated and the B3LYP/6-31G* values were taken from the "Frequency analysis of 1,5-hexadiene Conformers" earlier in this module.

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 7: Energy data for chair, boat and the anti2 conformation from the HF/3-21G and B3LYP/6-31G* methods of optimisation '''
|-
| ''' '''
!colspan="3"|'''HF/3-21G'''
!colspan="3"|'''B3LYP/6-31G*'''
|-
| ''' '''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
| width="125" align="center" | '''Electronic energy (a.u)'''
| width="125" align="center" | '''Sum of electronic and zero-point energies (a.u)'''
| width="125" align="center" | '''Sum of electronic and thermal energies (a.u)'''
|-
| ''' '''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| ''' '''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
|-
| '''Chair TS'''
| align="center" | -231.619322
| align="center" | -231.466677
| align="center" | -231.461320
| align="center" | -234.556983
| align="center" | -234.414929
| align="center" | -234.409008
|-
| '''Boat TS'''
| align="center" | -231.602802
| align="center" | -231.450936
| align="center" | -231.445303
| align="center" | -234.543093
| align="center" | -234.402342
| align="center" | -234.396008
|-
| '''Reactant (''anti2'')'''
| align="center" | -231.692535
| align="center" | -231.539524
| align="center" | -231.532562
| align="center" | -234.611716
| align="center" | -234.469177
| align="center" | -234.461835
|}
Digital repositories

Chair:
HF/3-21G: http://hdl.handle.net/10042/to-5569
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5568

Boat:
HF/3-21G: http://hdl.handle.net/10042/to-5570
B3LYP/6-31G*: http://hdl.handle.net/10042/to-5571

{| border="1" cellspacing="1" cellpadding="10" align=center
|+ '''Table 8: Summary of activation energies for the chair and boat conformations (ΔE = chair TS or boat TS minus anti2)'''
|-
| ''' '''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''HF/3-21G'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''B3LYP/6-31G*'''
| width="125" align="center" | '''Expt.'''
|-
| align="center" | 1 a.u = 627.509 kcal mol-1
| width="125" align="center" | '''at 0 K '''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
| width="125" align="center" | '''at 298.15 K'''
| width="125" align="center" | '''at 0 K'''
|-
| '''ΔE, chair (a.u)'''
| align="center" | 0.072847
| align="center" | 0.071242
| align="center" | 0.054248
| align="center" | 0.052827
| align="center" | -
|-
| '''ΔE, chair (kcal mol-1)'''
| align="center" | 45.71
| align="center" | 44.70
| align="center" | 34.04
| align="center" | 33.15
| align="center" | 33.5 ± 0.5
|-
| '''ΔE, boat (a. u)'''
| align="center" | 0.088588
| align="center" | 0.087259
| align="center" | 0.066835
| align="center" | 0.065827
| align="center" | -
|-
| '''ΔE, boat (kcal mol-1)'''
| align="center" | 55.59
| align="center" | 54.76
| align="center" | 41.94
| align="center" | 41.31
| align="center" | 44.7 ± 2.0
|}

By examining table 8 it is clear to see that the B3LYP/6-31G* method gives values which are much closer to the experimental ones. This is due to the B3LYP/6-31G* giving a higher level of optimisation than the HF/3-21G method. Table 8 also shows that the chair transition state has a lower activation energy to that of the boat transition state by 8.16 kcal mol-1. This suggests that the Cope rearrangement mechanism would most likely proceed through the chair transition state to that of the boat. Therefore the reaction would take place from the anti2 conformer to the chair transition state and then onto the gauche2 conformer within the Cope rearrangement.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 9: Geometric parameters for the optimised chair and boat structures '''
| || ''' HF/3-21G chair TS''' || ''' B3LYP/6-31G* chair TS''' || '''HF/3-21G boat TS''' || '''B3LYP/6-31G* boat TS'''
|-
| Optimised Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Freezedh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Chair_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Realone_BPdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> ||<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>Boat_b3lypdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Terminal C-C bond length (Å) || 2.02 (C5-C10 or C3-C14)|| 1.97 (C5-C10 or C3-C14) || 2.14 (C3-C4 or C1-C6) || 2.21 (C3-C4 or C1-C6)
|-
| Fragment angle || 120 o (C3-C1-C5) || 120 o (C3-C1-C5) || 122 o (C4-C5-C6) || 122 o (C4-C5-C6)
|-
| Dihedral angle || 55 o (C1-C3-C14-C11)|| 54 o (C1-C3-C14-C11)|| 0 o (C2-C1-C6-C5) || 0 o (C2-C1-C6-C5)
|-
| Numbering of carbons || [[Image:Numbering_chairdh08.jpg|200x200px]]|| [[Image:Numbering_chairdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]] || [[Image:Numbering_boatdh08.jpg|200x200px]]
|}

From studying table 9 it can be seen that the C-C bond breaking/forming length has changed between the two different types of optimisation methods. However the angles between the bonds remains unchanged.

=='''The Diels Alder Cycloaddition'''==

The Diels-Alder reaction is a cycloaddition between a conjugated diene and a dienophile (alkene system), which forms a cyclohexene system. It is often referred to as a [4s+2s] cycloaddition and proceeds through a single cyclic, concerted transition state where 2 σ bonds are formed in between the termini carbons of the 2 conjugated п systems. This is shown in the reaction scheme below where ethene is reacting with cis-butadiene.

[[Image:Reaction_scheme2dh08.jpg|300x300px|center]]

The π orbitals of the dienophile form new σ bonds with the π orbitals of the diene. The number of π electrons involved determine whether the reaction proceeds in a concerted stereospecific fashion (allowed) or not (forbidden). The HOMO/LUMO of one fragment interacts with the HOMO/LUMO of the other fragment to form two new bonding and anti-bonding MOs. The nodal properties allow one to predict how the reaction will occur.

Semi-Empirical/AM1 method and the B3LYP/6-31G* method of optimisation will be used on Gaussian to determine particular properites of the reactants to predict the way in which the reaction will proceed.

==='''Reagents'''===

<jmol>
<jmolAppletButton>
<uploadedFileContents>Cis_butadienedh08.mol</uploadedFileContents>
<text>Cis-butadiene</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Ethenedh08.mol</uploadedFileContents>
<text>ethylene</text>
<color>white</color>
</jmolAppletButton>
</jmol> were optimised separately using the AM1 semi-empirical method on Gaussview.The cis-butadiene had an energy of 0.04879725 a. u (1.28 x 102 KJ mol-1) and ethylene had an energy of 0.02619027 a. u (68.76 KJ mol-1).

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 10: Frontier molecular orbitals of cis-butadiene and ethylene '''
| '''Reagent''' || '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| cis-butadiene || LUMO || [[Image:LUMO1_cis_butadieneDH08.jpg|100x100px]] || [[Image:LUMO2_cis_butadienedh08.jpg|100x100px]]|| Symmetric || +0.017
|-
| cis-butadiene || HOMO || [[Image:HOMO1_cis_butadienedh08.jpg|100x100px]] || [[Image:HOMO2_cis_butadienedh08.jpg|100x100px]]|| Anti-symmetric || -0.343
|-
| ethylene || LUMO || [[Image:LUMO1_ethenedh08.jpg|100x100px]] || [[Image:LUMO2_ethenedh08.jpg|100x100px]]|| Anti-symmetric || +0.052
|-
| ethylene || HOMO || [[Image:HOMO1_ethenedh08.jpg|100x100px]] || [[Image:HOMO2_ethenedh08.jpg|100x100px]]|| Symmetric || -0.387
|}

 
The symmetries of the LUMO and HOMO orbitals for cis-butadiene are opposite of those for the ethylene. This therefore means that the only interactions that can take place are those between ethylene's LUMO and cis-butadiene's HOMO and in addition to this the interaction between cis-butadiene's LUMO and ethylene's HOMO. This is because orbital symmetry has to be conserved (the conservation of orbital symmetry states that the symmetry of an new orbital formed by the overlap of the 2 original orbitals must be the same as the symmetry of the initial orbitals).

==='''Transition State'''===

A <jmol>
<jmolAppletButton>
<uploadedFileContents>Bicyclic_startingdh08.mol</uploadedFileContents>
<text>bicyclo system</text>
<color>white</color>
</jmolAppletButton>
</jmol> was drawn and one of the -CH2-CH2- fragments were removed. Three C=C bonds were added and the previous one was taken away so a <jmol>
<jmolAppletButton>
<uploadedFileContents>Frozen_coorddh08.mol</uploadedFileContents>
<text>guess transition state</text>
<color>white</color>
</jmolAppletButton>
</jmol> was produced.

After this an optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while bonds between the termini of the fragments were frozen at 2.1Å. Following this the resulting file had two separate optimisation and frequency calculations run on it using the TS (Berny) with one using the semi-empirical/AM1 method and the other using the DFT/B3LYP/6-31G* level of theory. Both of these optimisations had the bonds unfrozen and the "derivative" option was chosen instead.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 11: Comparison of two methods of optimsations for the transition state '''
| || '''semi-empirical/AM1 method''' || '''DFT/B3LYP/6-31G* method'''
|-
| Structure || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>SE_AM1_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || <jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 100</script>
<uploadedFileContents>DFT_TSdh08.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| Energy (a. u) || 0.11165608 || -234.54389573
|-
| Bond lengths || [[Image:DistancesSEdh08.jpg|150x150px]]|| [[Image:DistancesDFTdh08.jpg|150x150px]]
|-
|Imaginary frequency (cm-1) || -957 || -525
|-
|Animation || [[Image:-957dh08.gif]] || [[Image:-525dh08.gif]]
|-
| Lowest real frequency (cm-1) || 148 || 136
|-
|Animation || [[Image: 148dh08.gif]] || [[Image: 136dh08.gif]]

|}

The C-C bond forming length between the termini carbons is 0.15 Å larger for the DFT/B3LYP/6-31G* level of theory and this is because this method gives carries out the calculation to a greater degree of accuracy. This larger bond length leads a lower potential energy gradient to the transition state, which gives a lower total energy for the transition state. Therefore the given values below will be the DFT/B3LYP/6-31G* and not the semi-empirical/AM1 method.

A typical sp3 C-C bond length is 1.54 Å and a typical sp2 C=C bond length is 1.34 Å<ref name="ref 2">G. Schultz, I. Hargitta, J. Mol. Struc., 1995, 346, pp. 63-69.[http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TGS-3YMWN3T-B6&_user=217827&_coverDate=02%2F15%2F1995&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000011279&_version=1&_urlVersion=0&_userid=217827&md5=9568533c3b0299cb0622ad03c88c9e58&searchtype=a]</ref>. Within the optimised transition molecule above the C-C bond lengths are 1.40 Å and the C=C bond length is 1.38 Å. This suggests that the π-bonds are breaking in order for the C-C bonds to form and additionally to this a π-bond is forming in between the two sp3 carbons.

The Van der Waals radius of a C-atom is 1.72 Å<ref name="ref 3">M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer, D. G. Truhlar, 1995, 346, pp. 63-69.[http://pubs.acs.org/doi/abs/10.1021/jp8111556]</ref>. The partly formed C-C length is 2.27 Å, which is shorter than two times the the van der Waals radius of a carbon atom. Therefore the carbon atoms are within a close enough distance to each other to share electron density and yet the bond has not formed at this point.

From looking at the animations for the imaginary frequency above it can be seen that the formation of the two bonds between the termini of the reactants is synchronous. This is expected of the Diels-Alder reaction since the mechanism is concerted. However the lowest real frequency is asynchronous with respect to the two bonds forming between the termini.

{| class="wikitable" border="1" align=center style="text-align:center"
|+ '''Table 12: Frontier molecular orbitals of the transition state '''
| '''Molecular Orbital''' || '''View 1''' || '''View 2'''|| '''Symmetry with respect to the plane''' || '''Energy (a. u)'''
|-
| LUMO || [[Image:LUMO1TSdh08.jpg|100x100px]] || [[Image:LUMO2TSdh08.jpg|100x100px]]|| Symmetric || +0.023
|-
| HOMO || [[Image:HOMO1TSdh08.jpg|100x100px]] || [[Image:HOMO2TSdh08.jpg|100x100px]]|| Anti-symmetric || -0.323
|}

From examining the images from table 12 above it can be deduced that the LUMO of the transition state originates from the cis-butadiene's LUMO and the ethylene's HOMO, which represents the σ*-antibonding orbitals. However the HOMO of the transition state originates from the cis-butadiene's HOMO and the ethylene's LUMO, which represents the σ-bonding orbitals within the two single C-C bonds that are formed within the reaction.

==='''Cyclohexa-1,3-diene reaction with maleic anhydride'''===

The <jmol>
<jmolAppletButton>
<uploadedFileContents>FrozenTS22dh08.mol</uploadedFileContents>
<text>exo</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_frozenTS22dh08.mol</uploadedFileContents>
<text>endo</text>
<color>white</color>
</jmolAppletButton>
</jmol> transition states were drawn on Gaussview and then the optimisation and frequency (Opt+Freq) calculation was run to a minimum using the semi-empirical/AM1 method while the distance of the forming bonds were frozen at 2.1Å between the respective carbons. After this the structures changed slightly (<jmol>
<jmolAppletButton>
<uploadedFileContents>TS2_afteropt1_exo1dh08.mol</uploadedFileContents>
<text>exo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol> and <jmol>
<jmolAppletButton>
<uploadedFileContents>Endo_TS2_afteropt1dh08.mol</uploadedFileContents>
<text>endo-after first calculation</text>
<color>white</color>
</jmolAppletButton>
</jmol>) and the resulting files had an optimisation and frequency calculation run on it using the TA (Berny) with using the semi-empirical/AM1 level of theory while the unfrozen option as changed to "derivative". After this the endo and exo optimised transition state structures were re-optimised as they were before but using the DFT/B3LYP/6-31G* level of theory instead with the additional keyword "Opt=NoEigen".

{| class="wikitable" border="1" align="center"
|+ '''Table 13: Comparison of the exo and endo transition states '''
! rowspan="2"|-!! colspan="2"|''exo'' !! colspan="2"|''endo''
|-
| Semi-empirical/AM1 || DFT/B3LYP/6-31G* || Semi-empirical/AM1 || DFT/B3LYP/6-31G*
|-
| align="center"|Structure ||align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>After_opt2_exodh08.mol</uploadedFileContents>
</jmolApplet>
</jmol> || align="center" colspan="2"|<jmol>
<jmolApplet>
<color>white</color>
<size>150</size>
<script>zoom 150</script>
<uploadedFileContents>Endo_TS2_afteropt2dh081.mol</uploadedFileContents>
</jmolApplet>
</jmol>
|-
| align="center"| Energy (a. u) ||align="center"| -0.05041975 ||align="center"| -612.67931095 ||align="center"| -0.05150470 ||align="center"| -612.68339684
|-
| C-C bond forming length (Å) ||align="center"| 2.17 ||align="center"| 2.29 ||align="center"| 2.16 ||align="center"| 2.27
|-align="center"
| Imaginary frequency (cm-1) ||align="center"| -812 ||align="center"| -448 ||align="center"| -806 ||align="center"| -447
|-align="center"
| Animation ||colspan="2"|[[Image:-812_SEdh08.gif]] || colspan="2"|[[Image:-806_cm-1dh08.gif]]
|}

Again the C-C bond forming length between the termini carbons is larger (by 0.11 and 0.12 Å) for the DFT/B3LYP/6-31G* level of theory and this is due the same reasons talked about above.

From looking at the animations for the imaginary frequency above it can also be seen that the formation of the two bonds between the carbons in this case as well is synchronous. Again this is expected of the Diels-Alder reaction since it is concerted.

{| class="wikitable" border="1" align="center"
|+ '''Table 14: Frontier molecular orbitals of the the transition state for the cyclohexa-1,3-diene reaction '''
! rowspan="2"|Molecular Orbital !! colspan="4"|''exo'' !! colspan="4"|''endo''
|- align="center"
| '''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''||'''View 1''' || '''View 2''' || '''Symmetry with respect to plane''' || '''Energy (a. u)'''
|- align="center"
| LUMO || [[Image:LUMO1exoh081.jpg|150x150px]]|| [[Image:LUMO2exodh081.jpg|150x150px]]|| Anti-symmetric || -0.078 || [[Image:LUMO1ENDOdh08.jpg|150x150px]]||[[Image:LUMO2ENDOdh08.jpg|150x150px]] || Anti-symmetric || -0.067
|- align="center"
| HOMO || [[Image:HOMO1exodh081.jpg|150x150px]]||[[Image:HOMO2exodh081.jpg|150x150px]] || Anti-symmetric|| -0.242 || [[Image:HOMO1ENDOdh08.jpg|150x150px]]|| [[Image:HOMO2ENDOdh08.jpg|150x150px]]||Anti-symmetric || -0.242
|}

Digital repositories:

exo: http://hdl.handle.net/10042/to-5596

endo: http://hdl.handle.net/10042/to-5597

{| align=center
|+ '''Strain diagrams for the endo and exo transition states'''
| [[Image:Exostraindh08.jpg|thumb|200x200px|'''Exo TS''']]
| [[Image:Strainendodh08.jpg|thumb|200x200px|'''Endo TS''']]
|}

Why do you think that the exo form could be more strained?

Examine carefully the nodal properties of the HOMO between the -(C=O)-O-(C=O)- fragment and the remainder of the system.

What can you conclude about the so called “secondary orbital overlap effect”? (There is some discussion of this in Ian Fleming's book 'Frontier Orbitals and Organic Chemical Reactions').

Further discussion:
What effects have been neglected in these calculations of Diels Alder transition states?

=='''Conclusion'''==

='''References'''=
<references/>